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C22×ΓL(1,𝔽8)×C23 #14515389

order1932
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C22×C3×C23 · Schur multiplier (H2) = C2

Aliases

direct productC22×ΓL(1,𝔽8)×C23

Splittings

Fitting, socle, HallC22×C7×C23 ⋊ C3
derived, SylowC7 ⋊ C22×C3×C23
HallC7×C23 ⋊ C22×C3
HallC22×C7 ⋊ C3×C23

Subgroups

subgroupG/H
center, hypercenterC22×C23ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C22×C3×C23
FrattiniC1G
Fitting, socleC22×C7×C23C3
LCSG ▷ C7
UCSC1 ◁ C22×C23

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 644

A minimal faithful permutation representation — 2 generators on 34 points:

(1,2)(6,7,8)(9,11,10)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34);(3,4)(5,6,7,9,8,10,11)

Conjugacy classes & character table

460 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC22×C23×C49⋊C3, C2×C23×#64423, C2×C23×#64422, …
maximal normal subgroup ofC22×C7×ΓL(1,𝔽8)×C23, C22×ΓL(1,𝔽8)×C23⋊C11, C22×C11×ΓL(1,𝔽8)×C23, …
maximal quotient ofC22×C23×C49⋊C3, C2×C23×#64423, C2×C23×#64422, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1932, 24);