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C22×C13×ΓL(1,𝔽8) #11764014

order1092
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C22×C3×C13 · Schur multiplier (H2) = C2

Aliases

direct productC22×C13×ΓL(1,𝔽8)

Splittings

Fitting, socle, HallC22×C7×C13 ⋊ C3
derived, SylowC7 ⋊ C22×C3×C13
HallC7×C13 ⋊ C22×C3
HallC22×C7 ⋊ C3×C13

Subgroups

subgroupG/H
center, hypercenterC22×C13ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C22×C3×C13
FrattiniC1G
Fitting, socleC22×C7×C13C3
LCSG ▷ C7
UCSC1 ◁ C22×C13

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 364

A minimal faithful permutation representation — 2 generators on 24 points:

(1,2)(5,6,7,8,9,10,11,12,13,14,15,16,17)(19,20,21)(22,24,23);(3,4)(18,19,20,22,21,23,24)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC22×ΓL(1,𝔽8)×C13⋊C3, A4×C13×ΓL(1,𝔽8), ΓL(1,𝔽8)×C22⋊(C13⋊C3), …
maximal normal subgroup ofC22×ΓL(1,𝔽8)×C13⋊C3, A4×C13×ΓL(1,𝔽8), ΓL(1,𝔽8)×C22⋊(C13⋊C3), …
maximal quotient ofC22×C13×C7⋊C9, C2×D3×C13×ΓL(1,𝔽8), C22×C3×C13×ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1092, 62);