Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C97 · Schur multiplier (H2) = C2 |
| direct product | C97×#1346 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C97 | #99 |
| commutator | C22×C4 | C2×C4×C97 |
| Frattini, cosocle | C2×C4⋊C4 | C22×C97 |
| prime radical | C23×C97 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C97 ◁ C23×C97 ◁ C2×C4⋊C4×C97 ◁ G | |
124 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 2 generators on 121 points:
(1,2)(3,5)(4,7)(6,8)(9,10,12,16,13,17,21,15)(11,18,19,22,20,23,24,14)(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);(2,5)(3,8)(6,7)(9,11,13,20)(10,14,17,22)(12,19,21,24)(15,18,16,23)2231 conjugacy classes. The character table is too large to display.
| maximal quotient of | C97×#7081, C97×#7083, C97×#7089, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 78);