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C2×D13559 #72290446

order54236
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD27118
direct productC2×D13559

Splittings

Fitting, socleC2×C7×C13×C149 ⋊ C2
derived, HallC7×C13×C149 ⋊ C22
HallC13×C149 ⋊ C2×D7
HallC7×C149 ⋊ C2×D13
SylowC149 ⋊ C2×D91
HallC7×C13 ⋊ C2×D149
SylowC13 ⋊ C2×D1043
SylowC7 ⋊ C2×D1937

Subgroups

subgroupG/H
center, hypercenterC2D13559
commutator, cosocle, hypocenterC7×C13×C149C22
FrattiniC1G
Fitting, socleC2×C7×C13×C149C2
LCSG ▷ C7×C13×C149
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 171
transitive degreeμt(G) = 27118

A minimal faithful permutation representation — 2 generators on 171 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);(4,9)(5,8)(6,7)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(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

13562 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((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), (4,9)(5,8)(6,7)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(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));