Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #12954 |
380 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 380 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)(80,83,86,102)(81,88,92,119)(82,94,98,130)(84,105,107,138)(85,109,110,101)(87,112,116,126)(89,121,123,136)(90,118,91,117)(93,124,113,135)(95,132,96,133)(97,134,108,129)(99,103,137,122)(100,106,139,131)(104,128,111,127)(114,143,115,142)(120,141,125,140);(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)(80,82,86,98)(81,87)(83,94,102,130)(84,104,107,111)(85,108,110,97)(88,112)(89,120)(90,115)(91,114)(92,116)(93,103)(95,131,96,106)(99,135)(100,133,139,132)(101,134,109,129)(105,128,138,127)(113,122)(117,143)(118,142)(119,126)(121,141)(123,125)(124,137)(136,140);(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)(80,81,85,90,86,92,110,91)(82,93,97,125,98,113,108,120)(83,99,101,121,102,137,109,136)(84,103,106,89,107,122,131,123)(87,111,114,95,116,104,115,96)(88,105,117,100,119,138,118,139)(94,126,129,143,130,112,134,142)(124,133,141,128,135,132,140,127)Polycyclic presentation:
2^8,79|a^2=f,b^2=fg,c^2=h,d^2=g,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,i^a=i^78,d^b=dg,e^b=egh,f^b=fh,g^b=gh,i^b=i^78,d^c=dgh,e^c=egh,f^c=fh,g^c=gh,i^c=i^781756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 86296);