Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C59 · Schur multiplier (H2) = C25 |
| direct product | C59×#22693 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C59 | D42 |
| commutator | C2×C4 | C23×C4×C59 |
| Frattini, cosocle | C22×C4 | C24×C59 |
| prime radical | C2×#135×C59 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C59 ◁ C22×C4×C59 ◁ G | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 79 |
| transitive degree | μt(G) = 3776 |
A minimal faithful permutation representation — 4 generators on 79 points:
(1,2,3,4)(5,6,12,15)(7,16,18,9)(8,10,14,11)(13,19,20,17)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(5,7)(6,13)(8,9)(10,19)(11,17)(12,18)(14,16)(15,20);(5,8,11,6,12,14,10,15)(7,16,17,20,18,9,19,13);(5,9)(6,7)(8,19)(10,20)(11,13)(12,16)(14,17)(15,18)3068 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 15681);