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C5×D3507 #46856386

order35070
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D3507
unambiguous semidirect productC3⋊(C5×D1169)
unambiguous semidirect productC7⋊(C5×D501)
unambiguous semidirect productC167⋊(C5×D21)

Splittings

Fitting, socle, HallC3×C5×C7×C167 ⋊ C2
derived, HallC3×C7×C167 ⋊ C2×C5
HallC5×C7×C167 ⋊ D3
HallC3×C5×C167 ⋊ D7
HallC7×C167 ⋊ C5×D3
HallC5×C167 ⋊ D21
HallC3×C167 ⋊ C5×D7
SylowC167 ⋊ C5×D21
HallC3×C5×C7 ⋊ D167
HallC5×C7 ⋊ D501
HallC3×C7 ⋊ C5×D167
HallC3×C5 ⋊ D1169
SylowC7 ⋊ C5×D501
SylowC3 ⋊ C5×D1169

Subgroups

subgroupG/H
center, hypercenterC5D3507
commutator, cosocle, hypocenterC3×C7×C167C2×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C167C2
LCSG ▷ C3×C7×C167
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 182
transitive degreeμt(G) = 17535

A minimal faithful permutation representation — 2 generators on 182 points:

(1,2,3,4,5)(6,7,8)(9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182);(7,8)(10,15)(11,14)(12,13)(17,182)(18,181)(19,180)(20,179)(21,178)(22,177)(23,176)(24,175)(25,174)(26,173)(27,172)(28,171)(29,170)(30,169)(31,168)(32,167)(33,166)(34,165)(35,164)(36,163)(37,162)(38,161)(39,160)(40,159)(41,158)(42,157)(43,156)(44,155)(45,154)(46,153)(47,152)(48,151)(49,150)(50,149)(51,148)(52,147)(53,146)(54,145)(55,144)(56,143)(57,142)(58,141)(59,140)(60,139)(61,138)(62,137)(63,136)(64,135)(65,134)(66,133)(67,132)(68,131)(69,130)(70,129)(71,128)(72,127)(73,126)(74,125)(75,124)(76,123)(77,122)(78,121)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)

Conjugacy classes & character table

8775 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35070, 13);