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Cayley diagram

C2×C3×#157 #6331

order216
isoclinismisoclinic class of #157
automorphismAut(G) = C22×AΓL(1,𝔽9) (order 576) · Inn(G) = #157 · Out(G) = C24
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×#157

Splittings

Fitting, socleC2×C33 ⋊ C4
derivedC32 ⋊ C2×C3×C4
SylowC33 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC2×C3#157
commutator, hypocenterC32C2×C3×C4
FrattiniC1G
Fitting, socleC2×C33C4
prime radicalC2×C3×C3⋊D3C2
cosocleC3⋊D3C22×C3
LCSG ▷ C32
UCSC1 ◁ C2×C3

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 11 points:

(3,4,5)(6,7)(8,9,10,11);(1,2)(6,10,8)

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13165090, #13165091, #13165092, …
maximal subgroup ofDic3×#157, (C2×D3)⋊#157, Dic3⋊#157, …
maximal normal subgroup ofDic3×#157, (C2×D3)⋊#157, Dic3⋊#157, …
maximal quotient ofC3×#3787, C3×#3788, C3×C4×#157, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 168);