Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C61 · Schur multiplier (H2) = C25 |
| direct product | C22×C61×#364 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C61 | D4 |
| commutator | C4 | C23×C8×C61 |
| Frattini, cosocle | C2×C8 | C24×C61 |
| socle | C23×C61 | C4⋊C8 |
| prime radical | C24×C61 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C61 ◁ C23×C8×C61 ◁ G | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 81 points:
(1,2)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(3,4);(5,6,9,14,10,13,18,20,11,15,17,12,8,19,7,16);(5,7,8,17,11,18,10,9)(6,12,13,16,15,14,19,20)6832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 26387);