Finite Group DatabasebetaFinite Group Databasebeta| order | 16768 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C131 · Schur multiplier (H2) = C25 |
| direct product | #1789×C131 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C131 | C2×D4 |
| commutator | C2×C4 | C22×C4×C131 |
| Frattini, cosocle | C22×C4 | C23×C131 |
| socle | C22×C131 | C2×C22⋊C4 |
| prime radical | C22×#41×C131 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C131 ◁ C23×C4×C131 ◁ G | |
618 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 618 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 4192 |
A minimal faithful permutation representation — 3 generators on 143 points:
(1,2)(3,4)(5,6,8,11,9,7,12,10)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143);(5,7)(6,9)(8,10)(11,12);(2,4)4978 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C131×#8359, C131×#8361, C131×#8364, … |
| maximal normal subgroup of | C131×#8359, C131×#8361, C131×#8364, … |
| maximal quotient of | C131×#8310, C131×#8311, C131×#8312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16768, 521);