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Cayley diagram

C2×ΓL(1,𝔽25) #900

order96
isoclinismisoclinic class of Dic6
automorphismAut(G) = C2×D3×#120 (order 384) · Inn(G) = C2×D3 · Out(G) = C22×D4
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22

Aliases

direct productC2×ΓL(1,𝔽25)

Splittings

FittingC2×C3×C8 ⋊ C2
SylowC3 ⋊ C2×M16

Subgroups

subgroupG/H
centerC2×C4C2×D3
commutatorC2×C3C22×C4
FrattiniC4C22×D3
FittingC2×C3×C8C2
socleC22×C3C2×C4
prime radicalC22×D3C4
cosocleC3×C4C23
hypercenterC2×C8D3
hypocenterC3C2×M16
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C8

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#4514, C4×ΓL(1,𝔽25), #4688, …
maximal normal subgroup of#4514, C4×ΓL(1,𝔽25), #4688, …
maximal quotient of#4679, #4685, C4×ΓL(1,𝔽25), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 107);