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D7×#5032 #72244275

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

Aliases

direct productD7×#5032

Splittings

FittingC3×C7×#120 ⋊ C22
HallC3×C7 ⋊ C2×B(4,𝔽2)
SylowC7 ⋊ C2×#5032
SylowC3 ⋊ D7×B(4,𝔽2)

Subgroups

subgroupG/H
centerC2D7×#937
commutator, cosocleC23×C3×C7C24
FrattiniC23C22×D3×D7
FittingC3×C7×#120C22
socleC2×C3×C7C2×#120
hypercenter#120D3×D7
hypocenterC3×C7C2×B(4,𝔽2)
LCSG ▷ C23×C3×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C23 ◁ #120

2036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 168

A minimal faithful permutation representation — 4 generators on 18 points:

(1,2,4,6,7,5,3)(8,9,10)(11,13)(12,15)(14,17)(16,18);(2,3)(4,5)(6,7);(15,18);(9,10)(11,12,14,16)(13,15,17,18)

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×D7×B(4,𝔽2), AGL(1,𝔽7)×#5032
maximal normal subgroup ofD3×D7×B(4,𝔽2), AGL(1,𝔽7)×#5032

GAP

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

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