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Cayley diagram

C22×C4×ΓL(1,𝔽8) #66695

order336
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC22×C4×ΓL(1,𝔽8)

Splittings

Fitting, HallC22×C4×C7 ⋊ C3
derived, SylowC7 ⋊ C22×C3×C4

Subgroups

subgroupG/H
center, hypercenterC22×C4ΓL(1,𝔽8)
commutator, hypocenterC7C22×C3×C4
FrattiniC2C23×ΓL(1,𝔽8)
FittingC22×C4×C7C3
socleC23×C7C2×C3
prime radicalC23×ΓL(1,𝔽8)C2
cosocleC2×C7C23×C3
LCSG ▷ C7
UCSC1 ◁ C22×C4

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C22⋊(C7⋊C8))⋊C3, ΓL(1,𝔽8)⋊#95, ΓL(1,𝔽8)×#95, …
maximal normal subgroup of(C22⋊(C7⋊C8))⋊C3, ΓL(1,𝔽8)⋊#95, ΓL(1,𝔽8)×#95, …
maximal quotient ofC2×C42×ΓL(1,𝔽8), C2×C22⋊C4×ΓL(1,𝔽8), C2×C4⋊C4×ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 164);