Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C79 ⋊ #12721 |
694 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 694 conjugacy classes of subgroups — too many to draw. Only its 103 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:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(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)(81,88,92,90)(83,99)(85,102)(87,103,105,107)(89,106,98,91)(93,104,109,108)(94,110)(96,111);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(80,82)(81,87)(83,94)(84,101)(85,96)(86,97)(88,103)(89,93)(90,107)(91,108)(92,105)(95,100)(98,109)(99,110)(102,111)(104,106);(80,81,85,91)(82,93,96,103)(83,98,100,88)(84,90,99,89)(86,92,102,106)(87,101,104,110)(94,105,95,108)(97,109,111,107)Polycyclic presentation:
2^8,79|a^2=f,c^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,e^b=eh,f^b=fg,d^c=dh,f^c=fh,i^c=i^78,f^d=fh931 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 85162);