Finite Group Databasebeta

← Prev (#72269737) Next (#72269739) →

AGL(1,𝔽7)×#92302 #72269738

order20412
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productAGL(1,𝔽7)×#92302

Splittings

FittingC7×#6891 ⋊ C22×C3
derivedC7×C32⋊C9 ⋊ C22×C32
HallΓL(1,𝔽8)×#6891 ⋊ C22
SylowC7 ⋊ C2×C3×#92302

Subgroups

subgroupG/H
center, hypercenterC3AGL(1,𝔽7)×C3⋊AGL(1,ℤ/9)
commutator, cosocle, hypocenterC7×C32⋊C9C22×C32
FrattiniC32C3×C3⋊D3×AGL(1,𝔽7)
FittingC7×#6891C22×C3
socleC32×C7C2×C32×C3⋊D3
LCSG ▷ C7×C32⋊C9
UCSC1 ◁ C3

4700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 126

A minimal faithful permutation representation — 3 generators on 25 points:

(1,2)(3,4)(5,7)(8,9,11)(13,14,16)(18,21,22)(20,24,25);(2,4,6)(3,5,7)(9,11)(10,13)(12,14)(15,16)(18,20)(21,25)(22,24);(8,15,13,11,12,14,9,10,16)(17,18,20)(19,22,24)(21,25,23)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofAGL(1,𝔽7)×#13091120

GAP

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

gap> G := Group((1,2)(3,4)(5,7)(8,9,11)(13,14,16)(18,21,22)(20,24,25), (2,4,6)(3,5,7)(9,11)(10,13)(12,14)(15,16)(18,20)(21,25)(22,24), (8,15,13,11,12,14,9,10,16)(17,18,20)(19,22,24)(21,25,23));