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D3×D7×ΓL(1,𝔽32) #27189129

order13020
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productD3×D7×ΓL(1,𝔽32)
unambiguous semidirect product(D3×C7)⋊(C2×ΓL(1,𝔽32))
unambiguous semidirect product(C3×D7)⋊(C2×ΓL(1,𝔽32))
unambiguous semidirect productD21⋊(C2×ΓL(1,𝔽32))
unambiguous semidirect product(D3×D7×C31)⋊C5

Splittings

Fitting, socle, derived, HallC3×C7×C31 ⋊ C22×C5
HallC3×C7×ΓL(1,𝔽32) ⋊ C22
HallD3×D7×C31 ⋊ C5
HallC7×ΓL(1,𝔽32) ⋊ C2×D3
HallC3×ΓL(1,𝔽32) ⋊ C2×D7
HallC7×C31 ⋊ C2×C5×D3
HallC3×C31 ⋊ C2×C5×D7
SylowC31 ⋊ C5×D3×D7
HallC3×C7 ⋊ C22×ΓL(1,𝔽32)
SylowC7 ⋊ C2×D3×ΓL(1,𝔽32)
SylowC3 ⋊ C2×D7×ΓL(1,𝔽32)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C7×C31C22×C5
LCSG ▷ C3×C7×C31
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 41
transitive degreeμt(G) = 651

A minimal faithful permutation representation — 2 generators on 41 points:

(2,3,5,8,4)(6,10,15,18,7)(9,13,14,12,11)(16,22,26,24,17)(19,23,21,25,20)(27,31,30,29,28)(33,34)(35,36)(37,38)(39,40,41);(1,2,4,7,8,12,18,24,5,9,14,20,15,21,26,30,3,6,11,17,13,19,25,29,10,16,23,28,22,27,31)(32,33,35,37,38,36,34)(40,41)

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13020, 96);