Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C22×C42 |
| direct product | C61×#29812 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C61 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C61 |
| prime radical | C24×C61 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C2×#117×C61 ◁ G | |
824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 105 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,21,18,22,30,36)(15,19,25,33,24,32,40,43)(16,26,27,34,28,35,29,20)(23,37,39,42,38,44,41,31)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(1,2)(3,4)(6,9)(10,12)(13,15)(14,19)(16,23)(17,25)(18,24)(20,31)(21,33)(22,32)(26,37)(27,39)(28,38)(29,41)(30,40)(34,42)(35,44)(36,43);(2,4)(7,11)(10,12)(13,16,18,28)(14,20,22,34)(15,23,24,38)(17,29,30,27)(19,31,32,42)(21,35,36,26)(25,41,40,39)(33,44,43,37);(5,7,8,11)(6,10,9,12)(15,24)(19,32)(23,38)(25,40)(31,42)(33,43)(37,44)(39,41)2440 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 22800);