Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C25 |
| direct product | C61×#24336 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C61 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C61 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C2×C4×Q8×C61 ◁ G | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 15616 |
A minimal faithful permutation representation — 4 generators on 89 points:
(1,2)(3,4)(5,6,9,12)(7,11,8,10)(13,14)(15,23)(16,20)(17,21)(18,22)(19,26)(24,27)(25,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(5,7,9,8)(6,10,12,11)(13,14)(15,24)(16,20)(17,21)(18,22)(19,25)(23,27)(26,28);(2,4)(13,15,18,27)(14,19,22,28)(16,25,17,26)(20,23,21,24);(5,8,9,7)(6,11,12,10)(13,16,18,17)(14,20,22,21)(15,25,27,26)(19,23,28,24)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 17324);