Finite Group DatabasebetaFinite Group Databasebeta| order | 16768 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C131 · Schur multiplier (H2) = C22×C4 |
| direct product | #1969×C131 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C131 | C2×D4 |
| commutator | C2×C4 | C22×C4×C131 |
| Frattini, cosocle | C22×C4 | C23×C131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C131 ◁ C2×C42×C131 ◁ G | |
312 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 16768 |
A minimal faithful permutation representation — 3 generators on 155 points:
(2,4)(5,6,7,8)(9,10)(11,16)(12,21)(13,23)(14,18)(15,19)(17,22)(20,24)(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);(1,2,3,4)(6,8)(9,11,13,19,14,20,22,12)(10,15,17,16,18,21,23,24);(9,12,22,20,14,19,13,11)(10,16,23,15,18,24,17,21)5764 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C131×#7113, C131×#13014, C131×#13015, … |
| maximal normal subgroup of | C131×#7113, C131×#13014, C131×#13015, … |
| maximal quotient of | C131×#9656, C131×#9662, C131×#9718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16768, 701);