Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C97 ⋊ #1858 |
252 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,8)(4,7)(5,6)(9,18)(10,17)(11,12)(13,16)(14,15)(19,32)(20,31)(21,22)(23,30)(24,25)(26,29)(27,28)(33,50)(34,49)(35,36)(37,48)(38,39)(40,47)(41,42)(43,46)(44,45)(51,72)(52,71)(53,54)(55,70)(56,57)(58,69)(59,60)(61,68)(62,63)(64,67)(65,66)(73,89)(74,87)(75,76)(77,88)(78,79)(80,85)(81,82)(83,84)(90,95)(91,92)(93,94)(96,97)(98,101)(99,106)(100,110)(102,117)(103,113)(104,114)(105,118)(107,122)(108,123)(109,124)(111,128)(112,127)(115,121)(116,126)(119,129)(120,125);(2,4,5,3)(6,10,11,9)(7,8,14,13)(12,20,21,19)(15,18,24,23)(16,26,27,17)(22,34,35,33)(25,32,38,37)(28,40,41,31)(29,30,44,43)(36,52,53,51)(39,50,56,55)(42,58,59,49)(45,48,62,61)(46,64,65,47)(54,74,75,73)(57,72,78,77)(60,80,81,71)(63,70,83,76)(66,85,86,69)(67,68,87,82)(79,89,91,90)(84,88,93,92)(94,95,97,96)(98,100,103,111,104,112,102,116)(99,105,107,120,108,119,115,109)(101,110,113,128,114,127,117,126)(106,118,122,125,123,129,121,124);(2,3,5,4)(6,9,11,10)(7,13,14,8)(12,19,21,20)(15,23,24,18)(16,17,27,26)(22,33,35,34)(25,37,38,32)(28,31,41,40)(29,43,44,30)(36,51,53,52)(39,55,56,50)(42,49,59,58)(45,61,62,48)(46,47,65,64)(54,73,75,74)(57,77,78,72)(60,71,81,80)(63,76,83,70)(66,69,86,85)(67,82,87,68)(79,90,91,89)(84,92,93,88)(94,96,97,95)(98,99,103,107,104,108,102,115)(100,109,111,105,112,120,116,119)(101,106,113,122,114,123,117,121)(110,125,128,129,127,124,126,118)Polycyclic presentation:
2^7,97|a^2=f,b^2=f,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,h^a=h^75,d^b=dg,e^b=eg,h^b=h^22,d^c=dg,h^c=h^96,h^f=h^96296 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 20468);