Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2082 |
184 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 184 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) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2)(4,7,8,23,10,24,9,25)(5,12,13,21,15,29,14,30)(6,17,18,33,20,22,19,34)(11,26,16,31,28,32,27,35);(2,3)(4,6,10,20)(5,11,15,28)(7,22,24,17)(8,19,9,18)(12,26,29,32)(13,27,14,16)(21,35,30,31)(23,33,25,34);(2,3)(4,5,9,14,10,15,8,13)(6,16,19,11,20,27,18,28)(7,21,23,29,24,30,25,12)(17,32,33,35,22,26,34,31)Polycyclic presentation:
2^7,3|a^2=e,b^2=g,c^2=d,d^2=g,e^2=g,b^a=bd,c^a=ce,f^a=fg,h^a=h^2,c^b=cf,d^b=dg,e^b=eg,h^b=h^2,e^c=eg,f^c=fg,h^c=h^245 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2082, C2×#69023, #10824136, … |
| maximal normal subgroup of | D3×#2082, C2×#69023, #10824136, … |
| maximal quotient of | #10699984, #10699990, #10700001, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1599);