Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C61 · Schur multiplier (H2) = C2×C4 |
| direct product | C61×#12665 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C61 | C2×D4 |
| commutator | C4 | C2×C4×C8×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| socle | C22×C61 | C8×D4 |
| prime radical | C23×C61 | #131 |
| LCS | G ▷ 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:
(5,6,10,14,9,13,22,8,11,15,27,23,25,32,36,24)(7,16,19,17,20,12,28,29,21,30,26,35,18,31,34,33)(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);(1,2,3,4)(5,7)(6,12)(8,17)(9,20)(10,26)(11,21)(13,30)(14,33)(15,31)(16,32)(18,25)(19,27)(22,34)(23,29)(24,35)(28,36);(5,8,22,32,25,14,10,15,11,24,36,13,9,23,27,6)(7,17,34,16,18,33,26,31,21,35,28,30,20,29,19,12)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5653);