Finite Group DatabasebetaFinite Group Databasebeta| order | 16768 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C131 · Schur multiplier (H2) = C2×C4 |
| direct product | #1496×C131 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C131 | C2×D4 |
| commutator | C2×C4 | C22×C4×C131 |
| Frattini, cosocle | C42 | C23×C131 |
| socle | C22×C131 | C2×C22⋊C4 |
| prime radical | #532×C131 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C131 ◁ C2×C42×C131 ◁ G | |
246 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 155 |
| transitive degree | μt(G) = 8384 |
A minimal faithful permutation representation — 3 generators on 155 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,16,13,15,19,22)(11,14,18,20,17,21,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155);(2,6)(3,4)(7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(1,3,5,4)(2,7,8,6)(10,15)(11,17)(16,22)(18,23)4192 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C131×#8489, C131×#8500, C131×#8503, … |
| maximal normal subgroup of | C131×#8489, C131×#8500, C131×#8503, … |
| maximal quotient of | C131×#7698, C131×#7699, C131×#7700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16768, 228);