Finite Group Databasebeta

← Prev (#72244354) Next (#72244356) →

C2×#99×AGL(1,𝔽7) #72244355

order2688
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#99×AGL(1,𝔽7)

Splittings

FittingC2×C7×#99 ⋊ C2×C3
HallC2×D7×#99 ⋊ C3
HallΓL(1,𝔽8) ⋊ C22×#99
SylowC7 ⋊ C22×C3×#99

Subgroups

subgroupG/H
centerC22C22⋊C4×AGL(1,𝔽7)
commutatorC22×C7C23×C3×C4
FrattiniC23C23×AGL(1,𝔽7)
FittingC2×C7×#99C2×C3
socleC22×C7C2×C3×C22⋊C4
prime radicalC22×D4×AGL(1,𝔽7)C2
cosocleC23×C7C24×C3
hypercenterC2×#99AGL(1,𝔽7)
hypocenterC7C22×C3×#99
LCSG ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

2224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×AGL(1,𝔽7)×B(4,𝔽2), D3×#99×AGL(1,𝔽7), C2×AGL(1,𝔽7)×AΣL(1,𝔽16), …
maximal normal subgroup ofC2×AGL(1,𝔽7)×B(4,𝔽2), C2×AGL(1,𝔽7)×AΣL(1,𝔽16), AGL(1,𝔽7)×#1781, …
maximal quotient ofD3×#99×AGL(1,𝔽7), C2×AGL(1,𝔽7)×AΣL(1,𝔽16), AGL(1,𝔽7)×#1781, …

GAP

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

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