Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C23 |
| direct product | C61×#12926 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | #447 |
| commutator | C2×C8 | C22×C4×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| prime radical | C2×#41×C61 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C22×C4×C61 ◁ C2×#131×C61 ◁ G | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 3 generators on 97 points:
(1,2,4,3)(5,6,11,17)(7,19,8,22)(9,25,28,34)(10,18,12,16)(13,23,32,20)(14,21,33,24)(15,27,30,29)(26,35,36,31)(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);(1,3,4,2)(5,7,11,8)(6,13,17,32)(9,26,28,36)(10,21,12,24)(14,30,33,15)(16,31,18,35)(19,34,22,25)(20,27,23,29);(5,8,11,7)(6,14,17,33)(9,23,28,20)(10,21,12,24)(13,30,32,15)(16,19,18,22)(25,31,34,35)(26,29,36,27)2806 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5914);