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Cayley diagram

C2×#4207 #66455

order324
isoclinismisoclinic class of #4207
automorphismAut(G) = C2×(#4207⋊C2)⋊C3 (order 1944) · Inn(G) = #4207 · Out(G) = C22×C3
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×#4207

Splittings

FittingC2×#737 ⋊ C2
derivedC3×C9 ⋊ C22×C3
Sylow#737 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2#4207
commutator, cosocle, hypocenterC3×C9C22×C3
FrattiniC3C2×C3×C3⋊D3
FittingC2×#737C2
socleC2×C3C3×C3⋊D3
LCSG ▷ C3×C9
UCSC1 ◁ C2

93 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#4207⋊C4, #10619907, #10619909, …
maximal normal subgroup of#4207⋊C4, #10619907, #10619909, …
maximal quotient of#10619965, C4×#4207, C4⋊#4207, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 147);