Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊(C3⋊C32) |
| Sylow | C3 ⋊ C4⋊C32 |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,3)(4,5,6,7);(2,3)(5,7)(8,9,10,14,11,15,18,24,12,16,19,25,21,27,30,34,13,17,20,26,22,28,31,35,23,29,32,36,33,37,38,39)Polycyclic presentation:
2^7,3|a^2=d,b^2=c,d^2=e,e^2=f,f^2=g,b^a=bc,h^a=h^2144 conjugacy classes — view the full character table.
| maximal subgroup of | D4⋊(C3⋊C32), #10682626, #10682628, … |
| maximal normal subgroup of | D4⋊(C3⋊C32), #10682626, #10682628, … |
| maximal quotient of | #10682853, #10682860, #10682874, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 85);