Finite Group Databasebeta

← Prev (#72228337) Next (#72228339) →

C2×C3×SL(2,𝔽8) #72228338

order3024
isoclinismisoclinic class of SL(2,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC2×C3×SL(2,𝔽8)

Subgroups

subgroupG/H
center, Fitting, hypercenterC2×C3SL(2,𝔽8)
commutator, hypocenterSL(2,𝔽8)C2×C3
FrattiniC1G
LCSG ▷ SL(2,𝔽8)
UCSC1 ◁ C2×C3

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

socle ofDic3×SL(2,𝔽8)
maximal subgroup ofSL(2,𝔽3)×SL(2,𝔽8), Dic3×SL(2,𝔽8)
maximal normal subgroup ofDic3×SL(2,𝔽8)

GAP

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

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