Finite Group Databasebeta

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Cayley diagram

C2×C4⋊(C3⋊(C3⋊D3)) #89075

order432
isoclinismisoclinic class of #6308
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23×C33

Aliases

direct productC2×C4⋊(C3⋊(C3⋊D3))

Splittings

FittingC2×C33×C4 ⋊ C2
SylowC33 ⋊ C2×D4

Subgroups

subgroupG/H
centerC22C2×C3⋊(C3⋊D3)
commutator, cosocleC2×C33C23
FrattiniC2C22×C3⋊(C3⋊D3)
FittingC2×C33×C4C2
socleC22×C33C22
hypercenterC2×C4C3⋊(C3⋊D3)
hypocenterC33C2×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ 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:

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

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11722027, #11722035, #11722036, …
maximal normal subgroup of#11722027, #11722035, #11722036, …
maximal quotient of#11723910, C4×C4⋊(C3⋊(C3⋊D3)), #11723915, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 722);