Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C97 · Schur multiplier (H2) = C29 |
| direct product | C97×#3449 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C97 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C97 |
| socle | C22×C97 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C97 ◁ G | |
1556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 5 generators on 117 points:
(1,2)(3,4)(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19);(2,4)(7,14)(10,13)(15,20)(17,19);(6,11)(10,13)(12,18)(17,19);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)4268 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 2181);