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C24×C4×ΓL(1,𝔽8) #13062293

order1344
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C210

Aliases

direct productC24×C4×ΓL(1,𝔽8)

Splittings

Fitting, HallC24×C4×C7 ⋊ C3
derived, SylowC7 ⋊ C24×C3×C4

Subgroups

subgroupG/H
center, hypercenterC24×C4ΓL(1,𝔽8)
commutator, hypocenterC7C24×C3×C4
FrattiniC2C25×ΓL(1,𝔽8)
FittingC24×C4×C7C3
socleC25×C7C2×C3
prime radicalC25×ΓL(1,𝔽8)C2
cosocleC2×C7C25×C3
LCSG ▷ C7
UCSC1 ◁ C24×C4

2724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 448

A minimal faithful permutation representation — 5 generators on 19 points:

(1,2)(14,15,16)(17,19,18);(3,4)(13,14,15,17,16,18,19);(5,6);(7,8);(9,10,11,12)

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC24×AGL(1,𝔽5)×ΓL(1,𝔽8), C24×C4×AGL(1,𝔽7), C4×ΓL(1,𝔽8)×#120, …
maximal normal subgroup ofC24×C4×AGL(1,𝔽7), C4×ΓL(1,𝔽8)×#120, C4×#10620789
maximal quotient ofC24×AGL(1,𝔽5)×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11582);