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Cayley diagram

C22×ΓL(1,𝔽25) #5737

order192
isoclinismisoclinic class of Dic6
automorphismAut(G) = C2×D3×#72209641 (order 18432) · Inn(G) = C2×D3 · Out(G) = C2×#11715850
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC22×ΓL(1,𝔽25)

Splittings

FittingC22×C3×C8 ⋊ C2
SylowC3 ⋊ C22×M16

Subgroups

subgroupG/H
centerC22×C4C2×D3
commutatorC2×C3C23×C4
FrattiniC4C23×D3
FittingC22×C3×C8C2
socleC23×C3C2×C4
prime radicalC23×D3C4
cosocleC3×C4C24
hypercenterC22×C8D3
hypocenterC3C22×M16
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C8

298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#68649, #68650, #68660, …
maximal normal subgroup of#68649, #68650, #68660, …
maximal quotient ofC2×#4679, #78447, C2×C4×ΓL(1,𝔽25), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1296);