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Cayley diagram

C22×C3×C3⋊D3 #6338

order216
isoclinismisoclinic class of C3⋊D3
automorphismAut(G) = C2×S4×AGL(2,𝔽3) (order 20736) · Inn(G) = C3⋊D3 · Out(G) = C2×S42
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23×C3

Aliases

direct productC22×C3×C3⋊D3

Splittings

Fitting, socleC22×C33 ⋊ C2
derivedC32 ⋊ C23×C3
SylowC33 ⋊ C23

Subgroups

subgroupG/H
center, hypercenterC22×C3C3⋊D3
commutator, cosocle, hypocenterC32C23×C3
FrattiniC1G
Fitting, socleC22×C33C2
LCSG ▷ C32
UCSC1 ◁ C22×C3

232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13155012, #13155013, #13155014, …
maximal subgroup ofC3×#3722, #88803, #88814, …
maximal normal subgroup ofC3×#3722, #88803, #88814, …
maximal quotient ofC2×C3×#634, C2×C3×C4×C3⋊D3, C2×C3×C4⋊(C3⋊D3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 175);