Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #13657 |
182 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 67 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)(80,83,86,100,87,101,110,85)(81,89,92,91,93,108,111,123)(82,95,112,134,97,132,98,109)(84,104,106,105,107,99,139,133)(88,113,124,122,114,115,116,127)(90,119,120,125,121,117,138,102)(94,126,137,143,128,118,129,130)(96,103,140,142,135,131,136,141);(80,82)(81,88)(83,95)(84,103)(85,109)(86,112)(87,97)(89,113)(90,118)(91,122)(92,124)(93,114)(94,125)(96,133)(98,110)(99,136)(100,134)(101,132)(102,128)(104,140)(105,135)(106,142)(107,131)(108,115)(111,116)(117,137)(119,129)(120,130)(121,126)(123,127)(138,143)(139,141);(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)(82,94,97,128)(83,91)(84,102)(85,108)(86,111)(87,93)(88,96,114,135)(89,100)(90,105)(92,110)(95,130,132,143)(98,137,112,129)(99,138)(101,123)(103,122,131,127)(104,120)(106,117)(107,125)(109,126,134,118)(113,141,115,142)(116,140,124,136)(119,139)(121,133)Polycyclic presentation:
2^8,79|a^2=e,c^2=eg,d^2=fgh,f^2=gh,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fg,g^a=gh,i^a=i^78,d^b=dfh,e^b=eh,f^b=fg,g^b=gh,d^c=dh,i^c=i^781048 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90187);