Finite Group DatabasebetaFinite Group Databasebeta| order | 16768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C131 · Schur multiplier (H2) = C4 |
| direct product | #2242×C131 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C131 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C131 |
| prime radical | C2×D8×C131 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C42×C131 ◁ C4×C8×C131 ◁ G | |
138 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 171 |
| transitive degree | μt(G) = 8384 |
A minimal faithful permutation representation — 3 generators on 171 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,23)(12,17)(13,31)(14,34)(15,35)(16,22)(18,27)(19,24)(20,33)(21,32)(25,37)(26,36)(28,40)(29,39)(30,38)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171);(2,5)(6,8)(9,11,24,29,15,26,33,30,16,27,14,25,32,28,13,12)(10,17,31,40,21,37,34,18,22,38,20,36,35,39,19,23);(1,3,4,7)(2,6,5,8)(9,12,13,28,32,25,14,27,16,30,33,26,15,29,24,11)(10,18,19,37,35,40,20,17,22,23,34,39,21,36,31,38)4978 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#2242×C131, C131×#33697, C131×#33702, … |
| maximal normal subgroup of | C2×#2242×C131, C131×#33697, C131×#33702, … |
| maximal quotient of | C131×#12889, C131×#13147, C131×#13148, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16768, 974);