Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #13491 |
272 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 272 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)(81,89,90,114)(82,94)(84,101)(85,104)(86,107)(87,100)(88,110,93,123)(91,119,117,143)(92,122,108,116)(95,127)(96,129)(97,130)(98,103)(99,105)(102,135)(106,126)(109,125)(111,140,124,142)(112,141,121,139)(113,136,131,118)(115,137,138,120)(128,133)(132,134);(80,82,84,95)(81,88,90,93)(83,98,101,132)(85,103,102,134)(86,106,87,97)(89,113,114,131)(91,118,117,136)(92,121,108,112)(94,125,127,99)(96,109,128,105)(100,133,107,129)(104,130,135,126)(110,138,123,115)(111,137,124,120)(116,142,122,140)(119,141,143,139);(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)(80,81,87,108,84,90,86,92)(82,93,106,112,95,88,97,121)(83,89,100,116,101,114,107,122)(85,91,109,137,102,117,105,120)(94,123,126,141,127,110,130,139)(96,124,134,118,128,111,103,136)(98,131,129,142,132,113,133,140)(99,115,104,119,125,138,135,143)Polycyclic presentation:
2^8,79|a^2=eg,b^2=h,c^2=d,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,e^c=eh,f^c=fg,g^c=gh1282 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89420);