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Cayley diagram

C7×ΓL(1,𝔽25) #66606

order336
isoclinismisoclinic class of Dic6
automorphismAut(G) = C24×C3×D3 (order 288) · Inn(G) = C2×D3 · Out(G) = C23×C3
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×ΓL(1,𝔽25)

Splittings

FittingC3×C7×C8 ⋊ C2
HallC3×C7 ⋊ M16
SylowC3 ⋊ C7×M16

Subgroups

subgroupG/H
centerC4×C7C2×D3
commutatorC2×C3C2×C4×C7
FrattiniC4C2×D3×C7
FittingC3×C7×C8C2
socleC2×C3×C7C2×C4
prime radicalC2×D3×C7C4
cosocleC3×C4C22×C7
hypercenterC7×C8D3
hypocenterC3C7×M16
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C7 ◁ C7×C8

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 168

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2,3)(4,11,10,8,7,9,6,5)(12,13,14,15,16,17,18);(2,3)(5,8)(9,11)

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×ΓL(1,𝔽25), #10620808, #10620815, …
maximal normal subgroup ofD7×ΓL(1,𝔽25), #10620808, #10620815, …
maximal quotient ofC7×Dic3⋊C8, C7×#815, C7×(C2×D3)⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 75);