Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #12237 |
334 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 334 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) = 5056 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,22,19)(18,24,26,32)(21,27,31,29)(23,30,28,25)(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)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,22,20)(18,23,26,28)(21,29,31,27)(24,30,32,25);(1,2,6,9)(3,10,12,15)(4,8,5,13)(7,14,16,11)(17,18)(19,23)(20,28)(21,30)(22,26)(24,29)(25,31)(27,32)(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=f,b^2=h,c^2=fh,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,d^b=df,e^b=egh,d^c=dg,e^c=eh,e^d=eh1276 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 82499);