Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#336 ⋊ C2 |
| Sylow | C3 ⋊ C8×C22⋊C4 |
384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 384 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) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(2,3)(12,14,15,19,16,13,17,18);(1,2,3)(4,5,6,7)(9,11);(8,9)(10,11)(12,13,15,18,16,14,17,19)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=e,e^2=g,b^a=bd,h^c=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C8×C22⋊C4, C2×#68647, #10792774, … |
| maximal normal subgroup of | D3×C8×C22⋊C4, C2×#68647, #10792774, … |
| maximal quotient of | #10685651, #10685659, #10685674, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1223);