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Q8×SL(2,𝔽8) #72228337

order4032
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

direct productQ8×SL(2,𝔽8)

Subgroups

subgroupG/H
center, Frattini, cosocleC2C22×SL(2,𝔽8)
commutator, socle, prime radicalC2×SL(2,𝔽8)C22
Fitting, hypercenterQ8SL(2,𝔽8)
hypocenterSL(2,𝔽8)Q8
LCSG ▷ C2×SL(2,𝔽8) ▷ SL(2,𝔽8)
UCSC1 ◁ C2 ◁ Q8

140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

derived subgroup ofSL(2,𝔽3)×SL(2,𝔽8)
maximal subgroup ofSL(2,𝔽3)×SL(2,𝔽8)
maximal normal subgroup ofSL(2,𝔽3)×SL(2,𝔽8)

GAP

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

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