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Cayley diagram

C2×#4179 #66381

order324
isoclinismisoclinic class of #4179
automorphismAut(G) = C2×(#4178⋊C2)⋊C3 (order 1944) · Inn(G) = #4179 · Out(G) = C22×C3
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×#4179

Splittings

FittingC2×#732 ⋊ C2
derivedC3×C9 ⋊ C22×C3
Sylow#732 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2#4179
commutator, cosocle, hypocenterC3×C9C22×C3
FrattiniC32C2×C3×D3
FittingC2×#732C2
socleC2×C3(C3⋊D3)⋊C3
LCSG ▷ C3×C9
UCSC1 ◁ C2

56 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:

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

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#4179⋊C4, #10619564, #10619565, …
maximal normal subgroup of#4179⋊C4, #10619564, #10619565, …
maximal quotient of#10619620, C4×#4179, C4⋊#4179, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 73);