Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C59 · Schur multiplier (H2) = C2 |
| direct product | C59×#7378 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C59 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C8×C59 |
| Frattini, cosocle | #408 | C22×C59 |
| socle | C2×C59 | #1326 |
| prime radical | C2×#41×C59 | C2×C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C59 ◁ C2×C4×C59 ◁ C22×C4×C59 ◁ C59×#408 ◁ G | |
170 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 46 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:
(1,2,5,11,7,12,26,42,8,13,27,43,6,28,30,41)(3,14,16,44,18,46,24,57,19,36,51,60,17,45,53,63)(4,20,21,54,22,10,35,37,23,38,25,58,29,39,56,64)(9,31,32,50,33,59,61,52,34,55,62,15,47,48,40,49)(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);(1,3,6,17,8,19,7,18)(2,9,12,33,13,34,28,47)(4,15,22,49,23,50,29,52)(5,24,26,51,27,53,30,16)(10,36,20,46,39,14,38,45)(11,40,41,62,43,61,42,32)(21,55,56,59,25,31,35,48)(37,63,58,44,64,57,54,60)2006 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 366);