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Cayley diagram

C2×D3×ΓL(1,𝔽8) #6990

order252
isoclinismisoclinic class of D3×ΓL(1,𝔽8)
automorphismAut(G) = C2×D3×AGL(1,𝔽7) (order 504) · Inn(G) = D3×ΓL(1,𝔽8) · Out(G) = C22
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×D3×ΓL(1,𝔽8)
unambiguous semidirect productC3⋊(C22×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C3)⋊(C2×ΓL(1,𝔽8))
unambiguous semidirect product(C2×D3×C7)⋊C3

Splittings

Fitting, socleC2×C3×C7 ⋊ C2×C3
derivedC3×C7 ⋊ C22×C3
HallC3×ΓL(1,𝔽8) ⋊ C22
SylowC7 ⋊ C2×C3×D3

Subgroups

subgroupG/H
center, hypercenterC2D3×ΓL(1,𝔽8)
commutator, cosocle, hypocenterC3×C7C22×C3
FrattiniC1G
Fitting, socleC2×C3×C7C2×C3
LCSG ▷ C3×C7
UCSC1 ◁ C2

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 42

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×ΓL(1,𝔽8)⋊C4, #4245⋊C3, #4247⋊C3, …
maximal normal subgroup ofD3×ΓL(1,𝔽8)⋊C4, #4245⋊C3, #4247⋊C3, …
maximal quotient ofΓL(1,𝔽8)×Dic6, C4×D3×ΓL(1,𝔽8), ΓL(1,𝔽8)×D12, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(252, 29);