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Cayley diagram

C3×ΓL(1,𝔽8)⋊C4 #6977

order252
isoclinismisoclinic class of AGL(1,𝔽7)
automorphismAut(G) = C2×D3×AGL(1,𝔽7) (order 504) · Inn(G) = AGL(1,𝔽7) · Out(G) = C2×D3
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C3

Aliases

direct productC3×ΓL(1,𝔽8)⋊C4
unambiguous semidirect productΓL(1,𝔽8)⋊(C3×C4)
unambiguous semidirect productDic7⋊C32
unambiguous semidirect product(C3×Dic7)⋊C3

Splittings

derived, SylowC7 ⋊ C32×C4
HallC3×ΓL(1,𝔽8) ⋊ C4
HallDic7 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC2×C3AGL(1,𝔽7)
commutator, hypocenterC7C32×C4
FrattiniC2C3×AGL(1,𝔽7)
Fitting, socleC2×C3×C7C2×C3
prime radicalC2×C3×ΓL(1,𝔽8)C2
cosocleC2×C7C2×C32
LCSG ▷ C7
UCSC1 ◁ C2×C3

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 84

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

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×ΓL(1,𝔽8)⋊C4, (C3⋊AGL(1,𝔽7))⋊C4, #4247⋊C3, …
maximal normal subgroup ofD3×ΓL(1,𝔽8)⋊C4, (C3⋊AGL(1,𝔽7))⋊C4, #4247⋊C3, …
maximal quotient ofC3×ΓL(1,𝔽8)⋊C8, C2×C3×ΓL(1,𝔽8)⋊C4, C9×ΓL(1,𝔽8)⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(252, 16);