Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1307 |
130 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 130 conjugacy classes of subgroups — too many to draw. Only its 57 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)(4,6,7,9,8,10,11,5)(12,14)(13,17,19,20)(15,22)(16,21)(18,25,26,27)(23,24);(2,3)(4,5,7,10,8,9,11,6)(12,13,15,18,16,19,23,26)(14,20,22,27,21,17,24,25)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,d^2=g,e^2=f,b^a=bc,c^a=cf,e^a=eg,h^a=h^2,c^b=cg,d^b=df54 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1307, C2×#67758, #10685570, … |
| maximal normal subgroup of | D3×#1307, C2×#67758, #10685570, … |
| maximal quotient of | #10681759, #10681760, #10681762, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 334);