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C2×D13699 #72290706

order54796
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD27398
direct productC2×D13699

Splittings

Fitting, socleC2×C7×C19×C103 ⋊ C2
derived, HallC7×C19×C103 ⋊ C22
HallC19×C103 ⋊ C2×D7
HallC7×C103 ⋊ C2×D19
HallC7×C19 ⋊ C2×D103
SylowC103 ⋊ C2×D133
SylowC19 ⋊ C2×D721
SylowC7 ⋊ C2×D1957

Subgroups

subgroupG/H
center, hypercenterC2D13699
commutator, cosocle, hypocenterC7×C19×C103C22
FrattiniC1G
Fitting, socleC2×C7×C19×C103C2
LCSG ▷ C7×C19×C103
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 131
transitive degreeμt(G) = 27398

A minimal faithful permutation representation — 2 generators on 131 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);(4,9)(5,8)(6,7)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(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

13702 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), (4,9)(5,8)(6,7)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(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));