Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C61 · Schur multiplier (H2) = C24×C4 |
| direct product | C61×#21098 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C22×D4 |
| commutator | C2×C4 | C23×C4×C61 |
| Frattini, cosocle | C22×C4 | C24×C61 |
| socle | C22×C61 | C2×C4×D4 |
| prime radical | C22×#41×C61 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C22×M16×C61 ◁ G | |
1022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 97 points:
(1,2)(3,4)(5,6,10,14,11,15,9,25)(7,16,20,27,18,30,19,12)(8,21,23,28,24,35,22,13)(17,31,34,36,32,26,33,29)(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);(5,7)(6,12,15,27)(8,17)(9,19)(10,20)(11,18)(13,26,28,31)(14,16,25,30)(21,29,35,36)(22,33)(23,34)(24,32);(2,4)(5,8,9,22,11,24,10,23)(6,13,14,21,15,28,25,35)(7,17,19,33,18,32,20,34)(12,26,16,29,27,31,30,36);(7,18)(12,27)(16,30)(17,32)(19,20)(26,31)(29,36)(33,34)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 14086);