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C22×C11×ΓL(1,𝔽8) #11745123

order924
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C22×C3×C11 · Schur multiplier (H2) = C2

Aliases

direct productC22×C11×ΓL(1,𝔽8)

Splittings

Fitting, socle, HallC22×C7×C11 ⋊ C3
derived, SylowC7 ⋊ C22×C3×C11
HallC7×C11 ⋊ C22×C3
HallC22×C7 ⋊ C3×C11

Subgroups

subgroupG/H
center, hypercenterC22×C11ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C22×C3×C11
FrattiniC1G
Fitting, socleC22×C7×C11C3
LCSG ▷ C7
UCSC1 ◁ C22×C11

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 308

A minimal faithful permutation representation — 2 generators on 22 points:

(1,2)(5,6,7,8,9,10,11,12,13,14,15)(17,18,19)(20,22,21);(3,4)(16,17,18,20,19,21,22)

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×ΓL(1,𝔽8)×Dic11, ΓL(1,𝔽8)×C22⋊D11, C2×C11×ΓL(1,𝔽8)⋊C4, …
maximal normal subgroup ofC2×ΓL(1,𝔽8)×Dic11, ΓL(1,𝔽8)×C22⋊D11, C2×C11×ΓL(1,𝔽8)⋊C4, …
maximal quotient ofC2×C4×C11×ΓL(1,𝔽8), D4×C11×ΓL(1,𝔽8), Q8×C11×ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(924, 24);