Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #67645 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #1346 |
124 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 37 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)(5,8)(6,11)(9,10)(12,14,16,23)(13,17,20,25)(15,22,24,27)(18,21,19,26);(2,3)(4,5)(6,8)(7,10)(9,11)(12,13,15,19,16,20,24,18)(14,21,22,25,23,26,27,17)Polycyclic presentation:
2^7,3|a^2=d,b^2=g,c^2=f,d^2=g,b^a=bc,c^a=ce,e^a=eg,h^a=h^2,c^b=cf,d^b=def,d^c=dg45 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1346, C2×#67650, #10705388, … |
| maximal normal subgroup of | D3×#1346, C2×#67650, #10705388, … |
| maximal quotient of | #10681875, #10681881, #10681899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 226);