Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C79 ⋊ #13034 |
462 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 462 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 143 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)(80,83,87,104)(81,89,93,119)(82,95,99,134)(84,106,85,110)(86,113,107,103)(88,114,116,140)(90,121,91,125)(92,128,122,118)(94,129,130,142)(96,135,97,136)(98,137,112,133)(100,126,117,123)(101,108,102,111)(105,132,109,131)(115,141,127,139)(120,143,124,138);(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)(80,82)(81,88,93,116)(83,95)(84,105)(85,109)(86,112)(87,99)(89,114,119,140)(90,120,91,124)(92,127,122,115)(94,126,130,123)(96,108)(97,111)(98,107)(100,129,117,142)(101,136)(102,135)(103,137)(104,134)(106,132)(110,131)(113,133)(118,141,128,139)(121,143,125,138);(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)(80,81,85,90,86,92,111,123,87,93,84,91,107,122,108,126)(82,94,97,115,98,120,109,116,99,130,96,127,112,124,105,88)(83,100,102,128,103,121,106,89,104,117,101,118,113,125,110,119)(95,114,132,138,133,139,135,142,134,140,131,143,137,141,136,129)Polycyclic presentation:
2^8,79|a^2=f,b^2=dfg,c^2=h,d^2=g,e^2=g,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,i^a=i^78,d^b=dg,e^b=egh,f^b=fg,g^b=gh,i^b=i^78,d^c=dgh,e^c=egh,f^c=fgh,g^c=gh,i^c=i^781288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 86782);