Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ C8×Q16 |
164 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 73 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) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(2,3)(4,6,7,11,8,5,9,10);(1,2,3)(12,14,16,22)(13,17,19,26)(15,24,25,21)(18,27,23,20);(4,5,7,10,8,6,9,11)(12,13,16,19)(14,20,22,27)(15,23,25,18)(17,24,26,21)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=ef,d^2=f,e^2=g,b^a=bd,d^a=df,d^b=df,h^c=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C8×Q16, C2×#68213, #10764561, … |
| maximal normal subgroup of | D3×C8×Q16, C2×#68213, #10764561, … |
| maximal quotient of | #10685957, #10685961, #10686415, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 789);