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:
(1,2,12,58,95,90,29,11)(3,19,71,50,96,77,56,44)(4,13,61,28,59,32,5,26)(6,10,48,85,75,82,37,18)(7,30,57,94,92,87,45,9)(8,20,14,63,53,83,81,41)(15,39,67,79,51,74,23,55)(16,40,24,47,68,54,34,60)(17,38,33,70,22,49,65,36)(21,73,42,35,91,27,86,62)(25,84,93,52,69,66,72,46)(31,43,78,88,76,97,89,80)(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);(2,3,5,9,6,4,8,11)(7,14,16,37,17,15,36,18)(10,21,23,46,24,22,45,26)(12,27,29,20,30,28,56,44)(13,31,33,42,34,32,19,41)(25,50,52,68,53,51,82,40)(35,62,48,77,59,60,63,54)(38,43,65,39,66,64,72,55)(47,69,67,88,75,74,73,70)(49,78,79,83,80,61,57,87)(58,89,81,71,84,90,86,91)(76,92,94,97,95,93,96,85)(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^33,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, 21728);