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C2×#90013⋊C3 #72229488

order2688
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productC2×#90013⋊C3

Splittings

FittingC2×C7×H(5,𝔽2) ⋊ C2×C3
HallC2×#90013 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×#453
SylowC7 ⋊ C2×C3×#453

Subgroups

subgroupG/H
centerC22#10620789
commutator, cosocleC2×C4×C7C24×C3
FrattiniC2×C4C23×AGL(1,𝔽7)
FittingC2×C7×H(5,𝔽2)C2×C3
socleC22×C7C3×#120
hypercenterC2×H(5,𝔽2)AGL(1,𝔽7)
hypocenterC7C2×C3×#453
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×H(5,𝔽2)

1688 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)(4,5,6)(11,15)(12,17);(10,11,13,15)(12,14,17,16);(8,9);(2,3)(4,5)(6,7)(11,14)(12,17)(15,16)

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#72214048, C2×AGL(1,𝔽7)×#453, D3×#90013⋊C3
maximal normal subgroup ofC2×AGL(1,𝔽7)×#453
maximal quotient ofD3×#90013⋊C3

GAP

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

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