Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2135 |
230 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 73 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) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,7,10,8)(5,12,14,21)(6,16,19,17)(9,20,24,22)(11,15,27,28)(13,29,31,30)(18,33,23,25)(26,32,35,34);(4,6,10,19)(5,11,14,27)(7,17,8,16)(9,23,24,18)(12,28,21,15)(13,26,31,35)(20,33,22,25)(29,34,30,32);(2,3)(4,5,9,13,10,14,24,31)(6,15,18,32,19,28,23,34)(7,12,20,29,8,21,22,30)(11,25,26,16,27,33,35,17)Polycyclic presentation:
2^7,3|a^2=e,b^2=g,c^2=g,d^2=g,e^2=g,b^a=bd,d^a=df,f^a=fg,h^a=h^2,c^b=cg,d^b=dg,e^b=efg,e^d=eg45 conjugacy classes — view the full character table.
| maximal subgroup of | #10710738, #10710739, #10710743, … |
| maximal normal subgroup of | #10710738, #10710739, #10710743, … |
| maximal quotient of | #10704303, #10704355, #10704481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4449);