Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C59 · Schur multiplier (H2) = C27 |
| direct product | C59×#21005 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C59 | C2×#121 |
| commutator | C2×C4 | C23×C4×C59 |
| Frattini, cosocle | C22×C4 | C24×C59 |
| prime radical | C22×AGL(1,ℤ/8)×C59 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C59 ◁ C23×C4×C59 ◁ G | |
1782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 79 |
| transitive degree | μt(G) = 1888 |
A minimal faithful permutation representation — 4 generators on 79 points:
(1,2)(3,4)(5,6,10,14,11,13,9,7)(8,16,17,12,18,15,19,20)(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);(5,7,10,13,11,14,9,6)(8,15,19,20,18,16,17,12);(2,4)(5,8)(6,12)(7,15)(9,17)(10,19)(11,18)(13,20)(14,16);(5,9,11,10)(6,7,13,14)(8,17,18,19)(12,15,20,16)3422 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 13993);