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Cayley diagram

C22⋊C4×ΓL(1,𝔽8) #66580

order336
isoclinismisoclinic class of D4×ΓL(1,𝔽8)
automorphismAut(G) = #120×AGL(1,𝔽7) (order 1344) · Inn(G) = C22×ΓL(1,𝔽8) · Out(G) = C2×D4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC22⋊C4×ΓL(1,𝔽8)
unambiguous semidirect product(C7×C22⋊C4)⋊C3

Splittings

Fitting, HallC7×C22⋊C4 ⋊ C3
SylowC7 ⋊ C3×C22⋊C4

Subgroups

subgroupG/H
center, FrattiniC22C22×ΓL(1,𝔽8)
commutatorC2×C7C2×C3×C4
FittingC7×C22⋊C4C3
socle, cosocleC22×C7C22×C3
prime radicalC23×ΓL(1,𝔽8)C2
hypercenterC22⋊C4ΓL(1,𝔽8)
hypocenterC7C3×C22⋊C4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22⋊C4

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 56

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

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6380⋊C3, ΓL(1,𝔽8)×#99, #6440⋊C3, …
maximal normal subgroup of#6380⋊C3, ΓL(1,𝔽8)×#99, #6440⋊C3, …
maximal quotient ofΓL(1,𝔽8)×#95, ΓL(1,𝔽8)×C22⋊C8, ΓL(1,𝔽8)×#99, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 49);