Finite Group DatabasebetaFinite Group Databasebeta| order | 16768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C131 · Schur multiplier (H2) = C26 |
| direct product | #2889×C131 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C131 | C22×C22⋊C4 |
| commutator | C22 | C23×C4×C131 |
| Frattini, cosocle | C2×C4 | C24×C131 |
| prime radical | C2×2−1+4×C131 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C131 ◁ C2×#41×C131 ◁ G | |
704 conjugacy classes of subgroups; only the 340 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 340 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 4192 |
A minimal faithful permutation representation — 4 generators on 163 points:
(1,2,5,10,6,9,18,23)(3,11,13,24,14,7,17,20)(4,8,16,22,15,21,28,31)(12,25,27,32,26,19,29,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(1,3,6,14)(2,7,9,11)(4,12,15,26)(5,17,18,13)(8,19,21,25)(10,24,23,20)(16,29,28,27)(22,32,31,30);(1,4)(2,8)(3,12)(5,16)(6,15)(7,19)(9,21)(10,22)(11,25)(13,27)(14,26)(17,29)(18,28)(20,30)(23,31)(24,32);(4,15)(8,21)(12,26)(16,28)(19,25)(22,31)(27,29)(30,32)5371 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C131×#13096, C131×#13098, C131×#21780, … |
| maximal normal subgroup of | C131×#13096, C131×#13098, C131×#21780, … |
| maximal quotient of | C131×#17351, C131×#17352, C131×#17368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16768, 1621);