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Cayley diagram

C2×(C3⋊(C3⋊C8))⋊C3 #88489

order432
isoclinismisoclinic class of (C3⋊D3)⋊C3
automorphismAut(G) = C2×D4×((C3⋊D3)⋊C3)⋊C2 (order 1728) · Inn(G) = (C3⋊D3)⋊C3 · Out(G) = C22×D4
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×(C3⋊(C3⋊C8))⋊C3
unambiguous semidirect product(C2×C3⋊(C3⋊C8))⋊C3

Splittings

derivedC32 ⋊ C2×C3×C8
SylowH(3,𝔽3) ⋊ C2×C8

Subgroups

subgroupG/H
center, hypercenterC2×C4(C3⋊D3)⋊C3
commutator, hypocenterC32C2×C3×C8
FrattiniC3×C4C2×C3×D3
FittingC2×C4×H(3,𝔽3)C2
socleC22×C3C3×Dic3
prime radicalC22×H(3,𝔽3)C4
cosocleC32×C4C22×C3
LCSG ▷ C32
UCSC1 ◁ C2×C4

93 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 2 generators on 19 points:

(3,4,5,7,6,8,9,10)(12,17,19,14,15,18)(13,16);(1,2)(11,12,14)(13,15,18)(16,19,17)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11720302, #11720303, #11720304, …
maximal normal subgroup of#11720302, #11720303, #11720304, …
maximal quotient ofC4×(C3⋊(C3⋊C8))⋊C3, C4⋊((C3⋊(C3⋊C8))⋊C3), C2×(C3⋊(C3⋊C16))⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 136);