Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #12902 |
278 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 119 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)(81,88)(82,92,93,115)(84,98)(85,99)(86,97)(87,103,104,114)(89,109)(90,110)(91,108)(94,116,117,127)(95,96,118,121)(100,125)(101,123)(102,124)(105,128,129,138)(106,107,130,120)(111,136)(112,134)(113,135)(119,122,139,132)(126,141)(131,133,142,140)(137,143);(80,82)(81,87)(83,96,97,121)(84,94)(85,95)(86,93)(88,107,108,120)(89,105)(90,106)(91,104)(92,99,115,124)(98,122,123,132)(100,119)(101,117)(102,118)(103,110,114,135)(109,133,134,140)(111,131)(112,129)(113,130)(116,125,127,141)(126,139)(128,136,138,143)(137,142);(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)(80,81,84,89,85,90,100,111,86,91,101,112,102,113,126,137)(82,87,94,105,95,106,119,131,93,104,117,129,118,130,139,142)(83,88,98,109,99,110,125,136,97,108,123,134,124,135,141,143)(92,114,116,138,96,120,122,140,115,103,127,128,121,107,132,133)Polycyclic presentation:
2^8,79|a^2=f,b^2=d,c^2=e,f^2=g,g^2=h,b^a=bd,c^a=ce,i^a=i^78,c^b=cg,e^b=eh,d^c=dh,i^c=i^781768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 85991);