Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1378 |
132 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 132 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 — 2 generators on 35 points:
(1,2)(4,6)(5,10,12,14)(7,16)(8,17)(9,15)(11,22,24,27)(13,23,26,28)(18,30)(19,21)(20,29)(25,33,34,35)(31,32);(2,3)(4,5,7,11,8,13,18,25,9,12,19,24,20,26,31,34)(6,14,16,27,17,28,30,35,15,10,21,22,29,23,32,33)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,d^2=f,f^2=g,b^a=bc,e^a=eg,h^a=h^2,c^b=cg,h^b=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1378, C2×#67814, #10710211, … |
| maximal normal subgroup of | D3×#1378, C2×#67814, #10710211, … |
| maximal quotient of | #10682024, #10682034, #10682041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 390);