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Cayley diagram

C2×#4184 #66386

order324
isoclinismisoclinic class of #4184
automorphismAut(G) = C2×(#4176⋊C2)⋊C3 (order 1944) · Inn(G) = #4184 · Out(G) = C22×C3
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

direct productC2×#4184

Splittings

FittingC2×#731 ⋊ C2
derived, Sylow#731 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2#4184
commutator, cosocle, hypocenter#731C22
FrattiniC32C2×C3⋊D3
FittingC2×#731C2
socleC2×C3#272
LCSG ▷ #731
UCSC1 ◁ C2

70 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:

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

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#4184⋊C4, #10619556, #10619558, …
maximal normal subgroup of#4184⋊C4, #10619556, #10619558, …
maximal quotient of#10619663, C4×#4184, C4⋊#4184, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 78);