Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3166 |
348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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)(14,21)(18,25)(20,27)(22,26);(2,3)(12,14,16,22,15,21,23,26)(13,18,19,20,17,25,24,27);(4,6,7,11)(5,8,9,10)(13,17)(18,25)(19,24)(20,27);(2,3)(4,5,7,9)(6,10,11,8)(12,13,15,17)(14,20,21,27)(16,24,23,19)(18,26,25,22)Polycyclic presentation:
2^7,3|a^2=e,b^2=eg,c^2=f,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 | #10692808, #10696095, #10696097, … |
| maximal normal subgroup of | #10692808, #10696095, #10696097, … |
| maximal quotient of | #10795623, #10798578, #10799777, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15918);