Finite Group DatabasebetaFinite Group Databasebeta| order | 2144 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C67 · Schur multiplier (H2) = C22 |
| direct product | C2×Q16×C67 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C67 | D4 |
| commutator, Frattini, cosocle | C4 | C23×C67 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C67 ◁ C2×C4×C67 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 2144 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(3,4,5,6,7,8,9,10)(11,18,17,16,15,14,13,12);(3,11,7,15)(4,12,8,16)(5,13,9,17)(6,14,10,18)938 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2144, 41);