Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C97 · Schur multiplier (H2) = C1 |
| direct product | C97×#1338 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C97 | C22⋊C4 |
| commutator | C2×C4 | C2×C8×C97 |
| Frattini, cosocle | C4×C8 | C22×C97 |
| socle, prime radical | C22×C97 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C97 ◁ C4×C8×C97 ◁ G | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 12416 |
A minimal faithful permutation representation — 2 generators on 121 points:
(2,5)(3,7)(6,8)(9,10,12,17,14,15,20,22,13,16,21,23,11,18,19,24)(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);(1,2,4,6)(3,8,7,5)(9,11,13,14)(10,15,16,18)(12,20,21,19)(17,24,23,22)3686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 70);