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Cayley diagram

C4×ΓL(1,𝔽8)⋊C4 #66545

order336
isoclinismisoclinic class of AGL(1,𝔽7)
automorphismAut(G) = #120×AGL(1,𝔽7) (order 1344) · Inn(G) = AGL(1,𝔽7) · Out(G) = #120
homologyAbelianization (H1) = C3×C42 · Schur multiplier (H2) = C4

Aliases

direct productC4×ΓL(1,𝔽8)⋊C4
unambiguous semidirect productΓL(1,𝔽8)⋊C42
unambiguous semidirect product(C4×Dic7)⋊C3

Splittings

derived, SylowC7 ⋊ C3×C42
HallC4×Dic7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C42

Subgroups

subgroupG/H
center, hypercenterC2×C4AGL(1,𝔽7)
commutator, hypocenterC7C3×C42
FrattiniC22C2×AGL(1,𝔽7)
FittingC2×C4×C7C2×C3
socle, cosocleC22×C7C22×C3
prime radicalC22×ΓL(1,𝔽8)C22
LCSG ▷ C7
UCSC1 ◁ C2×C4

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×ΓL(1,𝔽8)⋊C4, (ΓL(1,𝔽8)⋊C4)⋊C8, #6389⋊C3, …
maximal normal subgroup ofC8×ΓL(1,𝔽8)⋊C4, (ΓL(1,𝔽8)⋊C4)⋊C8, #6389⋊C3, …
maximal quotient ofC4×ΓL(1,𝔽8)⋊C8, #6377⋊C3, C8×ΓL(1,𝔽8)⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 14);