Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C61 · Schur multiplier (H2) = C2×C4 |
| direct product | C61×#13122 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C61 | C2×D4 |
| commutator | C2×C4 | C22×C8×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| socle | C22×C61 | C8×D4 |
| prime radical | C2×#41×C61 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C61 ◁ C22×C16×C61 ◁ G | |
326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 97 points:
(2,3)(5,6,11,16,12,15,30,33,9,14,18,19,10,27,29,35)(7,17,22,34,23,28,32,36,20,8,24,26,21,25,13,31)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(1,2,4,3)(5,7,18,24,12,23,29,13,9,20,11,22,10,21,30,32)(6,8,19,34,15,25,35,36,14,17,16,26,27,28,33,31);(1,3,4,2)(5,8,9,17)(6,13,14,32)(7,19,20,16)(10,28,12,25)(11,26,18,34)(15,22,27,24)(21,33,23,35)(29,36,30,31)4636 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 6110);