Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #3135 |
392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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)(12,16,18,17)(13,21,22,26)(14,24,25,15)(19,23,27,20);(2,3)(5,9)(8,11)(12,15,17,25,18,24,16,14)(13,20,21,19,22,23,26,27);(2,3)(4,6,7,10)(5,8,9,11)(12,14,17,15,18,25,16,24)(13,19,21,23,22,27,26,20);(4,5)(6,8)(7,9)(10,11)(12,13)(14,23)(15,19)(16,21)(17,26)(18,22)(20,25)(24,27)Polycyclic presentation:
2^7,3|b^2=f,c^2=e,d^2=g,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,c^b=cg,h^b=h^2,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10690134, #10690136, #10690148, … |
| maximal normal subgroup of | #10690134, #10690136, #10690148, … |
| maximal quotient of | #10795388, #10797829, #10798849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15858);