Finite Group Databasebeta

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Cayley diagram

C2×C3×C4⋊Dic3 #64048

order288
isoclinismisoclinic class of Dic6
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×C3×C4⋊Dic3

Splittings

SylowC32 ⋊ C2×C4⋊C4

Subgroups

subgroupG/H
centerC23×C3C2×D3
commutatorC2×C3C22×C3×C4
FrattiniC22C22×C3×D3
FittingC22×C32×C4C2
socle, prime radicalC23×C32C22
cosocleC22×C3C23×C3
hypercenterC22×C3×C4D3
hypocenterC3C2×C3×C4⋊C4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

211 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 96

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

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10589163, #10589164, #10589167, …
maximal normal subgroup of#10589163, #10589164, #10589167, …
maximal quotient ofC2×C3×C4⋊(C3⋊C8), C3×#4924, C3×C4×C4⋊Dic3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 696);