Finite Group Databasebeta

← Prev (#72245783) Next (#72245785) →

C23×D7×ΓL(1,𝔽8) #72245784

order2352
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productC23×D7×ΓL(1,𝔽8)

Splittings

Fitting, socleC23×C72 ⋊ C2×C3
derived, SylowC72 ⋊ C24×C3
HallC23×C7×D7 ⋊ C3
HallC7×ΓL(1,𝔽8) ⋊ C24

Subgroups

subgroupG/H
center, hypercenterC23D7×ΓL(1,𝔽8)
commutator, cosocle, hypocenterC72C24×C3
FrattiniC1G
Fitting, socleC23×C72C2×C3
LCSG ▷ C72
UCSC1 ◁ C23

552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 336

A minimal faithful permutation representation — 4 generators on 20 points:

(5,6)(7,8,10)(9,11,12)(14,16,18,20,19,17,15);(3,4)(7,9,10,11,12,13,8);(1,2);(1,2)(3,4)(5,6)(15,16)(17,18)(19,20)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC23×D7×AGL(1,𝔽7), ΓL(1,𝔽8)×#6500, #72245937, …
maximal normal subgroup ofC23×D7×AGL(1,𝔽7), ΓL(1,𝔽8)×#6500, #72245937, …
maximal quotient ofΓL(1,𝔽8)×#6548

GAP

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

gap> G := Group((5,6)(7,8,10)(9,11,12)(14,16,18,20,19,17,15), (3,4)(7,9,10,11,12,13,8), (1,2), (1,2)(3,4)(5,6)(15,16)(17,18)(19,20));