Finite Group Databasebeta

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Cayley diagram

C22×C3×C3⋊C8 #64043

order288
isoclinismisoclinic class of D3
automorphism?
homologyAbelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C23

Aliases

direct productC22×C3×C3⋊C8

Splittings

derivedC3 ⋊ C22×C3×C8
SylowC32 ⋊ C22×C8

Subgroups

subgroupG/H
center, hypercenterC22×C3×C4D3
commutator, hypocenterC3C22×C3×C8
FrattiniC4C22×C3×D3
FittingC22×C32×C4C2
socle, prime radicalC23×C32C4
cosocleC3×C4C23×C3
LCSG ▷ C3
UCSC1 ◁ C22×C3×C4

179 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10589155, #10589156, #10589164, …
maximal normal subgroup of#10589155, #10589156, #10589164, …
maximal quotient ofC2×C3×C4×C3⋊C8, C2×C3×C4⋊(C3⋊C8), C3×#4925, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 691);