Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #7483 |
262 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 59 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 — 2 generators on 111 points:
(1,3)(2,7,10,23)(4,13)(5,19)(6,16)(8,12,26,22)(9,27,28,30)(11,29,25,24)(14,17)(15,21)(18,32)(20,31)(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);(1,2,4,9,6,10,18,28)(3,12,14,7,16,22,31,23)(5,11,20,8,21,25,17,26)(13,29,15,27,32,24,19,30)(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)Polycyclic presentation:
2^8,79|a^2=d,b^2=e,c^2=h,d^2=h,f^2=h,b^a=bc,c^a=cf,e^a=eh,f^a=fg,g^a=gh,i^a=i^78,d^b=df,f^b=fh,g^b=gh,d^c=dgh,f^c=fh,f^d=fh1276 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57301);