Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | Q8×#216 |
| Sylow | C3 ⋊ Q8×Q16 |
336 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 336 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) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(12,14,16,22)(13,17,19,26)(15,24,25,21)(18,27,23,20);(2,3)(12,13,16,19)(14,20,22,27)(15,23,25,18)(17,24,26,21);(4,6,7,11)(5,8,9,10);(4,5,7,9)(6,10,11,8)60 conjugacy classes — view the full character table.
| maximal subgroup of | #10785337, #10785340, #10785345, … |
| maximal normal subgroup of | #10785337, #10785340, #10785345, … |
| maximal quotient of | Q8×#810, Q8×#835, C4⋊C4×#216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16278);