Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12952 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C23 |
| direct product | C61×#12954 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C61 | #1864 |
| commutator | C2×C8 | C22×C4×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| prime radical | C23×C61 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C61 ◁ C22×C4×C61 ◁ C61×#405 ◁ G | |
380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 125 points:
(1,2,6,11,7,13,31,12)(3,14,18,46,19,34,29,41)(4,20,22,42,23,58,30,57)(5,24,27,10,28,43,52,44)(8,32,35,16,37,25,36,17)(9,26,38,21,40,59,39,60)(15,47,50,64,51,33,55,63)(45,54,62,49,56,53,61,48)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(1,3,7,19)(2,8)(4,15,23,51)(5,25,28,32)(6,29,31,18)(9,33)(10,41)(11,36)(12,35)(13,37)(14,24)(16,52,17,27)(20,56)(21,54,60,53)(22,55,30,50)(26,49,59,48)(34,43)(38,64)(39,63)(40,47)(42,62)(44,46)(45,58)(57,61);(1,4,7,23)(2,9,13,40)(3,15,19,51)(5,26,28,59)(6,30,31,22)(8,33,37,47)(10,42,44,57)(11,39,12,38)(14,45,34,56)(16,53,17,54)(18,55,29,50)(20,24,58,43)(21,27,60,52)(25,49,32,48)(35,64,36,63)(41,62,46,61)2440 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5942);