Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3363 |
336 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 139 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 — 4 generators on 27 points:
(1,2,3)(4,6,7,11)(5,9,8,10);(2,3)(12,14,16,22)(13,17,19,26)(15,24,25,21)(18,27,23,20);(4,5,7,8)(6,10,11,9);(5,8)(9,10)(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,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=de,f^c=fg,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10785383, #10785392, #10785442, … |
| maximal normal subgroup of | #10785383, #10785392, #10785442, … |
| maximal quotient of | Q8×#810, #10811994, #10814136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16272);