Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #13513 |
228 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 87 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)(80,83,84,99,85,101,87,110)(81,89,90,116,91,119,102,118)(82,94,95,126,97,127,109,128)(86,107,103,135,104,138,108,100)(88,111,93,123,114,141,113,125)(92,122,120,117,105,139,136,134)(96,129,106,98,130,132,137,133)(112,131,124,142,140,143,121,115);(80,82,84,95,85,97,87,109)(81,88,90,93,91,114,102,113)(83,98,99,132,101,133,110,129)(86,106,103,130,104,137,108,96)(89,115,116,131,119,142,118,143)(92,121,120,112,105,124,136,140)(94,107,126,135,127,138,128,100)(111,139,123,134,141,122,125,117);(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)(80,81,84,102,85,91,87,90)(82,93,95,88,97,113,109,114)(83,89,99,118,101,119,110,116)(86,105,103,120,104,92,108,136)(94,125,126,141,127,123,128,111)(96,121,106,140,130,124,137,112)(98,131,132,115,133,143,129,142)(100,134,107,139,135,117,138,122)Polycyclic presentation:
2^8,79|a^2=eg,b^2=d,c^2=d,d^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,i^a=i^78,c^b=cf,e^b=eh,f^b=fg,g^b=gh,e^c=eh,f^c=fg,g^c=gh,i^c=i^781288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89551);