Finite Group Databasebeta

← Prev (#72249174) Next (#72249176) →

ΓL(1,𝔽8)×#4207⋊C3 #72249175

order10206
automorphism?
homologyAbelianization (H1) = C2×C33 · Schur multiplier (H2) = C34

Aliases

direct productΓL(1,𝔽8)×#4207⋊C3

Splittings

FittingC7×3−1+4 ⋊ C2×C3
derivedC3×C7×C9 ⋊ C2×C33
HallΓL(1,𝔽8)×3−1+4 ⋊ C2
SylowC7 ⋊ C3×#4207⋊C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC3×C7×C9C2×C33
FrattiniC3C32×C3⋊D3×ΓL(1,𝔽8)
FittingC7×3−1+4C2×C3
socleC3×C7C33×C3⋊D3
LCSG ▷ C3×C7×C9
UCSC1

3400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 189

A minimal faithful permutation representation — 3 generators on 34 points:

(1,2,3,5,4,6,7)(9,12)(10,15)(11,28,14,18,19,20)(13,21)(16,22)(17,33,23,26,25,32)(24,34,31,29,30,27);(8,11,18,12,19,20,9,14,28)(10,23,27,16,17,34,13,25,29)(15,30,33,21,31,26,22,24,32);(2,3,4)(5,7,6)(8,10,15)(9,13,22)(11,17,24)(12,16,21)(14,23,31)(18,29,26)(19,25,30)(20,27,32)(28,34,33)

Conjugacy classes & character table

275 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofΓL(1,𝔽8)×#13091120, AGL(1,𝔽7)×#4207⋊C3
maximal normal subgroup ofAGL(1,𝔽7)×#4207⋊C3

GAP

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

gap> G := Group((1,2,3,5,4,6,7)(9,12)(10,15)(11,28,14,18,19,20)(13,21)(16,22)(17,33,23,26,25,32)(24,34,31,29,30,27), (8,11,18,12,19,20,9,14,28)(10,23,27,16,17,34,13,25,29)(15,30,33,21,31,26,22,24,32), (2,3,4)(5,7,6)(8,10,15)(9,13,22)(11,17,24)(12,16,21)(14,23,31)(18,29,26)(19,25,30)(20,27,32)(28,34,33));