Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C97 ⋊ #2916 |
440 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 198 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 — 4 generators on 129 points:
(1,2,5,10,19,32,49,70,64,45,62,80,72,47,67,85,92,83,63,42,25,12,3,7,14,23,36,53,74,60,43,28,41,58,78,68,87,94,96,90,81,59,38,21,8,16,27,40,57,77,56,39,26,15,24,37,54,75,89,95,97,93,86,79,55,34,17,29,44,61,73,52,35,22,13,6,11,20,33,50,71,88,91,84,66,82,76,51,30,46,65,69,48,31,18,9,4)(101,114)(106,120)(110,124)(115,116)(118,128)(121,127)(122,129)(125,126);(98,101)(99,106)(100,110)(102,115)(103,116)(104,114)(105,118)(107,121)(108,120)(109,122)(111,125)(112,126)(113,124)(117,127)(119,128)(123,129);(2,4)(3,6)(5,9)(7,13)(8,15)(10,18)(11,12)(14,22)(16,26)(17,28)(19,31)(20,25)(21,24)(23,35)(27,39)(29,43)(30,45)(32,48)(33,42)(34,41)(36,52)(37,38)(40,56)(44,60)(46,64)(47,66)(49,69)(50,63)(51,62)(53,73)(54,59)(55,58)(57,77)(61,74)(65,70)(67,84)(68,86)(71,83)(72,82)(75,81)(76,80)(78,79)(85,91)(87,93)(88,92)(89,90)(94,97)(95,96)(98,100,102,111,104,113,103,112)(99,105,107,109,108,119,117,123)(101,110,115,125,114,124,116,126)(106,118,121,122,120,128,127,129);(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,86)(53,72,73,82)(57,76,77,80)(61,79,74,78)(65,81,70,75)(67,83,84,71)(85,90,91,89)(87,92,93,88)(94,96,97,95)(98,99,103,107,104,108,102,117)(100,109,112,119,113,123,111,105)(101,106,116,121,114,120,115,127)(110,122,126,128,124,129,125,118)Polycyclic presentation:
2^7,97|a^2=f,b^2=e,e^2=g,f^2=g,b^a=be,e^a=eg,h^a=h^22,h^b=h^96,d^c=dg,h^f=h^96524 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21172);