Finite Group DatabasebetaFinite Group Databasebeta| order | 2144 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C67 · Schur multiplier (H2) = C22 |
| direct product | C22⋊C8×C67 |
| unambiguous semidirect product | C22⋊(C8×C67) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C67 | C22 |
| commutator | C2 | C2×C8×C67 |
| Frattini, cosocle | C2×C4 | C22×C67 |
| socle | C22×C67 | C2×C4 |
| prime radical | C23×C67 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C67 ◁ G | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 79 |
| transitive degree | μt(G) = 1072 |
A minimal faithful permutation representation — 2 generators on 79 points:
(1,2)(3,4)(5,6,7,9,8,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(2,4)1340 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2144, 5);