Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1311 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4×C8 | C2×D3 |
| commutator | C2×C3 | C82 |
| Fitting | C3×C82 | C2 |
| socle, prime radical | C22×C3 | C4×C8 |
| cosocle | C3×C4×C8 | C22 |
| hypercenter | C82 | D3 |
| hypocenter | C3 | #1311 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4×C8 ◁ C82 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3)(4,5,6,8,7,9,10,11)(13,17)(18,23)(19,24)(25,27);(2,3)(12,13,14,18,15,19,20,25,16,17,21,23,22,24,26,27)Polycyclic presentation:
2^7,3|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,b^a=bg,h^a=h^2144 conjugacy classes — view the full character table.
| maximal subgroup of | #10681804, #10681853, #10681854, … |
| maximal normal subgroup of | #10681804, #10681853, #10681854, … |
| maximal quotient of | #10681802, #10682015, #10682517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 13);