Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C97 · Schur multiplier (H2) = C25 |
| direct product | C97×#3276 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C97 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C97 |
| prime radical | #547×C97 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C97 ◁ C4×D4×C97 ◁ G | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 4 generators on 117 points:
(1,2)(3,4)(5,6,11,14)(7,15,18,20)(8,13,10,9)(12,16,19,17)(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);(5,7,11,18)(6,12,14,19)(8,17,10,16)(9,15,13,20);(2,4)(5,8,11,10)(6,13,14,9)(7,16,18,17)(12,15,19,20);(5,9)(6,8)(7,12)(10,14)(11,13)(15,16)(17,20)(18,19)2813 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 2008);