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Cayley diagram

C3×ΓL(1,𝔽25) #3727

order144
isoclinismisoclinic class of Dic6
automorphismAut(G) = C24×D3 (order 96) · Inn(G) = C2×D3 · Out(G) = C23
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×ΓL(1,𝔽25)

Splittings

FittingC32×C8 ⋊ C2
SylowC32 ⋊ M16

Subgroups

subgroupG/H
centerC3×C4C2×D3
commutatorC2×C3C2×C3×C4
FrattiniC4C2×C3×D3
FittingC32×C8C2
socleC2×C32C2×C4
prime radicalC2×C3×D3C4
cosocleC3×C4C22×C3
hypercenterC3×C8D3
hypocenterC3C3×M16
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C3×C8

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×ΓL(1,𝔽25), #63791, #63794, …
maximal normal subgroup ofD3×ΓL(1,𝔽25), #63791, #63794, …
maximal quotient ofC3×Dic3⋊C8, C3×#815, C3×(C2×D3)⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 70);