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Cayley diagram

C2×C11×A4 #63167

order264
isoclinismisoclinic class of A4
automorphismAut(G) = C2×C5×S4 (order 240) · Inn(G) = A4 · Out(G) = C22×C5
homologyAbelianization (H1) = C2×C3×C11 · Schur multiplier (H2) = C2

Aliases

direct productC2×C11×A4
unambiguous semidirect productC23⋊(C3×C11)
unambiguous semidirect product(C23×C11)⋊C3

Splittings

Fitting, socle, HallC23×C11 ⋊ C3
derivedC22 ⋊ C2×C3×C11
SylowC23 ⋊ C3×C11

Subgroups

subgroupG/H
center, hypercenterC2×C11A4
commutator, cosocle, hypocenterC22C2×C3×C11
FrattiniC1G
Fitting, socleC23×C11C3
LCSG ▷ C22
UCSC1 ◁ C2×C11

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 66

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11761325, #11761328, #11761330, …
maximal subgroup ofC11×SL(2,ℤ/4), C11⋊SL(2,ℤ/4), A4×Dic11, …
maximal normal subgroup ofC11×SL(2,ℤ/4), C11⋊SL(2,ℤ/4), A4×Dic11, …
maximal quotient ofC4×C11×A4, C2×C11×SL(2,𝔽3), C11×#41⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(264, 35);