Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#12589 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C61 | C2×#99 |
| commutator | C2×C4 | C2×C42×C61 |
| Frattini, cosocle | C2×#41 | C23×C61 |
| socle | C2×C61 | C2×#342 |
| prime radical | C22×#41×C61 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C61 ◁ C22×C4×C61 ◁ C22×#41×C61 ◁ G | |
752 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)(3,11,12,23)(4,15,16,8)(5,18)(6,10)(7,13)(9,19)(14,28,26,24)(17,22,30,29)(20,25)(21,32)(27,31)(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);(1,3,5,14,6,12,19,26)(2,7,9,21,10,20,18,31)(4,13,17,27,16,25,30,32)(8,11,22,24,15,23,29,28);(3,12)(7,20)(11,23)(13,25)(14,26)(21,31)(24,28)(27,32)2806 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5577);