Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1287 |
126 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 126 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) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3)(4,6,8,15,9,16,7,17)(5,10,13,18,11,14,12,19)(20,22)(21,25,27,32)(23,29)(24,30)(26,28,33,34)(31,35);(2,3)(4,5,7,12,9,11,8,13)(6,14,15,19,16,10,17,18)(20,21)(22,28,30,34)(23,26)(24,27)(25,29,32,35)(31,33)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,c^2=fg,d^2=f,e^2=f,b^a=bc,c^a=cf,e^a=efg,h^a=h^2,d^b=dg72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1287, C2×#67747, #10684997, … |
| maximal normal subgroup of | D3×#1287, C2×#67747, #10684997, … |
| maximal quotient of | #10681721, #10681727, #10681733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 323);