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C2×C3⋊(C3⋊(D127⋊C7)) #44544808

order32004
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3⋊(C3⋊(D127⋊C7))
unambiguous semidirect product(C2×C3)⋊(C3⋊(D127⋊C7))
unambiguous semidirect product(C2×C127⋊C7)⋊(C3⋊D3)
unambiguous semidirect product(C2×C3⋊D381)⋊C7

Splittings

Fitting, socleC2×C32×C127 ⋊ C2×C7
derived, HallC32×C127 ⋊ C22×C7
HallC32×C127⋊C7 ⋊ C22
HallC2×C3⋊D381 ⋊ C7
HallC127⋊C7 ⋊ C2×C3⋊D3
SylowC127 ⋊ C2×C7×C3⋊D3
SylowC32 ⋊ C2×D127⋊C7

Subgroups

subgroupG/H
center, hypercenterC2C3⋊(C3⋊(D127⋊C7))
commutator, cosocle, hypocenterC32×C127C22×C7
FrattiniC1G
Fitting, socleC2×C32×C127C2×C7
LCSG ▷ C32×C127
UCSC1 ◁ C2

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 135
transitive degreeμt(G) = 2286

A minimal faithful permutation representation — 3 generators on 135 points:

(4,5)(7,8)(10,14,15,31,13,27,28,11,12,20,22,17,18,44)(16,38,39,89,49,115,46,24,25,64,58,79,80,75)(19,33,50,57,26,70,71,21,30,55,34,41,42,103)(23,59,37,92,76,47,48,35,36,61,62,110,29,77)(32,86,85,87,105,45,73,52,53,54,122,123,43,74)(40,98,97,121,129,130,78,67,68,128,126,99,116,113)(51,120,66,124,108,102,111,83,84,96,91,135,104,112)(56,117,118,60,100,101,109,81,88,82,90,69,131,132)(63,95,119,125,114,106,107,65,94,127,93,133,72,134);(1,2)(6,7,8)(9,10,13,19,12,16,26,43,18,29,49,81,30,51,85,52,15,23,37,63,25,40,69,102,42,72,108,134,73,109,123,103,28,46,76,111,48,78,114,131,80,116,129,98,117,128,96,64,50,82,119,95,84,121,125,92,122,124,90,59,86,89,57,31,22,34,58,53,36,60,91,87,62,93,126,120,94,127,118,55,39,66,97,88,68,99,130,115,101,133,113,77,112,110,75,44,71,105,132,74,107,135,106,70,104,100,67,38,65,61,35,20,32,54,83,33,56,79,47,27,45,41,24,14,21,17,11);(3,4,5)

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32004, 190);