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C2×#89804⋊C3 #72244362

order2688
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C27

Aliases

direct productC2×#89804⋊C3

Splittings

FittingC2×C7×#120 ⋊ C2×C3
HallC2×#89804 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×B(4,𝔽2)
SylowC7 ⋊ C2×C3×B(4,𝔽2)

Subgroups

subgroupG/H
centerC22#6500⋊C3
commutator, cosocleC23×C7C24×C3
FrattiniC23C23×AGL(1,𝔽7)
FittingC2×C7×#120C2×C3
socleC22×C7C3×#120
hypercenterC2×#120AGL(1,𝔽7)
hypocenterC7C2×C3×B(4,𝔽2)
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24 ◁ C2×#120

2036 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)(10,12)(11,14)(13,16)(15,17);(1,2,5,6,4,7,3)(8,9);(14,17);(10,11,13,15)(12,14,16,17)

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×AGL(1,𝔽7)×B(4,𝔽2), D3×#89804⋊C3, #72276664, …
maximal normal subgroup ofC2×AGL(1,𝔽7)×B(4,𝔽2), #72276664, #72276695, …
maximal quotient ofD3×#89804⋊C3, #72276924, #72276984, …

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