Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ C4×Q32 |
150 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 39 points:
(2,3)(4,6,7,5);(1,2,3)(8,10,12,20)(9,14,16,30)(11,23,25,39)(13,28,24,21)(15,33,34,38)(17,35,27,31)(18,36,37,22)(19,32,29,26);(4,5,7,6)(8,9,12,16)(10,18,20,37)(11,22,25,36)(13,27,24,17)(14,29,30,19)(15,32,34,26)(21,38,28,33)(23,35,39,31)Polycyclic presentation:
2^7,3|a^2=e,b^2=g,c^2=eg,d^2=fg,f^2=g,b^a=bd,d^a=df,f^a=fg,d^b=df,f^b=fg,h^c=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D3×Q32, C2×#69207, #10948465, … |
| maximal normal subgroup of | C4×D3×Q32, C2×#69207, #10948465, … |
| maximal quotient of | #10706918, #10707100, #10707106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1783);