Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C61 · Schur multiplier (H2) = C27 |
| direct product | C2×C61×#1883 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C61 | #120 |
| commutator | C2×C4 | C23×C4×C61 |
| Frattini, cosocle | C22×C4 | C24×C61 |
| prime radical | C22×AGL(1,ℤ/8)×C61 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C23×C4×C61 ◁ G | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 75 points:
(1,2)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75);(4,6)(7,8,10,12,11,9,14,13);(3,4,5,6)(7,9)(8,11)(10,12)(13,14);(7,8)(9,11)(10,12)(13,14)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 13929);