Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #13489 |
278 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 81 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)(80,83)(81,89,90,115)(82,95)(84,101)(85,106)(86,110)(87,103)(88,111,94,124)(91,99,93,123)(92,120,117,143)(96,127)(97,130)(98,129)(100,105)(102,107)(104,136)(108,140)(109,128)(112,141,122,126)(113,133,125,132)(114,137,142,119)(116,118,139,121)(131,134)(135,138);(80,82)(81,88)(83,100)(84,96)(85,105)(86,109)(87,98)(89,114)(90,94)(91,112)(92,119)(93,122)(95,102)(97,107)(99,132)(101,134)(103,135)(104,131)(106,127)(108,138)(110,130)(111,116)(113,121)(115,142)(117,137)(118,125)(120,141)(123,133)(124,139)(126,143)(128,140)(129,136);(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)(82,94,98,88)(83,99,103,123)(84,91)(85,92)(86,93)(87,90)(89,101,115,110)(95,126)(96,122,109,112)(97,125,131,113)(100,133)(102,120,136,143)(104,118)(105,137,138,119)(106,139,140,116)(107,121)(108,117)(111,127)(114,134)(124,128)(129,141)(130,142)(132,135)Polycyclic presentation:
2^8,79|a^2=eg,c^2=d,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,i^a=i^78,c^b=cf,f^b=fg,g^b=gh,f^c=fg,g^c=gh1756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89406);