Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C61 · Schur multiplier (H2) = C24×C4 |
| direct product | C61×#10952 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | #386 |
| commutator | C2×C4 | C2×C42×C61 |
| Frattini, cosocle | C23×C4 | C23×C61 |
| prime radical | C24×C61 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C23×C4×C61 ◁ G | |
890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 97 points:
(2,4)(5,6)(7,12,18,27)(8,20)(9,16)(10,15)(11,14)(13,22)(17,31,33,34)(19,30,24,26)(21,28)(23,29)(25,35,36,32)(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);(5,7,10,19,11,18,9,24)(6,12,16,26,14,27,15,30)(8,17,23,32,21,33,22,35)(13,25,28,31,29,36,20,34);(1,2,3,4)(5,8,9,22,11,21,10,23)(6,13,15,20,14,29,16,28)(7,17,24,35,18,33,19,32)(12,25,30,34,27,36,26,31)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 3940);