Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C61 · Schur multiplier (H2) = C23×C4 |
| direct product | C61×#10811 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C61 | C4×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C61 |
| Frattini, cosocle | C23×C4 | C23×C61 |
| socle | C2×C61 | #1434 |
| prime radical | C24×C61 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C61 ◁ C22×M16×C61 ◁ G | |
644 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)(3,13,16,23)(4,18)(5,10)(6,11)(7,12)(8,15,24,17)(9,20)(14,28,31,22)(19,25)(21,26)(27,29,32,30)(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);(1,3,6,17,7,16,5,15)(2,8,10,13,12,24,11,23)(4,14,20,30,21,31,19,29)(9,22,18,27,25,28,26,32);(1,4,5,19,7,21,6,20)(2,9,10,18,12,25,11,26)(3,14,15,29,16,31,17,30)(8,22,13,27,24,28,23,32)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 3799);