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Cayley diagram

C2×C9⋊A4 #6267

order216
isoclinismisoclinic class of C9⋊A4
automorphismAut(G) = C3×D3×A4 (order 216) · Inn(G) = C3×A4 · Out(G) = C2×C3
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C2

Aliases

direct productC2×C9⋊A4
unambiguous semidirect product(C2×C9)⋊A4

Splittings

FittingC23×C9 ⋊ C3
SylowC23 ⋊ 3−1+2

Subgroups

subgroupG/H
centerC2×C3C3×A4
commutator, cosocleC22×C3C2×C32
FrattiniC3C2×C3×A4
FittingC23×C9C3
socleC23×C3C32
prime radicalC2×C3×A4C3
hypercenterC2×C9A4
hypocenterC22C2×3−1+2
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C2×C3 ◁ C2×C9

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic9⋊A4, C4×C9⋊A4, C2×D9⋊A4, …
maximal normal subgroup ofDic9⋊A4, C4×C9⋊A4, C2×D9⋊A4, …
maximal quotient ofC4×C9⋊A4, C2×C9⋊SL(2,𝔽3), #88683, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 104);