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Cayley diagram

C3×#1113 #66475

order324
isoclinismisoclinic class of #1113
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC3×#1113

Splittings

Fitting, socle, SylowC34 ⋊ C22
derivedC33 ⋊ C22×C3

Subgroups

subgroupG/H
center, hypercenterC3#1113
commutator, cosocle, hypocenterC33C22×C3
FrattiniC1G
Fitting, socleC34C22
LCSG ▷ C33
UCSC1 ◁ C3

174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 12

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC3⋊(D33⋊C3), D3×#1113⋊C3
maximal subgroup ofC3×D32⋊D3, C3×#157⋊D3, #10620203, …
maximal normal subgroup ofC3×D32⋊D3, C3×#157⋊D3, #10620203, …
maximal quotient ofC3×#6294, C3×#6295, C3×#6296, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 167);