Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4451 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #1302 |
144 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 192 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3)(5,8,10,9)(12,14,16,24,17,22,15,23)(13,18,19,26,20,21,25,27);(2,3)(4,5)(6,8)(7,10)(9,11)(12,13,15,19,17,20,16,25)(14,21,23,27,22,18,24,26)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,c^2=f,d^2=g,e^2=g,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×#1302, C2×#67662, #10685445, … |
| maximal normal subgroup of | D3×#1302, C2×#67662, #10685445, … |
| maximal quotient of | #10681743, #10681750, #10681752, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 238);