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Cayley diagram

C23×A4 #1021

order96
isoclinismisoclinic class of A4
automorphismAut(G) = S4×GL(3,𝔽2) (order 4032) · Inn(G) = A4 · Out(G) = C2×GL(3,𝔽2)
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C24

Aliases

direct productC23×A4

Splittings

Fitting, socle, SylowC25 ⋊ C3
derivedC22 ⋊ C23×C3

Subgroups

subgroupG/H
center, hypercenterC23A4
commutator, cosocle, hypocenterC22C23×C3
FrattiniC1G
Fitting, socleC25C3
LCSG ▷ C22
UCSC1 ◁ C23

178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 3 generators on 10 points:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11711501, #11711502, #11711610, …
maximal subgroup of#5432, A4×C22⋊C4, C22×SL(2,ℤ/4), …
maximal normal subgroup of#5432, A4×C22⋊C4, C22×SL(2,ℤ/4), …
maximal quotient ofC22×C4×A4, C2×D4×A4, C23×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 228);