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C2×D15301 #72295902

order61204
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD30602
direct productC2×D15301

Splittings

Fitting, socleC2×C11×C13×C107 ⋊ C2
derived, HallC11×C13×C107 ⋊ C22
HallC13×C107 ⋊ C2×D11
HallC11×C107 ⋊ C2×D13
HallC11×C13 ⋊ C2×D107
SylowC107 ⋊ C2×D143
SylowC13 ⋊ C2×D1177
SylowC11 ⋊ C2×D1391

Subgroups

subgroupG/H
center, hypercenterC2D15301
commutator, cosocle, hypocenterC11×C13×C107C22
FrattiniC1G
Fitting, socleC2×C11×C13×C107C2
LCSG ▷ C11×C13×C107
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 133
transitive degreeμt(G) = 30602

A minimal faithful permutation representation — 2 generators on 133 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);(4,13)(5,12)(6,11)(7,10)(8,9)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(28,133)(29,132)(30,131)(31,130)(32,129)(33,128)(34,127)(35,126)(36,125)(37,124)(38,123)(39,122)(40,121)(41,120)(42,119)(43,118)(44,117)(45,116)(46,115)(47,114)(48,113)(49,112)(50,111)(51,110)(52,109)(53,108)(54,107)(55,106)(56,105)(57,104)(58,103)(59,102)(60,101)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)

Conjugacy classes & character table

15304 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), (4,13)(5,12)(6,11)(7,10)(8,9)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(28,133)(29,132)(30,131)(31,130)(32,129)(33,128)(34,127)(35,126)(36,125)(37,124)(38,123)(39,122)(40,121)(41,120)(42,119)(43,118)(44,117)(45,116)(46,115)(47,114)(48,113)(49,112)(50,111)(51,110)(52,109)(53,108)(54,107)(55,106)(56,105)(57,104)(58,103)(59,102)(60,101)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81));