Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×Q8⋊C8 ⋊ C2 |
| Sylow | C3 ⋊ #1630 |
218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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:
(1,2,3)(4,5)(6,11)(7,9)(8,10)(12,15,17,16)(13,19,20,24)(14,22,23,25)(18,26,27,21);(2,3)(6,10)(8,11)(12,14)(13,18)(15,22)(16,25)(17,23)(19,26)(20,27)(21,24);(2,3)(5,8)(6,10)(9,11)(12,13,17,20)(14,21,23,26)(15,19,16,24)(18,25,27,22)Polycyclic presentation:
2^7,3|a^2=f,c^2=f,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2,d^b=df,e^b=eg,h^b=h^2,d^c=dg,e^c=eg54 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1630, C2×#68260, #10770832, … |
| maximal normal subgroup of | D3×#1630, C2×#68260, #10770832, … |
| maximal quotient of | #10683981, #10684005, #10684164, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 836);