Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C23 |
| direct product | C61×#12927 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C61 | #447 |
| commutator | C2×C8 | C22×C4×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| socle | C2×C61 | #1852 |
| prime radical | C2×#41×C61 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C61 ◁ C22×C4×C61 ◁ C2×#131×C61 ◁ G | |
378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 125 points:
(1,2,6,13,8,14,7,12)(3,15,18,42,19,43,30,37)(4,20,23,36,24,40,22,10)(5,25,27,39,29,38,28,11)(9,26,33,49,34,57,32,17)(16,44,47,62,48,60,46,41)(21,52,55,61,56,35,54,51)(31,45,50,53,59,64,58,63)(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);(1,3,6,18,8,19,7,30)(2,9,13,33,14,34,12,32)(4,16,23,47,24,48,22,46)(5,26,27,49,29,57,28,17)(10,31,20,50,36,59,40,58)(11,37,25,15,39,42,38,43)(21,53,55,64,56,63,54,45)(35,60,51,41,52,44,61,62);(1,4)(2,10)(3,16)(5,21)(6,23)(7,22)(8,24)(9,31)(11,35)(12,40)(13,20)(14,36)(15,41)(17,45)(18,47)(19,48)(25,51)(26,53)(27,55)(28,54)(29,56)(30,46)(32,58)(33,50)(34,59)(37,60)(38,61)(39,52)(42,44)(43,62)(49,64)(57,63)2806 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5915);