Finite Group DatabasebetaFinite Group Databasebeta| order | 16768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C131 · Schur multiplier (H2) = C26 |
| direct product | C2×#456×C131 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C131 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C24×C131 |
| prime radical | C2×2−1+4×C131 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C22×C4×C131 ◁ G | |
708 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 708 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 149 |
| transitive degree | μt(G) = 4192 |
A minimal faithful permutation representation — 4 generators on 149 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149);(3,4,8,6)(5,11,14,18)(7,15,17,10)(9,16,12,13);(3,5)(4,9)(6,12)(7,16)(8,14)(10,18)(11,15)(13,17);(3,6,8,4)(5,12,14,9)(7,10,17,15)(11,16,18,13)4192 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C131×#13056, C131×#13058, C131×#13358, … |
| maximal normal subgroup of | C131×#13056, C131×#13058, C131×#13358, … |
| maximal quotient of | C2×#1603×C131, C2×#1609×C131, C2×#1604×C131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16768, 1749);