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ΓL(1,𝔽8)×#2403 #72259827

order2688
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C28

Aliases

direct productΓL(1,𝔽8)×#2403

Splittings

Fitting, HallC7×#2403 ⋊ C3
SylowC7 ⋊ C3×#2403

Subgroups

subgroupG/H
center, FrattiniC23C24×ΓL(1,𝔽8)
commutator, socle, cosocleC23×C7C24×C3
FittingC7×#2403C3
hypercenter#2403ΓL(1,𝔽8)
hypocenterC7C3×#2403
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2403

3128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#2403, ΓL(1,𝔽8)×#32898, ΓL(1,𝔽8)×#10336, …
maximal normal subgroup ofAGL(1,𝔽7)×#2403, ΓL(1,𝔽8)×#32898, ΓL(1,𝔽8)×#10336, …
maximal quotient ofΓL(1,𝔽8)×#23900, ΓL(1,𝔽8)×#15947

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((10,12)(14,15,16)(17,19,18), (2,4)(9,10)(11,12)(13,14,16,18,15,17,19), (6,8), (1,2)(3,4)(5,6)(7,8));