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C2×D13755 #72290812

order55020
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD27510
direct productC2×D13755

Splittings

Fitting, socleC2×C3×C5×C7×C131 ⋊ C2
derived, HallC3×C5×C7×C131 ⋊ C22
HallC5×C7×C131 ⋊ C2×D3
HallC3×C7×C131 ⋊ C2×D5
HallC3×C5×C131 ⋊ C2×D7
HallC7×C131 ⋊ C2×D15
HallC5×C131 ⋊ C2×D21
HallC3×C131 ⋊ C2×D35
SylowC131 ⋊ C2×D105
HallC3×C5×C7 ⋊ C2×D131
HallC5×C7 ⋊ C2×D393
HallC3×C7 ⋊ C2×D655
HallC3×C5 ⋊ C2×D917
SylowC7 ⋊ C2×D1965
SylowC5 ⋊ C2×D2751
SylowC3 ⋊ C2×D4585

Subgroups

subgroupG/H
center, hypercenterC2D13755
commutator, cosocle, hypocenterC3×C5×C7×C131C22
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C131C2
LCSG ▷ C3×C5×C7×C131
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 148
transitive degreeμt(G) = 27510

A minimal faithful permutation representation — 2 generators on 148 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);(4,5)(7,10)(8,9)(12,17)(13,16)(14,15)(19,148)(20,147)(21,146)(22,145)(23,144)(24,143)(25,142)(26,141)(27,140)(28,139)(29,138)(30,137)(31,136)(32,135)(33,134)(34,133)(35,132)(36,131)(37,130)(38,129)(39,128)(40,127)(41,126)(42,125)(43,124)(44,123)(45,122)(46,121)(47,120)(48,119)(49,118)(50,117)(51,116)(52,115)(53,114)(54,113)(55,112)(56,111)(57,110)(58,109)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)

Conjugacy classes & character table

13758 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), (4,5)(7,10)(8,9)(12,17)(13,16)(14,15)(19,148)(20,147)(21,146)(22,145)(23,144)(24,143)(25,142)(26,141)(27,140)(28,139)(29,138)(30,137)(31,136)(32,135)(33,134)(34,133)(35,132)(36,131)(37,130)(38,129)(39,128)(40,127)(41,126)(42,125)(43,124)(44,123)(45,122)(46,121)(47,120)(48,119)(49,118)(50,117)(51,116)(52,115)(53,114)(54,113)(55,112)(56,111)(57,110)(58,109)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84));