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Cayley diagram

C2×#4176 #66378

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×#4176

Splittings

FittingC2×#731 ⋊ C2
Sylow#731 ⋊ C22

Subgroups

subgroupG/H
center, socle, hypercenterC2×C3(C3⋊D3)⋊C3
commutator, cosocle, hypocenterH(3,𝔽3)C22×C3
FrattiniC32C2×C3×D3
FittingC2×#731C2
prime radicalC2×#272C3
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3

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:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#4176⋊C4, #10619556, #10619557, …
maximal normal subgroup of#4176⋊C4, #10619556, #10619557, …
maximal quotient of#10619613, C4×#4176, C4⋊#4176, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 70);