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C4×ΓL(1,𝔽8)×C43 #16813409

order3612
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C4×C43 · Schur multiplier (H2) = C1

Aliases

direct productC4×ΓL(1,𝔽8)×C43
unambiguous semidirect product(C4×C7)⋊(C3×C43)

Splittings

Fitting, HallC4×C7×C43 ⋊ C3
derived, SylowC7 ⋊ C3×C4×C43
HallC7×C43 ⋊ C3×C4
HallC4×C7 ⋊ C3×C43

Subgroups

subgroupG/H
center, hypercenterC4×C43ΓL(1,𝔽8)
commutator, hypocenterC7C3×C4×C43
FrattiniC2C2×ΓL(1,𝔽8)×C43
FittingC4×C7×C43C3
socleC2×C7×C43C2×C3
prime radicalC2×ΓL(1,𝔽8)×C43C2
cosocleC2×C7C2×C3×C43
LCSG ▷ C7
UCSC1 ◁ C4×C43

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 54
transitive degreeμt(G) = 1204

A minimal faithful permutation representation — 2 generators on 54 points:

(1,2,3,4)(6,7,8)(9,11,10)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54);(5,6,7,9,8,10,11)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC4×ΓL(1,𝔽8)×C43⋊C3, Dic3×ΓL(1,𝔽8)×C43, C3×C4×ΓL(1,𝔽8)×C43, …
maximal normal subgroup ofC4×ΓL(1,𝔽8)×C43⋊C3, C3×C4×ΓL(1,𝔽8)×C43, C4×C5×ΓL(1,𝔽8)×C43, …
maximal quotient ofC4×C43×C7⋊C9, Dic3×ΓL(1,𝔽8)×C43, C3×C4×ΓL(1,𝔽8)×C43, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3612, 21);