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 Incoming links: Character table of G6n

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 19I / b / iii / Solution Created 2026-09-24 Updated 2026-10-03

Every element has a unique form

a^{s} b^{t}

. Since

a^{2}

is central, the conjugacy classes, indexed by

0 \leq r < n

, are

C_{r} = {a^{2 r}}, D_{r} = {a^{2 r} b, a^{2 r} b^{2}}, E_{r} = {a^{2 r + 1}, a^{2 r + 1} b, a^{2 r + 1} b^{2}} .

These have sizes one, two, and three and account for all

6 n

elements.

Take

ρ_{k} = ρ_{ξ^{k}}

for

0 \leq k < n

. On an even power of

a

, its representing matrix is scalar; on an odd power it is off-diagonal. Since

ω + ω^{2} = - 1

, the full Character table of G6n is

λ_{j} χ_{ρ_{k}} C_{r} ξ^{2 j r} 2 ξ^{2 k r} D_{r} ξ^{2 j r} - ξ^{2 k r} E_{r} ξ^{j (2 r + 1)} 0

for

0 \leq j < 2 n

,

0 \leq k < n

, and

0 \leq r < n

.

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