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C3×C53×C22⋊ΓL(1,𝔽8) #27211323

order13356
isoclinismisoclinic class of C22⋊ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C32×C53 · Schur multiplier (H2) = C2×C3

Aliases

direct productC3×C53×C22⋊ΓL(1,𝔽8)
unambiguous semidirect product(C22×C3)⋊(ΓL(1,𝔽8)×C53)
unambiguous semidirect product(C3×C7)⋊(A4×C53)
unambiguous semidirect product(C22×C3×C53)⋊ΓL(1,𝔽8)
unambiguous semidirect product(C3×C7×C53)⋊A4

Splittings

Fitting, socleC22×C3×C7×C53 ⋊ C3
derived, HallC22×C7 ⋊ C32×C53
HallC22×C7×C53 ⋊ C32
HallC7×C53 ⋊ C3×A4
HallC22×C53 ⋊ C3×ΓL(1,𝔽8)
SylowC7 ⋊ C3×A4×C53
SylowC22 ⋊ C3×ΓL(1,𝔽8)×C53

Subgroups

subgroupG/H
center, hypercenterC3×C53C22⋊ΓL(1,𝔽8)
commutator, cosocle, hypocenterC22×C7C32×C53
FrattiniC1G
Fitting, socleC22×C3×C7×C53C3
LCSG ▷ C22×C7
UCSC1 ◁ C3×C53

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 67
transitive degreeμt(G) = 4452

A minimal faithful permutation representation — 2 generators on 67 points:

(2,3,4)(5,7,6)(9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64);(1,2,3,5,4,6,7)(8,11)(9,10)(65,66,67)

Conjugacy classes & character table

1908 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13356, 116);