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D3×C3⋊(D127⋊C7) #44544786

order32004
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2

Aliases

direct productD3×C3⋊(D127⋊C7)
unambiguous semidirect product(C3×D3)⋊(D127⋊C7)
unambiguous semidirect product(D3×D381)⋊C7
unambiguous semidirect product(D3×C127⋊C7)⋊D3

Splittings

Fitting, socle, derived, HallC32×C127 ⋊ C22×C7
HallC32×C127⋊C7 ⋊ C22
HallD3×D381 ⋊ C7
HallC127⋊C7 ⋊ D32
SylowC127 ⋊ D32×C7
SylowC32 ⋊ C2×D127⋊C7

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C127C22×C7
LCSG ▷ C32×C127
UCSC1

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 133
transitive degreeμt(G) = 762

A minimal faithful permutation representation — 2 generators on 133 points:

(1,2,3)(5,6)(8,12,13,29,11,25,26,9,10,18,20,15,16,42)(14,36,37,87,47,113,44,22,23,62,56,77,78,73)(17,31,48,55,24,68,69,19,28,53,32,39,40,101)(21,57,35,90,74,45,46,33,34,59,60,108,27,75)(30,84,83,85,103,43,71,50,51,52,120,121,41,72)(38,96,95,119,127,128,76,65,66,126,124,97,114,111)(49,118,64,122,106,100,109,81,82,94,89,133,102,110)(54,115,116,58,98,99,107,79,86,80,88,67,129,130)(61,93,117,123,112,104,105,63,92,125,91,131,70,132);(2,3)(4,5,6)(7,8,11,17,10,14,24,41,16,27,47,79,28,49,83,50,13,21,35,61,23,38,67,100,40,70,106,132,71,107,121,101,26,44,74,109,46,76,112,129,78,114,127,96,115,126,94,62,48,80,117,93,82,119,123,90,120,122,88,57,84,87,55,29,20,32,56,51,34,58,89,85,60,91,124,118,92,125,116,53,37,64,95,86,66,97,128,113,99,131,111,75,110,108,73,42,69,103,130,72,105,133,104,68,102,98,65,36,63,59,33,18,30,52,81,31,54,77,45,25,43,39,22,12,19,15,9)

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32004, 168);