Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3231 |
326 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(5,11)(9,12)(10,19)(17,18)(20,23,25,24)(21,27,28,32)(22,30,31,33)(26,34,35,29);(2,3)(4,7,8,16)(5,10,11,19)(6,13,14,15)(9,17,12,18)(20,22)(21,26)(23,30)(24,33)(25,31)(27,34)(28,35)(29,32);(4,6,8,14)(5,9,11,12)(7,15,16,13)(10,18,19,17);(2,3)(4,5)(6,12)(7,10)(8,11)(9,14)(13,18)(15,17)(16,19)(20,21,25,28)(22,29,31,34)(23,27,24,32)(26,33,35,30)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,c^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,h^a=h^2,c^b=ce,f^c=fg,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10691780, #10691782, #10691822, … |
| maximal normal subgroup of | #10691780, #10691782, #10691822, … |
| maximal quotient of | #10804683, #10804899, #10805725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11849);