Finite Group DatabasebetaFinite Group Databasebeta| order | 16768 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C131 · Schur multiplier (H2) = C23×C4 |
| direct product | #3135×C131 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C131 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C131 |
| socle | C22×C131 | C22×D4 |
| prime radical | #532×C131 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C131 ◁ C2×C42×C131 ◁ G | |
392 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 184 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 — 4 generators on 155 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,20)(12,16)(13,18)(14,23)(15,19)(17,22)(21,24)(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);(2,5)(6,8)(9,11,14,12,15,22,13,21)(10,16,18,20,19,24,23,17);(1,3,4,7)(2,6,5,8)(9,12,14,22,15,21,13,11)(10,17,18,16,19,20,23,24);(9,13,15,14)(10,18,19,23)(11,21,22,12)(16,20,24,17)4192 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C131×#9005, C131×#9008, C131×#10033, … |
| maximal normal subgroup of | C131×#9005, C131×#9008, C131×#10033, … |
| maximal quotient of | C131×#20762, C131×#21086, C131×#21199, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16768, 1867);