Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C97 · Schur multiplier (H2) = C28 |
| direct product | C97×#3529 |
| 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 | |
1058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 5 generators on 121 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23);(2,6)(3,7)(11,18)(14,17)(19,24)(21,23);(2,6)(5,8)(10,15)(11,18)(16,22)(19,24);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)3686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 2261);