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Cayley diagram

C2×C4×C3⋊(C3⋊D3) #89074

order432
isoclinismisoclinic class of C3⋊(C3⋊D3)
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C33

Aliases

direct productC2×C4×C3⋊(C3⋊D3)

Splittings

FittingC2×C33×C4 ⋊ C2
derived, SylowC33 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC2×C4C3⋊(C3⋊D3)
commutator, hypocenterC33C22×C4
FrattiniC2C22×C3⋊(C3⋊D3)
FittingC2×C33×C4C2
socleC22×C33C22
prime radicalC22×C3⋊(C3⋊D3)C2
cosocleC2×C33C23
LCSG ▷ C33
UCSC1 ◁ C2×C4

756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 216

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11722024, #11722190, C2×#88785, …
maximal normal subgroup of#11722024, #11722190, C2×#88785, …
maximal quotient ofC4×#6308, C42×C3⋊(C3⋊D3), #11723913, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 721);