Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C97 ⋊ #2170 |
146 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 3 generators on 129 points:
(1,2,9,42,92,67,20,7)(3,11,50,19,44,24,4,17)(5,18,52,90,87,82,30,15)(6,21,66,94,96,78,59,16)(8,23,74,56,93,81,43,40)(10,32,22,53,55,83,68,48)(12,29,73,77,63,86,38,36)(13,31,33,60,57,65,26,51)(14,45,25,76,37,62,72,28)(27,80,41,70,71,34,84,46)(35,39,69,95,64,58,61,75)(47,79,85,88,97,91,89,49)(98,101,103,110,102,115,117,121,104,116,119,123,118,129,122,105)(99,100,107,112,108,113,125,127,109,114,126,128,106,111,124,120);(98,100)(99,105)(101,107)(102,113)(103,112)(104,114)(106,123)(108,110)(109,121)(111,118)(115,125)(116,126)(117,127)(119,128)(120,122)(124,129);(2,3,4,7,5,8,10,16)(6,12,13,15,14,22,28,30)(9,19,20,40,21,41,43,32)(11,24,25,48,26,49,23,46)(17,33,34,59,35,18,37,38)(27,51,52,65,44,71,53,81)(29,45,54,72,36,61,47,75)(31,55,56,82,57,39,63,64)(42,67,68,80,69,91,70,74)(50,77,78,89,79,66,83,62)(58,84,85,60,86,73,76,87)(88,95,94,90,92,96,93,97)(98,99,103,107,102,108,117,125,104,109,119,126,118,106,122,124)(100,110,112,115,113,121,127,116,114,123,128,129,111,105,120,101)Polycyclic presentation:
2^7,97|a^2=e,c^2=e,d^2=g,e^2=f,f^2=g,b^a=bd,d^a=dg,h^a=h^64,d^b=dg,h^c=h^64,h^e=h^22,h^f=h^96140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21727);