Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #12277 |
318 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 127 points:
(4,14)(7,12)(9,16)(10,13)(17,20)(18,24,26,40)(19,27)(21,33)(22,32)(23,36,38,39)(25,30,34,45)(28,43)(29,31)(35,48)(37,42,41,46)(44,47)(49,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,106,108,110,112,114,116,118,120,122,124,126,127,125,123,121,119,117,115,113,111,109,107,105,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14)(17,19,22,29)(18,23,26,38)(20,31,32,27)(21,28,35,44)(24,39,40,36)(25,41,34,37)(30,42,45,46)(33,47,48,43)(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)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(104,105)(106,107)(108,109)(110,111)(112,113)(114,115)(116,117)(118,119)(120,121)(122,123)(124,125)(126,127);(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18)(19,23)(20,30)(21,34)(22,26)(24,33)(25,35)(27,42)(28,41)(29,38)(31,46)(32,45)(36,43)(37,44)(39,47)(40,48)(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)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(104,105)(106,107)(108,109)(110,111)(112,113)(114,115)(116,117)(118,119)(120,121)(122,123)(124,125)(126,127)Polycyclic presentation:
2^8,79|a^2=f,b^2=h,c^2=fgh,d^2=f,e^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gh,i^a=i^78,c^b=ch,d^b=df,e^b=egh,i^b=i^78,d^c=dg,e^d=eh1039 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 82765);