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Cayley diagram

C2×C22⋊H(3,𝔽3) #6270

order216
isoclinismisoclinic class of C9⋊A4
automorphismAut(G) = ((C3⋊D3)⋊C3)⋊S4 (order 1296) · Inn(G) = C3×A4 · Out(G) = D32
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C32

Aliases

direct productC2×C22⋊H(3,𝔽3)
unambiguous semidirect productC23⋊H(3,𝔽3)
unambiguous semidirect product(C2×C32)⋊A4
unambiguous semidirect product(C2×C3×A4)⋊C3

Splittings

FittingC23×C32 ⋊ C3
SylowC23 ⋊ H(3,𝔽3)

Subgroups

subgroupG/H
centerC2×C3C3×A4
commutator, cosocleC22×C3C2×C32
FrattiniC3C2×C3×A4
FittingC23×C32C3
socleC23×C3C32
hypercenterC2×C32A4
hypocenterC22C2×H(3,𝔽3)
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C2×C3 ◁ C2×C32

68 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)(7,8,11)(9,10,14)(12,13,15);(4,5,6)(8,10,13)(11,15,14)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11722271, #11722272, #11722273, …
maximal subgroup of(C3⋊SL(2,ℤ/4))⋊C3, #88613, (C3⋊Dic3)⋊A4, …
maximal normal subgroup of(C3⋊SL(2,ℤ/4))⋊C3, #88613, (C3⋊Dic3)⋊A4, …
maximal quotient ofC4×C22⋊H(3,𝔽3), C2×Q8⋊H(3,𝔽3), #41⋊H(3,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 107);