Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C61 · Schur multiplier (H2) = C1 |
| direct product | C61×#7462 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C61 | D16 |
| commutator | C16 | C2×C8×C61 |
| Frattini, cosocle | C4×C16 | C22×C61 |
| socle | C2×C61 | #1371 |
| prime radical | C22×C61 | #368 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C61 ◁ C2×C8×C61 ◁ C4×C8×C61 ◁ C4×C16×C61 ◁ G | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 2 generators on 125 points:
(1,2,5,11,7,12,26,38,8,14,27,39,29,13,6,28)(3,15,18,41,20,42,50,62,21,44,51,63,53,43,19,52)(4,22,23,35,24,54,47,37,25,56,57,36,46,55,16,10)(9,30,31,48,32,58,60,49,34,59,61,64,33,45,40,17)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(1,3,16,45,6,19,46,64,29,53,57,59,27,51,25,49,8,21,47,58,26,50,24,48,7,20,23,30,5,18,4,17)(2,9,22,41,11,31,35,42,12,32,54,62,38,60,37,44,14,34,56,63,39,61,36,43,13,33,55,52,28,40,10,15)4636 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 450);