Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #7421 |
140 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 151 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 2 generators on 151 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)(80,82)(81,84)(83,87)(85,86)(88,90,95,108)(89,96,102,128)(91,110,115,142)(92,116,118,105)(93,120,94,122)(97,130,134,138)(98,127,101,124)(99,135,100,136)(103,109,140,132)(104,113,143,114)(106,111,107,117)(112,150,151,148)(119,125,121,126)(123,129,141,149)(131,145,133,144)(137,147,139,146);(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)(82,86,87,84)(83,85)(88,89,92,98,95,102,118,101)(90,103,105,137,108,140,116,139)(91,109,112,146,115,132,151,147)(93,119,117,99,94,121,111,100)(96,123,124,104,128,141,127,143)(97,129,131,114,134,149,133,113)(106,144,120,138,107,145,122,130)(110,135,148,125,142,136,150,126)Polycyclic presentation:
2^8,79|a^2=d,b^2=h,c^2=fg,d^2=h,e^2=gh,f^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fg,g^a=gh,i^a=i^78,c^b=cf,d^b=defg,e^b=eg,f^b=fg,g^b=gh,i^b=i^783169 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57133);