Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1991 |
252 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 109 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:
(1,2,3)(8,10,11,15)(9,12,13,14)(17,22)(21,29)(23,28)(24,30);(4,6,7,5)(16,18,25,31,20,27,19,26)(17,21,28,30,22,29,23,24);(2,3)(4,5,7,6)(8,9,11,13)(10,14,15,12)(16,17)(18,24)(19,28)(20,22)(21,26)(23,25)(27,30)(29,31)Polycyclic presentation:
2^7,3|a^2=f,b^2=de,c^2=efg,d^2=g,b^a=bd,c^a=cef,d^a=dg,h^a=h^2108 conjugacy classes — view the full character table.
| maximal subgroup of | #10682012, #10687738, D3×#1991, … |
| maximal normal subgroup of | #10682012, #10687738, D3×#1991, … |
| maximal quotient of | #10695198, #10695214, #10695261, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2354);