Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C59 · Schur multiplier (H2) = C2 |
| direct product | C59×#7388 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C59 | #357 |
| commutator | C16 | C2×C8×C59 |
| Frattini, cosocle | C4×C16 | C22×C59 |
| socle | C2×C59 | #1331 |
| prime radical | C59×#508 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C59 ◁ C2×C8×C59 ◁ C4×C8×C59 ◁ C4×C16×C59 ◁ G | |
118 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 118 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 3776 |
A minimal faithful permutation representation — 2 generators on 123 points:
(1,2,5,11,7,12,29,39,8,14,24,44,32,13,6,30)(3,15,17,47,19,49,31,48,21,50,18,56,20,57,52,38)(4,22,25,40,27,58,51,42,28,46,62,41,63,59,26,10)(9,33,34,54,36,61,35,16,37,23,43,53,45,60,64,55)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(1,3)(2,9)(4,23)(5,18)(6,31)(7,19)(8,21)(10,38)(11,43)(12,36)(13,45)(14,37)(15,46)(16,51)(17,24)(20,32)(22,50)(25,54)(26,55)(27,60)(28,33)(29,52)(30,35)(34,44)(39,64)(40,47)(41,56)(42,48)(49,59)(53,62)(57,58)(61,63)3068 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 376);