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Cayley diagram

C2×C3×ΓL(1,𝔽25) #64023

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

Aliases

direct productC2×C3×ΓL(1,𝔽25)

Splittings

FittingC2×C32×C8 ⋊ C2
SylowC32 ⋊ C2×M16

Subgroups

subgroupG/H
centerC2×C3×C4C2×D3
commutatorC2×C3C22×C3×C4
FrattiniC4C22×C3×D3
FittingC2×C32×C8C2
socleC22×C32C2×C4
prime radicalC22×C3×D3C4
cosocleC3×C4C23×C3
hypercenterC2×C3×C8D3
hypocenterC3C2×C3×M16
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C8

147 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)(9,10,11,14,12,13,15,16);(1,2)(4,5)(10,13)(14,16);(1,2)

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10589109, C3×#4514, Dic3×ΓL(1,𝔽25), …
maximal normal subgroup of#10589109, C3×#4514, Dic3×ΓL(1,𝔽25), …
maximal quotient ofC3×#4679, C3×#4685, C3×C4×ΓL(1,𝔽25), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 671);