Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C23 |
| direct product | C61×#12956 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | #442 |
| commutator | C2×C8 | C22×C4×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| prime radical | C22×D8×C61 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C22×C4×C61 ◁ C2×#97×C61 ◁ G | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2)(3,4)(5,6,9,16,11,17,10,20)(7,15,18,13,19,12,8,14)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(2,4)(6,12,17,15)(7,8)(9,10)(13,20,14,16)(18,19);(6,13)(7,18)(8,19)(9,10)(12,20)(14,17)(15,16)2440 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5944);