Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C79×#1543 ⋊ C2 |
| Sylow | C79 ⋊ #13560 |
486 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 486 conjugacy classes of subgroups — too many to draw. Only its 71 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:
(1,4,6,17)(2,8,11,24)(3,13,15,28)(5,16,19,30)(7,20,21,31)(9,22,10,23)(12,26,25,32)(14,27,18,29)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,106,108,110,111,109,107,105,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(1,3)(2,7)(4,13)(5,18)(6,15)(8,20)(9,12)(10,25)(11,21)(14,19)(16,29)(17,28)(22,26)(23,32)(24,31)(27,30)(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)(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);(1,2)(3,12,14,7)(4,8)(5,10,19,9)(6,11)(13,26,27,20)(15,25,18,21)(16,23,30,22)(17,24)(28,32,29,31)(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)(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)Polycyclic presentation:
2^8,79|a^2=e,c^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,i^a=i^78,e^b=efh,f^b=fh,g^b=gh,i^b=i^78,e^d=egh,f^d=fh,f^e=fh1516 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89788);