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C5×D3423 #46803834

order34230
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D3423
unambiguous semidirect productC3⋊(C5×D1141)
unambiguous semidirect productC7⋊(C5×D489)
unambiguous semidirect productC163⋊(C5×D21)

Splittings

Fitting, socle, HallC3×C5×C7×C163 ⋊ C2
derived, HallC3×C7×C163 ⋊ C2×C5
HallC5×C7×C163 ⋊ D3
HallC3×C5×C163 ⋊ D7
HallC7×C163 ⋊ C5×D3
HallC5×C163 ⋊ D21
HallC3×C163 ⋊ C5×D7
SylowC163 ⋊ C5×D21
HallC3×C5×C7 ⋊ D163
HallC5×C7 ⋊ D489
HallC3×C7 ⋊ C5×D163
HallC3×C5 ⋊ D1141
SylowC7 ⋊ C5×D489
SylowC3 ⋊ C5×D1141

Subgroups

subgroupG/H
center, hypercenterC5D3423
commutator, cosocle, hypocenterC3×C7×C163C2×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C163C2
LCSG ▷ C3×C7×C163
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 178
transitive degreeμt(G) = 17115

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

Conjugacy classes & character table

8565 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34230, 16);