Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1461 |
248 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 248 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) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,6,8,7)(5,9,11,10)(12,15,16,23)(13,18,19,25)(14,21,22,27)(17,24,20,26);(12,14,16,22)(13,17,19,20)(15,21,23,27)(18,24,25,26);(2,3)(4,5,7,10,8,11,6,9)(12,13)(14,20,22,17)(15,18)(16,19)(21,26,27,24)(23,25)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=fg,e^2=g,b^a=bd,d^a=df,h^a=h^2,e^b=ef84 conjugacy classes — view the full character table.
| maximal subgroup of | #10682191, #10682192, #10682406, … |
| maximal normal subgroup of | #10682191, #10682192, #10682406, … |
| maximal quotient of | #10683159, #10683165, #10683234, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4200);