Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C25 |
| direct product | C61×#12766 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C61 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C61 |
| Frattini, cosocle | C2×#41 | C23×C61 |
| socle | C2×C61 | #1790 |
| prime radical | C61×#3529 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C61 ◁ C2×C4×C61 ◁ C2×#41×C61 ◁ G | |
760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2,5,9,6,10,21,28)(3,11,14,29,12,25,19,31)(4,15,18,27,17,8,20,26)(7,22,24,16,23,30,32,13)(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);(1,3)(2,7)(4,16)(5,19)(6,12)(8,25)(9,24)(10,23)(11,15)(13,17)(14,21)(18,22)(20,30)(26,31)(27,29)(28,32);(3,12)(7,23)(8,15)(13,16)(14,19)(22,30)(24,32)(26,27)2074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5754);