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Cayley diagram

C2×#4177 #66379

order324
isoclinismisoclinic class of #4177
automorphismAut(G) = C2×(#4176⋊C2)⋊C3 (order 1944) · Inn(G) = #4177 · Out(G) = C22×C3
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×#4177

Splittings

FittingC2×#731 ⋊ C2
derivedC3×C9 ⋊ C22×C3
Sylow#731 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2#4177
commutator, cosocle, hypocenterC3×C9C22×C3
FrattiniC32C2×C3×D3
FittingC2×#731C2
socleC2×C3(C3⋊D3)⋊C3
LCSG ▷ C3×C9
UCSC1 ◁ C2

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 2 generators on 29 points:

(4,14,16,7,8,26)(5,23,25,11,12,24)(6,13)(9,29,15,20,21,18)(10,27,28,22,19,17);(1,2)(3,10,7,6,20,9,13,4,22)(5,19,15,12,29,17,25,8,26)(11,16,14,24,28,21,23,18,27)

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#4177⋊C4, #10619557, #10619558, …
maximal normal subgroup of#4177⋊C4, #10619557, #10619558, …
maximal quotient of#10619614, C4×#4177, C4⋊#4177, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 71);