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Cayley diagram

C8×S4 #5399

order192
isoclinismisoclinic class of S4
automorphismAut(G) = C23×S4 (order 192) · Inn(G) = S4 · Out(G) = C23
homologyAbelianization (H1) = C2×C8 · Schur multiplier (H2) = C22

Aliases

direct productC8×S4

Splittings

FittingC22×C8 ⋊ D3
derivedA4 ⋊ C2×C8

Subgroups

subgroupG/H
center, hypercenterC8S4
commutator, hypocenterA4C2×C8
FrattiniC4C2×S4
FittingC22×C8D3
socleC23C4×D3
prime radicalC2×S4C4
cosocleC4×A4C22
LCSG ▷ A4
UCSC1 ◁ C8

79 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 12 points:

(1,2,4,3);(2,3,4)(5,6,7,9,8,10,11,12)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC16×S4, #73034, C2×C8×S4, …
maximal normal subgroup ofC16×S4, #73034, C2×C8×S4, …
maximal quotient ofC8×#226, #226⋊C8, C8×GL(2,𝔽3), …
outer automorphism group ofC22×D17

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 958);