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Cayley diagram

C2×#1094 #6269

order216
isoclinismisoclinic class of C9⋊A4
automorphismAut(G) = A4⋊#272 (order 648) · Inn(G) = C3×A4 · Out(G) = C3⋊D3
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C2

Aliases

direct productC2×#1094

Splittings

SylowC23 ⋊ 3−1+2

Subgroups

subgroupG/H
centerC2×C3C3×A4
commutator, cosocleC22×C3C2×C32
FrattiniC3C2×C3×A4
Fitting, prime radicalC23×C32C3
socleC23×C3C32
hypercenterC2×C32A4
hypocenterC22C2×3−1+2
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C2×C3 ◁ C2×C32

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 18

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

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#1094⋊C4, C4×#1094, C2×((C22⋊C9)⋊C2)⋊C3, …
maximal normal subgroup of#1094⋊C4, C4×#1094, C2×((C22⋊C9)⋊C2)⋊C3, …
maximal quotient ofC4×#1094, C2×#6204, #88691, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 106);