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C2×ΓL(1,𝔽8)×AΣL(1,𝔽16) #72276556

order2688
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24

Aliases

direct productC2×ΓL(1,𝔽8)×AΣL(1,𝔽16)

Splittings

Fitting, HallC2×C7×AΣL(1,𝔽16) ⋊ C3
SylowC7 ⋊ C2×C3×AΣL(1,𝔽16)

Subgroups

subgroupG/H
centerC22ΓL(1,𝔽8)×#99
commutatorC23×C7C22×C3×C4
FrattiniC2×D4C23×ΓL(1,𝔽8)
FittingC2×C7×AΣL(1,𝔽16)C3
socleC22×C7C3×#99
prime radicalC2×ΓL(1,𝔽8)×#120C2
cosocleC2×C7×D4C23×C3
hypercenterC2×AΣL(1,𝔽16)ΓL(1,𝔽8)
hypocenterC7C2×C3×AΣL(1,𝔽16)
LCSG ▷ C23×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×AΣL(1,𝔽16)

876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×AGL(1,𝔽7)×AΣL(1,𝔽16), #72276599, ΓL(1,𝔽8)×#12687, …
maximal normal subgroup ofC2×AGL(1,𝔽7)×AΣL(1,𝔽16), #72276599, ΓL(1,𝔽8)×#12687, …
maximal quotient ofΓL(1,𝔽8)×#12687

GAP

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

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