Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C28 |
| direct product | C2×C61×#3389 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C61 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C61 |
| prime radical | C22×D8×C61 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C2×C4×Q8×C61 ◁ G | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 5 generators on 87 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87);(3,4,7,6)(5,9,10,8)(11,12)(13,20)(14,18)(15,17)(16,21)(19,23)(22,24)(25,26);(3,5,7,10)(4,8,6,9)(11,13,14,21)(12,16,18,20)(15,22,23,26)(17,25,19,24);(12,17)(13,21)(15,23)(16,24)(18,19)(20,25);(3,6,7,4)(5,8,10,9)(13,21)(16,20)(22,26)(24,25)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 54351);