Finite Group Databasebeta

← Prev (#72276859) Next (#72276861) →

ΓL(1,𝔽8)×#1795 #72276860

order2688
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C25

Aliases

direct productΓL(1,𝔽8)×#1795

Splittings

Fitting, HallC7×#1795 ⋊ C3
SylowC7 ⋊ C3×#1795

Subgroups

subgroupG/H
centerC2ΓL(1,𝔽8)×#379
commutatorC2×C4×C7C22×C3×C4
FrattiniC22×C4C23×ΓL(1,𝔽8)
FittingC7×#1795C3
socleC2×C7C3×#379
prime radicalC2×ΓL(1,𝔽8)×H(5,𝔽2)C2
cosocleC22×C4×C7C23×C3
hypercenter#1795ΓL(1,𝔽8)
hypocenterC7C3×#1795
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1795

1176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

(2,3,4)(5,7,6)(10,17)(11,20)(13,21)(14,19)(16,22)(18,23);(1,2,4,6,3,5,7)(8,10,11,17,12,18,20,23)(9,13,14,22,15,16,19,21);(8,9)(10,16)(11,19)(12,15)(13,18)(14,20)(17,22)(21,23)

Conjugacy classes & character table

145 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#1795
maximal normal subgroup ofAGL(1,𝔽7)×#1795

GAP

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

gap> G := Group((2,3,4)(5,7,6)(10,17)(11,20)(13,21)(14,19)(16,22)(18,23), (1,2,4,6,3,5,7)(8,10,11,17,12,18,20,23)(9,13,14,22,15,16,19,21), (8,9)(10,16)(11,19)(12,15)(13,18)(14,20)(17,22)(21,23));