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Cayley diagram

C2×C3×C22⋊C9 #6268

order216
isoclinismisoclinic class of A4
automorphismAut(G) = S4×#272 (order 1296) · Inn(G) = A4 · Out(G) = C2×#272
homologyAbelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×C22⋊C9
unambiguous semidirect product(C23×C3)⋊C9

Splittings

derivedC22 ⋊ C2×C3×C9
SylowC23 ⋊ C3×C9

Subgroups

subgroupG/H
center, hypercenterC2×C32A4
commutator, hypocenterC22C2×C3×C9
FrattiniC3C2×C3×A4
Fitting, socle, prime radicalC23×C32C3
cosocleC22×C3C2×C32
LCSG ▷ C22
UCSC1 ◁ C2×C32

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2)(7,8,9)(10,14,18,12,17,13,15,11,16);(3,4,5)(6,9)(7,8)

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11722261, #11722264, #11722266, …
maximal subgroup ofC3×(C22⋊C9)⋊C4, C3⋊((C22⋊C9)⋊C4), Dic3×C22⋊C9, …
maximal normal subgroup ofC3×(C22⋊C9)⋊C4, C3⋊((C22⋊C9)⋊C4), Dic3×C22⋊C9, …
maximal quotient ofC3×C4×C22⋊C9, C2×C3×Q8⋊C9, C3×#41⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 105);