Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#462 ⋊ C2 |
| Sylow | C3 ⋊ #3209 |
384 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 123 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)(5,9)(8,10);(6,11)(8,10)(12,14,16,22)(13,17,19,26)(15,24,25,21)(18,27,23,20);(2,3)(4,6,7,11)(5,8,9,10);(4,5)(6,10)(7,9)(8,11)(12,13,16,19)(14,20,22,27)(15,23,25,18)(17,24,26,21)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,c^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ce,h^b=h^2,f^c=fg,h^d=h^251 conjugacy classes — view the full character table.
| maximal subgroup of | #10772435, #10772437, #10772453, … |
| maximal normal subgroup of | #10772435, #10772437, #10772453, … |
| maximal quotient of | #10804497, #10804868, #10805708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 13056);