Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #13522 |
240 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 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)(80,83,84,91,86,90,98,81)(82,88,93,92,96,106,95,105)(85,89,87,97,99,107,102,100)(94,109,101,104,103,108,110,111);(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)(80,82,84,93,86,96,98,95)(81,88,83,92,91,106,90,105)(85,94,87,101,99,103,102,110)(89,104,97,108,107,111,100,109);(80,81,87,100,98,83,85,89,86,91,102,97,84,90,99,107)(82,92,101,104,95,106,94,108,96,105,110,111,93,88,103,109)Polycyclic presentation:
2^8,79|a^2=eg,b^2=d,c^2=d,d^2=h,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fh,i^b=i^78,e^c=eg,f^c=fh,g^c=gh,i^c=i^781282 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89600);