Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C3×#97 ⋊ C2 |
| Sylow | C3 ⋊ #1532 |
218 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 69 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)(4,6,8,7)(5,9,10,11)(12,15,17,16)(13,19,20,24)(14,22,23,25)(18,26,21,27);(2,3)(4,5)(6,9)(7,11)(8,10)(12,14)(13,18)(15,22)(16,25)(17,23)(19,26)(20,21)(24,27);(5,9,10,11)(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=f,b^a=bd,c^a=cf,d^a=df,d^b=dg,e^b=efg,h^b=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10686706, #10686707, #10686711, … |
| maximal normal subgroup of | #10686706, #10686707, #10686711, … |
| maximal quotient of | #10683207, #10683212, #10683237, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2375);