Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#11216 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C61 | #121 |
| commutator | C22×C4 | C22×C4×C61 |
| Frattini, cosocle | C23×C4 | C23×C61 |
| prime radical | C24×C61 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C23×C4×C61 ◁ G | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2)(3,4)(6,7)(9,10,13,17,14,16,12,21)(11,15,20,23,19,24,18,22)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(5,6,8,7)(9,11)(10,15)(12,18)(13,20)(14,19)(16,24)(17,23)(21,22);(2,4)(5,7,8,6)(10,16)(11,18)(12,13)(15,22)(19,20)(23,24)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 4204);