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C2×D13029 #72289210

order52116
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD26058
direct productC2×D13029

Splittings

Fitting, socleC2×C3×C43×C101 ⋊ C2
derived, HallC3×C43×C101 ⋊ C22
HallC43×C101 ⋊ C2×D3
HallC3×C101 ⋊ C2×D43
HallC3×C43 ⋊ C2×D101
SylowC101 ⋊ C2×D129
SylowC43 ⋊ C2×D303
SylowC3 ⋊ C2×D4343

Subgroups

subgroupG/H
center, hypercenterC2D13029
commutator, cosocle, hypocenterC3×C43×C101C22
FrattiniC1G
Fitting, socleC2×C3×C43×C101C2
LCSG ▷ C3×C43×C101
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 149
transitive degreeμt(G) = 26058

A minimal faithful permutation representation — 2 generators on 149 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);(4,5)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(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

13032 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), (4,5)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(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));