Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #3218 |
386 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 115 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 — 4 generators on 27 points:
(1,2,3)(5,9)(8,10)(13,17)(18,26)(19,24)(20,27);(6,11)(8,10)(12,14,15,21)(13,18,17,26)(16,25,23,22)(19,27,24,20);(2,3)(4,6,7,11)(5,8,9,10)(13,17)(18,26)(19,24)(20,27);(4,5)(6,10)(7,9)(8,11)(12,13)(14,20)(15,17)(16,24)(18,22)(19,23)(21,27)(25,26)Polycyclic presentation:
2^7,3|b^2=eg,c^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ceg,h^b=h^2,f^c=fg45 conjugacy classes — view the full character table.
| maximal subgroup of | #10769463, #10769465, #10769479, … |
| maximal normal subgroup of | #10769463, #10769465, #10769479, … |
| maximal quotient of | #10804543, #10804919, #10805745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12611);