Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #13529 |
342 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 3 generators on 123 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)(81,83)(92,94,97,106)(93,99,102,115)(95,108,110,122)(96,111,109,105)(98,103,113,119)(100,116,112,123)(101,118,117,114)(104,120,121,107);(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)(80,81)(82,83)(84,86,87,91)(85,88,89,90)(93,98)(95,107)(96,109)(99,103)(100,101)(102,113)(104,108)(105,111)(110,120)(112,117)(114,123)(115,119)(116,118)(121,122);(84,85,87,89)(86,90,91,88)(92,93,95,100,109,117,120,113,97,102,110,112,96,101,107,98)(94,103,104,118,111,123,122,115,106,119,121,114,105,116,108,99)Polycyclic presentation:
2^8,79|a^2=d,b^2=deg,c^2=h,d^2=gh,e^2=gh,g^2=h,b^a=bd,c^a=ce,c^b=cf,d^b=dg,e^b=eg,g^b=gh,d^c=dg,e^c=eg,g^c=gh,i^c=i^781768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89643);