Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×#332 ⋊ C2 |
| Sylow | C3 ⋊ #1553 |
242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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 — 3 generators on 27 points:
(1,2)(5,9)(8,10)(12,15,17,16)(13,19,20,24)(14,22,23,25)(18,26,21,27);(2,3)(5,8)(6,11)(9,10)(12,14)(13,18)(15,22)(16,25)(17,23)(19,26)(20,21)(24,27);(4,5)(6,8)(7,9)(10,11)(12,13,16,19,17,20,15,24)(14,21,22,26,23,18,25,27)Polycyclic presentation:
2^7,3|a^2=e,c^2=fg,d^2=g,e^2=fg,b^a=bd,c^a=cf,d^a=df,d^b=dg,e^b=efg,h^b=h^2,h^c=h^251 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1553, C2×#68426, #10763576, … |
| maximal normal subgroup of | D3×#1553, C2×#68426, #10763576, … |
| maximal quotient of | #10683208, #10683218, #10683326, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1002);