Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C28 |
| direct product | C61×#61016 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C61 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C4×#41×C61 ◁ G | |
1728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 5 generators on 93 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(18,22)(20,27)(23,28)(24,29)(26,31)(30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(18,23)(19,25)(22,28)(26,31);(3,10)(6,9)(11,16)(13,15)(17,18)(19,24)(20,28)(21,23)(22,27)(25,30)(26,32)(29,31);(2,7)(6,9)(8,14)(13,15)(17,19)(18,24)(20,26)(21,25)(22,29)(23,30)(27,31)(28,32);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(19,25)(24,30)(26,31)(29,32)2806 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 54004);