Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#17835 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C22×D4 |
| commutator | C2×C4 | C23×C4×C61 |
| Frattini, cosocle | C42 | C24×C61 |
| socle | C22×C61 | C22×C4⋊C4 |
| prime radical | C2×#41×C61 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C4×#41×C61 ◁ G | |
612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 93 points:
(1,2,7,4)(3,10,14,16)(5,9,6,15)(8,12,11,13)(17,18,21,25,22,26,20,24)(19,27,29,32,30,31,28,23)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3,6,13,7,14,5,12)(2,8,9,10,4,11,15,16)(17,19,20,28,22,30,21,29)(18,23,25,27,26,32,24,31);(1,4,7,2)(3,11,14,8)(5,9,6,15)(10,13,16,12)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27);(1,5,7,6)(2,9,4,15)(3,12,14,13)(8,10,11,16)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27)4270 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 10823);