Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2070 |
196 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 65 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)(4,7,9,21)(5,11,13,26)(6,16,18,19)(8,22,15,20)(10,23,24,35)(12,27,28,25)(14,29,30,34)(17,33,31,32);(4,6,15,31,9,18,8,17)(5,10,12,14,13,24,28,30)(7,19,22,32,21,16,20,33)(11,23,25,34,26,35,27,29);(2,3)(4,5,8,12,9,13,15,28)(6,14,17,24,18,30,31,10)(7,11,20,25,21,26,22,27)(16,29,32,23,19,34,33,35)Polycyclic presentation:
2^7,3|a^2=d,b^2=de,c^2=g,d^2=g,b^a=bde,c^a=ce,e^a=eg,f^a=fg,h^a=h^2,c^b=cf,f^b=fg,d^c=dg45 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2070, C2×#70111, #10823912, … |
| maximal normal subgroup of | D3×#2070, C2×#70111, #10823912, … |
| maximal quotient of | #10699819, #10699821, #10699823, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2687);