Finite Group DatabasebetaFinite Group Databasebeta| order | 16768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C131 · Schur multiplier (H2) = C25 |
| direct product | C2×#470×C131 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C131 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C131 |
| prime radical | C24×C131 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C131 ◁ C23×C4×C131 ◁ G | |
492 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 492 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) = 153 |
| transitive degree | μt(G) = 8384 |
A minimal faithful permutation representation — 4 generators on 153 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153);(3,4)(5,6)(7,8,11,14)(9,15,17,22)(10,19,16,13)(12,20,21,18);(7,9,16,20,11,17,10,18)(8,12,19,22,14,21,13,15);(4,6)(9,17)(12,21)(15,22)(18,20)4192 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#1340×C131, C2×#1342×C131, C131×#13000, … |
| maximal normal subgroup of | C2×#1340×C131, C2×#1342×C131, C131×#13000, … |
| maximal quotient of | C2×#1689×C131, C2×#1695×C131, C2×#1703×C131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16768, 1785);