Finite Group DatabasebetaFinite Group Databasebeta| order | 16768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C131 · Schur multiplier (H2) = C25 |
| direct product | #3059×C131 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C131 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C131 |
| prime radical | C2×#41×C131 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C2×#41×C131 ◁ G | |
460 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 200 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 8384 |
A minimal faithful permutation representation — 4 generators on 167 points:
(1,2)(3,4)(5,6,10,14)(7,15,19,29)(8,12,20,26)(9,22,16,13)(11,23,25,18)(17,28,31,36)(21,33,30,27)(24,34,35,32)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(5,7,16,23,10,19,9,18)(6,11,22,29,14,25,13,15)(8,17,30,34,20,31,21,32)(12,24,33,36,26,35,27,28);(2,4)(5,8)(6,12)(7,17)(9,21)(10,20)(11,24)(13,27)(14,26)(15,28)(16,30)(18,32)(19,31)(22,33)(23,34)(25,35)(29,36);(8,20)(12,26)(17,31)(21,30)(24,35)(27,33)(28,36)(32,34)4192 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C131×#11869, C131×#11871, C131×#13075, … |
| maximal normal subgroup of | C131×#11869, C131×#11871, C131×#13075, … |
| maximal quotient of | C131×#19188, C131×#19193, C131×#19199, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16768, 1791);