Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #7480 |
166 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 2 generators on 143 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)(80,82,85,98,110,134,106,96,108,131,105,140,103,130,104,139,87,100,111,135,86,99,83,102,84,97,107,132,141,138,142,113)(81,88,91,115,123,133,121,95,90,114,101,137,112,126,120,127,94,118,125,136,92,116,89,119,93,117,124,128,122,143,109,129);(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)(80,81,84,90,87,94,108,93)(82,95,97,127,100,119,131,129)(83,101,104,125,106,124,142,91)(85,109,107,121,111,120,105,89)(86,112,103,92,110,122,141,123)(88,113,114,102,118,139,117,96)(98,133,132,126,135,116,140,143)(99,136,130,128,134,115,138,137)Polycyclic presentation:
2^8,79|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=fh,f^2=gh,g^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fg,g^a=gh,i^a=i^78,c^b=ch3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57292);