Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1746 |
240 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 31 points:
(2,3)(8,10,11,14)(9,13,12,15)(18,24)(21,27)(25,30)(28,31);(1,2,3)(4,5,6,7)(9,12)(13,15)(16,18)(17,21)(19,25)(20,24)(22,28)(23,27)(26,30)(29,31);(8,9)(10,13)(11,12)(14,15)(16,17,19,22,20,23,26,29)(18,21,25,28,24,27,30,31)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=d,e^2=g,b^a=bd,c^b=cg,h^c=h^296 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1746, C2×#68598, #10792664, … |
| maximal normal subgroup of | D3×#1746, C2×#68598, #10792664, … |
| maximal quotient of | #10685611, #10685615, #10685618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1174);