Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#12547 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C61 | C42 |
| commutator | C4 | C2×C4×C8×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| socle | C22×C61 | C2×#97 |
| prime radical | C24×C61 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C61 ◁ G | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 97 points:
(1,2)(3,4)(5,6,9,14,10,16,24,33,11,15,25,32,8,13,22,30)(7,17,20,34,21,12,26,28,18,29,23,36,19,27,35,31)(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);(2,4)(5,7)(6,12,15,27)(8,19)(9,23)(10,21)(11,18)(13,17,16,29)(14,31,32,28)(20,25)(22,26)(24,35)(30,36,33,34);(7,18)(12,27)(17,29)(19,21)(20,23)(26,35)(28,31)(34,36)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5535);