Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q32×C97 ⋊ C4 |
| Sylow | C97 ⋊ #2179 |
144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 3 generators on 129 points:
(1,2,7,3)(4,11,14,8)(5,16,12,6)(9,24,23,17)(10,29,25,15)(13,20,27,21)(18,41,36,30)(19,46,42,28)(22,37,40,34)(26,33,44,38)(31,62,53,47)(32,67,63,45)(35,58,57,51)(39,54,61,55)(43,50,65,59)(48,82,74,68)(49,87,83,66)(52,80,77,72)(56,78,73,76)(60,75,69,79)(64,71,85,81)(70,89,92,86)(84,88,94,90)(91,95,96,93)(98,101,103,116,104,117,120,102)(99,106,108,118,110,107,109,125)(100,112,113,121,115,129,114,119)(105,122,123,111,124,126,127,128);(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)(99,105)(101,112)(102,119)(103,113)(104,115)(106,122)(107,126)(108,123)(109,127)(110,124)(111,118)(114,120)(116,121)(117,129)(125,128);(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)(98,99,103,108,104,110,120,109)(100,111,113,126,115,128,114,122)(101,107,116,125,117,106,102,118)(105,121,123,129,124,119,127,112)Polycyclic presentation:
2^7,97|a^2=e,c^2=e,d^2=fg,e^2=g,f^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fg,h^a=h^22,d^b=df,f^b=fg,h^b=h^96,h^c=h^22,h^e=h^96296 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 20913);