Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #13599 |
312 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 79 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:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(82,92)(86,102)(91,108)(93,110)(94,98)(95,101)(103,111)(105,109);(80,82,85,92)(81,86)(83,95,97,101)(84,98,100,94)(87,91)(88,105)(89,108)(90,102)(93,96,110,99)(103,104)(106,111)(107,109);(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)(80,81,83,88,85,90,97,107)(82,91,93,109,92,108,110,105)(84,89,99,106,100,87,96,104)(86,101,103,94,102,95,111,98)Polycyclic presentation:
2^8,79|a^2=e,b^2=g,d^2=gh,e^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,i^a=i^78,c^b=ch,d^b=dg,e^b=efg,f^b=fh,e^d=egh,f^e=fh1276 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89931);