Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×#7547 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | D32 |
| commutator | C32 | C2×C4×C11 |
| Frattini, cosocle | C2×C32 | C22×C11 |
| socle | C2×C11 | #1423 |
| prime radical | C22×C11 | Q64 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C2×C4×C11 ◁ C2×C8×C11 ◁ C2×C11×C16 ◁ C2×C11×C32 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 1408 |
A minimal faithful permutation representation — 2 generators on 139 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,23,20,26,19,49)(14,27,30,64,34,67,33,85)(15,35,37,86,40,89,39,95)(17,43,41,96,46,98,45,99)(18,47,42,62,48,101,44,25)(21,50,51,76,54,75,102,29)(22,55,56,72,58,71,103,28)(24,60,59,91,61,90,97,36)(31,79,77,128,82,130,81,131)(32,83,78,118,84,132,80,66)(38,93,92,135,94,136,69,88)(52,108,107,138,109,139,114,120)(53,74,100,106,110,127,137,119)(57,70,87,111,112,124,134,123)(63,105,113,122,115,121,133,68)(65,104,116,126,117,125,129,73);(12,14,28,68,17,31,69,119,44,80,123,139,90,125,39,75,19,33,71,121,45,81,94,127,48,84,124,138,91,126,40,76,20,34,72,122,46,82,92,106,42,78,111,108,60,104,37,50,16,30,55,105,41,77,38,74,18,32,70,120,36,73,15,29)(13,21,35,65,24,52,87,83,47,100,93,128,96,113,56,64,23,51,86,116,59,107,112,118,62,110,135,130,98,115,58,67,26,54,89,117,61,109,134,132,101,137,136,131,99,133,103,85,49,102,95,129,97,114,57,66,25,53,88,79,43,63,22,27)770 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 535);