Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#13045 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C61 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C61 |
| Frattini, cosocle | C2×#41 | C23×C61 |
| socle | C2×C61 | #1889 |
| prime radical | C61×#3025 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C61 ◁ C2×C4×C61 ◁ C2×#41×C61 ◁ G | |
654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2,5,9,6,10,17,23)(3,11,14,24,15,25,28,32)(4,8,12,19,16,22,26,30)(7,18,21,27,20,29,31,13)(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);(1,3,12,18,6,15,26,29)(2,7,19,24,10,20,30,32)(4,13,5,14,16,27,17,28)(8,11,9,21,22,25,23,31);(4,16)(5,17)(13,27)(14,28)2440 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 6033);