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C7×A4⋊C8 #10620697

order672
isoclinismisoclinic class of S4
automorphism?
homologyAbelianization (H1) = C7×C8 · Schur multiplier (H2) = C2

Aliases

direct productC7×A4⋊C8
unambiguous semidirect productA4⋊(C7×C8)

Splittings

derivedA4 ⋊ C7×C8

Subgroups

subgroupG/H
center, hypercenterC4×C7S4
commutator, hypocenterA4C7×C8
FrattiniC4C7×S4
FittingC22×C4×C7D3
socleC23×C7Dic3
prime radicalC2×C7×A4C4
cosocleC4×A4C2×C7
LCSG ▷ A4
UCSC1 ◁ C4×C7

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 168

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C7⋊S4)⋊C8, #13057035, D7×A4⋊C8, …
maximal normal subgroup of(C7⋊S4)⋊C8, #13057035, D7×A4⋊C8, …
maximal quotient ofC7×SL(2,𝔽3)⋊C8, C7×A4⋊C16, C7×#4628, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 279);