Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #12216 |
388 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 388 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 — 3 generators on 111 points:
(17,19,21,27)(18,23,25,32)(20,29,28,26)(22,30,24,31)(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);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(18,22)(20,28)(23,30)(24,25)(26,29)(31,32)(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);(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,21,25)(19,23,27,32)(20,22,28,24)(26,30,29,31)Polycyclic presentation:
2^8,79|a^2=f,b^2=fh,c^2=h,d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=gh,d^b=df,e^b=egh,i^b=i^78,d^c=dg,e^d=eh1045 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 82364);