Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #13718 |
160 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 55 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:
(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)(80,82,85,108,86,93,105,96)(81,87,90,120,91,110,107,112)(83,99,102,138,98,127,134,136)(84,104,100,95,106,140,117,125)(88,115,116,129,114,123,132,143)(89,118,103,111,119,135,139,142)(92,122,130,97,121,133,124,131)(94,126,141,101,109,137,128,113);(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)(83,98)(88,114)(92,121)(94,109)(97,131)(99,127)(101,113)(102,134)(115,123)(116,132)(122,133)(124,130)(126,137)(128,141)(129,143)(136,138);(80,81,86,91)(82,92,93,121)(83,97,98,131)(84,103,106,139)(85,107,105,90)(87,109,110,94)(88,113,114,101)(89,117,119,100)(95,129,125,143)(96,130,108,124)(99,135,127,118)(102,122,134,133)(104,123,140,115)(111,136,142,138)(112,128,120,141)(116,137,132,126)Polycyclic presentation:
2^8,79|a^2=h,c^2=dg,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=ef,f^a=fg,g^a=gh,e^b=eh,d^c=dh,e^c=efh,f^c=fg,g^c=gh,i^c=i^781045 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90530);