Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | D16×C97 ⋊ C4 |
| Sylow | C97 ⋊ #2178 |
146 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 46 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:
(1,2,7,3)(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)(98,101,103,115,104,116,119,102)(99,106,107,124,109,125,108,117)(100,111,112,120,114,128,113,118)(105,121,122,127,123,129,126,110);(98,100)(99,105)(101,111)(102,118)(103,112)(104,114)(106,121)(107,122)(108,126)(109,123)(110,117)(113,119)(115,120)(116,128)(124,127)(125,129);(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)(98,99,103,107,104,109,119,108)(100,110,112,121,114,127,113,129)(101,106,115,124,116,125,102,117)(105,120,122,128,123,118,126,111)Polycyclic presentation:
2^7,97|a^2=e,c^2=e,d^2=fg,e^2=g,f^2=g,b^a=bd,d^a=df,f^a=fg,h^a=h^22,d^b=df,f^b=fg,h^c=h^22,h^e=h^96308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 20909);