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C5×D3297 #45682132

order32970
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D3297
unambiguous semidirect productC3⋊(C5×D1099)
unambiguous semidirect productC7⋊(C5×D471)
unambiguous semidirect productC157⋊(C5×D21)

Splittings

Fitting, socle, HallC3×C5×C7×C157 ⋊ C2
derived, HallC3×C7×C157 ⋊ C2×C5
HallC5×C7×C157 ⋊ D3
HallC3×C5×C157 ⋊ D7
HallC7×C157 ⋊ C5×D3
HallC5×C157 ⋊ D21
HallC3×C157 ⋊ C5×D7
SylowC157 ⋊ C5×D21
HallC3×C5×C7 ⋊ D157
HallC5×C7 ⋊ D471
HallC3×C7 ⋊ C5×D157
HallC3×C5 ⋊ D1099
SylowC7 ⋊ C5×D471
SylowC3 ⋊ C5×D1099

Subgroups

subgroupG/H
center, hypercenterC5D3297
commutator, cosocle, hypocenterC3×C7×C157C2×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C157C2
LCSG ▷ C3×C7×C157
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 172
transitive degreeμt(G) = 16485

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

Conjugacy classes & character table

8250 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32970, 16);