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Cayley diagram

C3×#1084 #66407

order324
isoclinismisoclinic class of #272
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×#1084

Splittings

derivedH(3,𝔽3) ⋊ C3×C4
SylowC3×H(3,𝔽3) ⋊ C4

Subgroups

subgroupG/H
center, socle, hypercenterC2×C32C3⋊D3
commutator, hypocenterH(3,𝔽3)C3×C4
FrattiniC2×C3C3×C3⋊D3
Fitting, prime radicalC2×C3×H(3,𝔽3)C2
cosocleC2×H(3,𝔽3)C2×C3
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C2×C32

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13048253, #13048254, #13048255, …
maximal subgroup ofC3×#6188, C3⋊#6188, C3×#6196, …
maximal normal subgroup ofC3×#6188, C3⋊#6188, C3×#6196, …
maximal quotient ofC3×#6180, C2×C3×#1084, C3×#66328, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 99);