Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C97 ⋊ #1379 |
136 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 136 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) = 133 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 2 generators on 133 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)(98,99,100,101)(102,104)(103,109)(105,118)(106,120)(107,115)(108,117)(110,126)(111,128)(112,114)(113,125)(116,122)(119,132)(121,123)(124,130)(127,131)(129,133);(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)(102,103,106,111,107,112,121,129,108,113,122,130,105,110,119,127)(104,114,116,131,115,125,132,128,117,126,120,133,118,109,123,124)Polycyclic presentation:
2^7,97|a^2=d,b^2=e,c^2=g,d^2=f,f^2=g,b^a=bc,h^a=h^22,c^b=cg,d^b=dg,h^b=h^96,h^d=h^96536 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 19565);