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Cayley diagram

C2×C4⋊(C3⋊Dic3) #64134

order288
isoclinismisoclinic class of #634
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C3

Aliases

direct productC2×C4⋊(C3⋊Dic3)

Splittings

SylowC32 ⋊ C2×C4⋊C4

Subgroups

subgroupG/H
centerC23C2×C3⋊D3
commutatorC2×C32C22×C4
FrattiniC22C22×C3⋊D3
FittingC22×C32×C4C2
socle, prime radicalC23×C32C22
cosocleC22×C32C23
hypercenterC22×C4C3⋊D3
hypocenterC32C2×C4⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4

276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 288

A minimal faithful permutation representation — 3 generators on 16 points:

(3,4,5)(13,14,15,16);(1,2)(4,5)(7,8)(9,10,11,12)(14,16);(1,2)(6,7,8)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10589164, #10589168, #10589395, …
maximal normal subgroup of#10589164, #10589168, #10589395, …
maximal quotient ofC2×C4⋊(C3⋊(C3⋊C8)), #10592623, C4×C4⋊(C3⋊Dic3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 782);