Finite Group DatabasebetaFinite Group Databasebeta| order | 16768 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C131 · Schur multiplier (H2) = C2×C4 |
| direct product | #1836×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 | |
224 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 151 |
| transitive degree | μt(G) = 16768 |
A minimal faithful permutation representation — 3 generators on 151 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,15)(16,17,20,19)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151);(6,9)(10,12)(13,15,17,20,18,14,19,16);(5,7,8,11)(6,10,9,12)(13,16,17,15,18,20,19,14)4192 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C131×#7266, C131×#7367, C131×#8911, … |
| maximal normal subgroup of | C131×#7266, C131×#7367, C131×#8911, … |
| maximal quotient of | C131×#8786, C131×#8789, C131×#8794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16768, 568);