Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #13693 |
196 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 59 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)(81,90)(82,96)(84,101)(85,106)(86,110)(87,111)(88,103)(89,112)(91,117)(92,121)(93,124)(94,125)(95,118)(97,128)(98,130)(99,127)(100,105)(102,107)(104,135)(108,139)(109,138)(113,133)(114,141)(115,140)(116,120)(119,126)(122,143)(123,142)(129,134)(131,136)(132,137);(80,82,87,98,88,99,84,86)(81,89,94,114,95,115,91,93)(83,96,111,130,103,127,101,110)(85,97,108,120,109,129,104,107)(90,112,125,141,118,140,117,124)(92,113,122,137,123,136,119,105)(100,121,133,143,132,142,131,126)(102,106,128,139,116,138,134,135);(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)(80,81,88,95)(82,93,99,114)(83,100,103,132)(84,94,87,91)(85,105,109,137)(86,89,98,115)(90,116,118,102)(92,120,123,107)(96,126,127,143)(97,119,129,122)(101,133,111,131)(104,113,108,136)(106,124,138,141)(110,121,130,142)(112,139,140,135)(117,134,125,128)Polycyclic presentation:
2^8,79|a^2=h,b^2=dh,d^2=h,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ef,f^a=fg,g^a=gh,i^a=i^78,e^c=ef,f^c=fg,g^c=gh1753 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90394);