Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #13412 |
430 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 115 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)(80,82,83,81)(85,92)(86,96)(87,102)(88,104)(89,103)(91,108)(93,94)(95,113)(97,100)(98,107)(99,101)(105,106)(110,114)(111,112);(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)(82,83)(84,86,90,100)(85,91,95,112)(87,101,104,98)(88,96,102,97)(89,107,103,99)(92,109,113,115)(93,110,105,114)(94,108,106,111);(81,82)(84,85)(86,91)(87,92)(88,105)(89,106)(90,95)(93,102)(94,103)(96,114)(97,110)(98,115)(99,108)(100,112)(101,109)(104,113)(107,111)Polycyclic presentation:
2^8,79|b^2=h,c^2=d,e^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fg,g^a=gh,c^b=cf,e^b=eg,f^b=fgh,g^b=gh,i^b=i^78,e^c=eg,f^c=fgh,g^c=gh1288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 88939);