Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #125×C97 ⋊ C4 |
| Sylow | C97 ⋊ #1414 |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 2 generators on 129 points:
(1,2,5,10,19,32,49,70,51,30,46,65,85,66,86,94,89,73,52,35,22,13,6,11,20,33,50,71,79,55,34,17,29,44,61,81,93,97,96,91,76,56,39,26,15,24,37,54,75,83,59,38,21,8,16,27,40,57,77,92,95,88,68,80,60,43,28,41,58,78,87,63,42,25,12,3,7,14,23,36,53,74,90,72,47,67,84,64,45,62,82,69,48,31,18,9,4)(98,100)(99,104,107,121)(101,113)(102,117)(103,111)(105,108,122,125)(106,119,116,120)(109,114)(110,118)(112,123,128,124)(115,127)(126,129);(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)(49,68,69,81)(53,72,73,86)(57,70,76,82)(61,79,80,78)(65,83,84,75)(67,87,85,71)(74,88,89,93)(77,90,91,94)(92,95,96,97)(98,99)(100,108,110,123)(101,112,114,128)(102,116,118,106)(103,107)(104,109,119,126)(105,115)(111,125,117,124)(113,120,127,121)(122,129)Polycyclic presentation:
2^7,97|a^2=d,b^2=f,c^2=f,e^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,h^a=h^75,c^b=cfg,d^b=defg,e^b=eg,d^c=dfg,e^d=eg,h^d=h^96353 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 19656);