Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#11281 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | #124 |
| commutator | C22×C4 | C22×C4×C61 |
| Frattini, cosocle | C23×C4 | C23×C61 |
| socle | C22×C61 | #397 |
| prime radical | C24×C61 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C23×C4×C61 ◁ G | |
594 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)(7,16)(8,20)(9,25)(10,14)(11,15)(12,18)(13,23)(17,30)(19,28)(21,35)(22,31)(24,34)(26,29)(27,36)(32,33)(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)(6,12)(8,21,23,36)(9,26)(10,18)(11,19)(13,30,20,32)(14,28)(15,29)(16,25)(17,24,33,22)(27,31,35,34);(5,8,9,22,11,23,10,24)(6,13,14,31,15,20,25,34)(7,17,18,21,19,33,26,36)(12,27,16,32,29,35,28,30)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 4269);