Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1658 |
182 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 45 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 — 3 generators on 35 points:
(1,2,3)(4,6,8,13)(5,10,11,19)(7,16,14,12)(9,17,15,18)(20,22)(21,25)(23,32)(24,29)(26,35)(27,34)(28,30)(31,33);(2,3)(4,5,8,11)(6,10,13,19)(7,15,14,9)(12,17,16,18)(20,21,24,27)(22,25,29,34)(23,31,30,26)(28,35,32,33);(5,9)(7,14)(10,18)(11,15)(12,16)(17,19)(22,28)(23,30)(25,33)(26,31)(29,32)(34,35)Polycyclic presentation:
2^7,3|b^2=g,c^2=f,d^2=f,e^2=fg,b^a=bd,c^a=ce,d^a=df,e^a=efg,d^b=df,e^b=eg,h^b=h^2,d^c=dg,e^c=efg36 conjugacy classes — view the full character table.
| maximal subgroup of | #10683029, #10683031, D3×#1658, … |
| maximal normal subgroup of | #10683029, #10683031, D3×#1658, … |
| maximal quotient of | #10683911, #10683912, #10683913, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3826);