Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #10644 |
298 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 119 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)(81,86)(83,84)(85,87)(88,91,92,103,93,104,106,117)(89,95,96,109,97,110,112,119)(90,99,100,115,101,102,105,116)(94,107,98,113,108,118,111,114);(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)(81,85)(86,87)(88,90,93,101)(89,94,97,108)(91,102,104,99)(92,105,106,100)(95,107,110,118)(96,111,112,98)(103,115,117,116)(109,114,119,113);(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)(80,81,84,86,82,85,83,87)(88,89)(90,98,101,111)(91,95)(92,96)(93,97)(94,100,108,105)(99,114,102,113)(103,109)(104,110)(106,112)(107,116,118,115)(117,119)Polycyclic presentation:
2^8,79|a^2=eh,b^2=g,c^2=dh,d^2=g,e^2=h,b^a=bd,c^a=ce,f^a=fg,i^a=i^78,c^b=cf,d^b=dg,e^b=eg,i^b=i^78,e^c=eh,f^c=fg,i^c=i^781519 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 74445);