Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C97 · Schur multiplier (H2) = C2 |
| direct product | C97×#1585 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C97 | C2×D4 |
| commutator | C2×C4 | C22×C4×C97 |
| Frattini, cosocle | C42 | C23×C97 |
| socle | C22×C97 | C4×D4 |
| prime radical | C2×D4×C97 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C97 ◁ C4×C8×C97 ◁ G | |
168 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 3 generators on 113 points:
(1,2,4,3,5,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(10,13)(11,16)(14,15);(1,3,4,6,5,7,8,2)(10,14)(13,15)3686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 317);