Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3162 |
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)(12,15,17,16)(13,19,20,25)(14,23,22,24)(18,21,26,27);(2,3)(12,14,16,24,17,22,15,23)(13,18,19,21,20,26,25,27);(4,6,7,11)(5,8,9,10)(14,22)(18,26)(21,27)(23,24);(4,5,7,9)(6,10,11,8)(12,13)(14,21)(15,19)(16,25)(17,20)(18,24)(22,27)(23,26)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,c^2=f,d^2=g,f^2=g,b^a=be,c^a=cf,f^a=fg,c^b=cg,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10690138, #10690139, #10690141, … |
| maximal normal subgroup of | #10690138, #10690139, #10690141, … |
| maximal quotient of | #10795609, #10798536, #10799763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15916);