Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4451 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1298 |
190 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3)(4,6)(5,8,9,10)(7,11)(12,14,16,24,17,25,15,23)(13,18,20,26,21,27,19,22);(2,3)(4,5)(6,10)(7,9)(8,11)(12,13,15,20,17,21,16,19)(14,22,23,18,25,26,24,27)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,c^2=f,d^2=f,e^2=f,b^a=bc,c^a=cf,e^a=efg,h^a=h^2,c^b=cg78 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1298, C2×#67680, #10685328, … |
| maximal normal subgroup of | D3×#1298, C2×#67680, #10685328, … |
| maximal quotient of | #10681741, #10681749, #10681750, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 256);