Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C59 · Schur multiplier (H2) = C2 |
| direct product | C59×#7377 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C59 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C2×C8×C59 |
| Frattini, cosocle | #408 | C22×C59 |
| socle | C2×C59 | #1326 |
| prime radical | C2×#41×C59 | C2×C4 |
| LCS | G ▷ C2×D4 ▷ 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:
(1,2,5,11,7,12,27,42,8,13,28,43,6,29,31,41)(3,14,16,44,18,46,50,64,19,47,51,60,17,45,25,57)(4,20,22,53,23,10,36,37,24,38,26,58,30,39,56,63)(9,32,33,15,34,48,40,49,35,59,62,21,54,55,61,52)(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)(2,9,12,34,13,35,29,54)(4,21,23,52,24,15,30,49)(5,25,27,16,28,50,31,51)(10,14,20,45,39,47,38,46)(11,40,41,33,43,61,42,62)(22,32,56,55,26,59,36,48)(37,57,58,44,63,64,53,60)2006 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 365);