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Cayley diagram

C24×ΓL(1,𝔽8) #66751

order336
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productC24×ΓL(1,𝔽8)

Splittings

Fitting, socle, HallC24×C7 ⋊ C3
derived, SylowC7 ⋊ C24×C3

Subgroups

subgroupG/H
center, hypercenterC24ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C24×C3
FrattiniC1G
Fitting, socleC24×C7C3
LCSG ▷ C7
UCSC1 ◁ C24

268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×(C22⋊Dic7)⋊C3, #10620789, C2×C22⋊C4×ΓL(1,𝔽8), …
maximal normal subgroup ofC2×(C22⋊Dic7)⋊C3, #10620789, C2×C22⋊C4×ΓL(1,𝔽8), …
maximal quotient ofC23×C4×ΓL(1,𝔽8), C22×D4×ΓL(1,𝔽8), C22×Q8×ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 220);