Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3181 |
366 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 125 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 — 4 generators on 27 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(14,21)(17,26)(20,27)(22,25);(2,3)(12,14,15,22,16,21,24,25)(13,17,18,20,19,26,23,27);(5,8)(9,11);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13,16,19)(14,20,21,27)(15,23,24,18)(17,25,26,22)Polycyclic presentation:
2^7,3|a^2=g,c^2=f,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,d^c=dg,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10692756, #10692758, #10776564, … |
| maximal normal subgroup of | #10692756, #10692758, #10776564, … |
| maximal quotient of | #10795779, #10799509, #10799520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15714);