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Cayley diagram

C4×ΓL(1,𝔽25) #4687

order192
isoclinismisoclinic class of Dic6
automorphismAut(G) = D3×#394058 (order 3072) · Inn(G) = C2×D3 · Out(G) = C2×#2021
homologyAbelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4

Aliases

direct productC4×ΓL(1,𝔽25)

Splittings

FittingC3×C4×C8 ⋊ C2
SylowC3 ⋊ C4×M16

Subgroups

subgroupG/H
centerC42C2×D3
commutatorC2×C3C2×C42
FrattiniC2×C4C22×D3
FittingC3×C4×C8C2
socleC22×C3C22×C4
prime radicalC22×D3C2×C4
cosocleC2×C3×C4C23
hypercenterC4×C8D3
hypocenterC3C4×M16
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C42 ◁ C4×C8

142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×ΓL(1,𝔽25), #68048, #68050, …
maximal normal subgroup ofC8×ΓL(1,𝔽25), #68048, #68050, …
maximal quotient ofC8×ΓL(1,𝔽25), #68048, #68050, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 246);