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Cayley diagram

C2×#4178 #66380

order324
isoclinismisoclinic class of #4174
automorphismAut(G) = C2×C3×((C3⋊D3)⋊C3)⋊C2 (order 648) · Inn(G) = (C3⋊D3)⋊C3 · Out(G) = C22×C3
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×#4178

Splittings

FittingC2×#732 ⋊ C2
Sylow#732 ⋊ C22

Subgroups

subgroupG/H
center, socle, hypercenterC2×C3(C3⋊D3)⋊C3
commutator, cosocle, hypocenterH(3,𝔽3)C22×C3
FrattiniC32C2×C3×D3
FittingC2×#732C2
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3

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:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#4178⋊C4, #10619563, #10619564, …
maximal normal subgroup of#4178⋊C4, #10619563, #10619564, …
maximal quotient of#10619619, C4×#4178, C4⋊#4178, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 72);