Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #13630 |
198 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 143 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 — 3 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)(82,94)(88,108)(93,116)(95,98)(96,119)(97,120)(107,123)(109,131)(110,132)(117,136)(118,137)(121,124)(122,139)(130,140)(133,142)(138,143);(80,82,83,95,87,94,86,98)(81,88,89,93,92,108,99,116)(84,96,100,107,104,119,103,123)(85,97,101,121,106,120,105,124)(90,109,111,117,114,131,125,136)(91,110,112,118,115,132,126,137)(102,122,127,130,129,139,128,140)(113,133,134,138,135,142,141,143);(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)(80,81,84,90,85,91,102,113,86,89,103,111,105,112,128,134,87,92,104,114,106,115,129,135,83,99,100,125,101,126,127,141)(82,93,96,117,97,118,122,138,98,108,123,131,124,132,140,142,94,116,119,136,120,137,139,143,95,88,107,109,121,110,130,133)Polycyclic presentation:
2^8,79|a^2=e,b^2=d,d^2=h,e^2=f,f^2=g,g^2=h,b^a=bd,d^a=dh,i^a=i^78,c^b=ch3208 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90061);