Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C97 · Schur multiplier (H2) = C25 |
| direct product | C2×#443×C97 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C97 | D4 |
| commutator | C4 | C23×C4×C97 |
| Frattini, cosocle | C2×C4 | C24×C97 |
| socle | C22×C97 | C4×D4 |
| prime radical | C2×#135×C97 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C97 ◁ C22×C8×C97 ◁ G | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 4 generators on 115 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(3,4,7,11,8,5,12,10)(6,9,14,17,15,13,18,16);(3,5)(4,8)(6,13)(7,10)(9,15)(11,12)(14,16)(17,18);(3,6,7,14,8,15,12,18)(4,9,11,17,5,13,10,16)5432 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 1685);