Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #67526 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #1348 |
116 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 39 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)(4,6)(5,8)(7,11)(9,10)(12,13,16,19)(14,21,23,27)(15,24,22,18)(17,25,26,20);(2,3)(4,5)(6,10,11,8)(7,9)(13,17)(14,20)(15,22)(18,27)(19,26)(21,24)(23,25)Polycyclic presentation:
2^7,3|a^2=d,b^2=g,c^2=fg,e^2=g,f^2=g,b^a=bc,c^a=ce,e^a=eg,f^a=fg,h^a=h^2,c^b=cf,d^b=defg,e^b=eg,f^b=fg,h^b=h^269 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1348, C2×#67528, #10705412, … |
| maximal normal subgroup of | D3×#1348, C2×#67528, #10705412, … |
| maximal quotient of | #10681864, #10681867, #10681900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 104);