Finite Group Databasebeta

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Cayley diagram

C2×C3×C4×C3⋊D3 #89064

order432
isoclinismisoclinic class of C3⋊D3
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23×C3

Aliases

direct productC2×C3×C4×C3⋊D3

Splittings

FittingC2×C33×C4 ⋊ C2
derivedC32 ⋊ C22×C3×C4
SylowC33 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC2×C3×C4C3⋊D3
commutator, hypocenterC32C22×C3×C4
FrattiniC2C22×C3×C3⋊D3
FittingC2×C33×C4C2
socleC22×C33C22
prime radicalC22×C3×C3⋊D3C2
cosocleC2×C32C23×C3
LCSG ▷ C32
UCSC1 ◁ C2×C3×C4

388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#63557, #11722023, #11722072, …
maximal normal subgroup ofC3×#63557, #11722023, #11722072, …
maximal quotient ofC3×C4×#634, C3×C42×C3⋊D3, C3×#64081, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 711);