Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C59 · Schur multiplier (H2) = C2×C4 |
| direct product | C59×#8827 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C59 | #127 |
| commutator | C42 | C22×C4×C59 |
| Frattini, cosocle | C2×C42 | C23×C59 |
| socle | C22×C59 | #384 |
| prime radical | C23×C59 | C4⋊Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C59 ◁ C2×C42×C59 ◁ G | |
306 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 166 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 91 |
| transitive degree | μt(G) = 7552 |
A minimal faithful permutation representation — 3 generators on 91 points:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32)(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);(1,3,4,12,6,11,5,13)(2,7,8,10,9,15,14,16)(18,22)(23,30)(24,29)(25,31);(3,11)(7,15)(10,16)(12,13)(17,19,21,27,20,26,28,32)(18,23,24,25,22,30,29,31)3068 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 1815);