Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C97 ⋊ #1470 |
204 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 6208 |
A minimal faithful permutation representation — 3 generators on 129 points:
(1,2)(3,11)(4,5)(6,7)(8,24)(9,10)(12,20)(13,14)(15,16)(17,41)(18,19)(21,37)(22,23)(25,33)(26,27)(28,29)(30,62)(31,32)(34,58)(35,36)(38,54)(39,40)(42,50)(43,44)(45,46)(47,82)(48,49)(51,80)(52,53)(55,78)(56,57)(59,75)(60,61)(63,71)(64,65)(66,67)(68,79)(69,70)(72,76)(73,74)(81,89)(83,88)(84,85)(86,87)(90,95)(91,92)(93,94)(96,97)(98,101,102,109)(99,104,105,111)(100,107,108,113)(103,110,106,112)(116,122)(119,125)(123,128)(126,129);(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)(98,100,102,108)(99,103,105,106)(101,107,109,113)(104,110,111,112)(114,116)(115,119)(117,123)(118,122)(120,126)(121,125)(124,128)(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)(100,106,108,103)(101,104)(102,105)(107,112,113,110)(109,111)(114,115,117,120,118,121,124,127)(116,119,123,126,122,125,128,129)Polycyclic presentation:
2^7,97|a^2=e,b^2=f,c^2=f,e^2=g,b^a=bd,d^a=df,h^a=h^22,c^b=cg,e^b=ef,h^b=h^96,h^c=h^96,h^e=h^96512 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 19784);