Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #7229 |
244 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 244 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) = 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)(80,82,85,97,101,125,103,127,87,99,83,95,86,98,108,130)(81,88,91,111,92,112,107,134,93,113,89,94,116,123,100,124)(84,96,104,109,132,135,121,141,106,129,102,126,105,128,114,136)(90,117,118,122,119,140,133,142,120,110,115,137,139,138,131,143);(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)(80,81,84,90,87,93,106,120)(82,94,96,122,99,111,129,137)(83,100,102,131,85,107,104,133)(86,92,105,119,101,116,132,139)(88,109,110,95,113,126,117,97)(89,114,115,103,91,121,118,108)(98,124,128,142,125,134,135,143)(112,136,138,127,123,141,140,130)Polycyclic presentation:
2^8,79|a^2=d,b^2=e,c^2=fgh,d^2=h,e^2=fgh,f^2=h,b^a=bc,c^a=cf,e^a=efgh,g^a=gh,i^a=i^78,d^b=dg,d^c=dh2224 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 56640);