Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C25 |
| direct product | C2×C61×#2264 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C2×D16 |
| commutator, Frattini, cosocle | C16 | C24×C61 |
| prime radical | C2×C61×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C22×C4×C61 ◁ C22×C8×C61 ◁ C22×C16×C61 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 127 points:
(1,2)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(3,4)(5,15)(6,20)(7,26)(8,30)(9,11)(10,16)(12,23)(13,22)(14,28)(17,47)(18,50)(19,42)(21,39)(24,58)(25,41)(27,37)(29,60)(31,43)(32,46)(33,54)(34,53)(35,44)(36,48)(38,65)(40,45)(49,66)(51,56)(52,57)(55,64)(59,62)(61,63);(3,5,9,19)(4,10,11,32)(6,21,25,55)(7,27,29,61)(8,31,28,18)(12,37,35,63)(13,39,36,64)(14,40,30,34)(15,41,42,20)(16,44,46,23)(17,48,49,22)(24,59,57,54)(26,33,60,62)(38,66,51,47)(43,52,50,58)(45,56,53,65);(4,11)(10,32)(12,35)(13,36)(14,30)(15,42)(20,41)(26,60)(33,62)(34,40)(37,63)(38,51)(39,64)(43,50)(47,66)(52,58)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 26968);