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C5×D3129 #44530977

order31290
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D3129
unambiguous semidirect productC3⋊(C5×D1043)
unambiguous semidirect productC7⋊(C5×D447)
unambiguous semidirect productC149⋊(C5×D21)

Splittings

Fitting, socle, HallC3×C5×C7×C149 ⋊ C2
derived, HallC3×C7×C149 ⋊ C2×C5
HallC5×C7×C149 ⋊ D3
HallC3×C5×C149 ⋊ D7
HallC7×C149 ⋊ C5×D3
HallC5×C149 ⋊ D21
HallC3×C149 ⋊ C5×D7
SylowC149 ⋊ C5×D21
HallC3×C5×C7 ⋊ D149
HallC5×C7 ⋊ D447
HallC3×C7 ⋊ C5×D149
HallC3×C5 ⋊ D1043
SylowC7 ⋊ C5×D447
SylowC3 ⋊ C5×D1043

Subgroups

subgroupG/H
center, hypercenterC5D3129
commutator, cosocle, hypocenterC3×C7×C149C2×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C149C2
LCSG ▷ C3×C7×C149
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 164
transitive degreeμt(G) = 15645

A minimal faithful permutation representation — 2 generators on 164 points:

(1,2,3,4,5)(6,7,8)(9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164);(7,8)(10,15)(11,14)(12,13)(17,164)(18,163)(19,162)(20,161)(21,160)(22,159)(23,158)(24,157)(25,156)(26,155)(27,154)(28,153)(29,152)(30,151)(31,150)(32,149)(33,148)(34,147)(35,146)(36,145)(37,144)(38,143)(39,142)(40,141)(41,140)(42,139)(43,138)(44,137)(45,136)(46,135)(47,134)(48,133)(49,132)(50,131)(51,130)(52,129)(53,128)(54,127)(55,126)(56,125)(57,124)(58,123)(59,122)(60,121)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)

Conjugacy classes & character table

7830 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31290, 13);