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C2×#100×AGL(1,𝔽7) #72229472

order2688
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×#100×AGL(1,𝔽7)

Splittings

FittingC2×C7×#100 ⋊ C2×C3
HallC2×D7×#100 ⋊ C3
HallΓL(1,𝔽8) ⋊ C22×#100
SylowC7 ⋊ C22×C3×#100

Subgroups

subgroupG/H
centerC22C22⋊C4×AGL(1,𝔽7)
commutatorC22×C7C23×C3×C4
FrattiniC2×C4C23×AGL(1,𝔽7)
FittingC2×C7×#100C2×C3
socleC22×C7C2×C3×C22⋊C4
prime radicalC22×D4×AGL(1,𝔽7)C2
cosocleC2×C4×C7C24×C3
hypercenterC2×#100AGL(1,𝔽7)
hypocenterC7C22×C3×#100
LCSG ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

1984 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:

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

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×AGL(1,𝔽7)×#453, D3×#100×AGL(1,𝔽7), C2×AGL(1,𝔽7)×AΣL(1,𝔽16), …
maximal normal subgroup ofC2×AGL(1,𝔽7)×#453, C2×AGL(1,𝔽7)×AΣL(1,𝔽16), AGL(1,𝔽7)×#1783, …
maximal quotient ofD3×#100×AGL(1,𝔽7), AGL(1,𝔽7)×#1783

GAP

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

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