Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #7476 |
162 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 162 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:
(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)(80,82,85,98,109,137,102,114,84,106,107,140,112,135,105,133,88,100,103,132,86,99,104,131,87,97,110,134,101,130,83,96)(81,89,92,116,123,142,121,128,113,115,139,141,111,136,90,95,94,119,125,143,93,117,124,127,91,118,122,138,120,126,108,129);(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)(80,81,84,91,88,94,87,113)(82,95,97,127,100,129,106,128)(83,92,102,122,105,125,104,139)(85,108,107,124,103,90,110,121)(86,111,101,123,109,120,112,93)(89,114,115,96,119,131,118,133)(98,136,134,117,132,126,140,142)(99,138,135,116,137,141,130,143)Polycyclic presentation:
2^8,79|a^2=d,b^2=e,c^2=fgh,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,d^b=dh,i^b=i^781768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57280);