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C8×#4427 #13094163

order1512
isoclinismisoclinic class of #4427
automorphism?
homologyAbelianization (H1) = C32×C8 · Schur multiplier (H2) = C1

Aliases

direct productC8×#4427

Splittings

FittingC7×C8×C9 ⋊ C3
HallC7×C8 ⋊ 3−1+2
SylowC7 ⋊ C8×3−1+2

Subgroups

subgroupG/H
centerC3×C8C3×ΓL(1,𝔽8)
commutatorC3×C7C32×C8
FrattiniC3×C4C2×C3×ΓL(1,𝔽8)
FittingC7×C8×C9C3
socleC2×C3×C7C32×C4
prime radicalC2×C3×ΓL(1,𝔽8)C3×C4
cosocleC3×C4×C7C2×C32
hypercenterC8×C9ΓL(1,𝔽8)
hypocenterC7C8×3−1+2
LCSG ▷ C3×C7 ▷ C7
UCSC1 ◁ C3×C8 ◁ C8×C9

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 504

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

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1512, 60);