Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3338 |
356 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 139 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)(6,11)(9,10)(14,22)(17,21)(23,27)(25,26);(4,7,8,5)(6,10,11,9)(13,18)(15,24)(17,25)(19,20)(21,26)(23,27);(4,6)(5,9)(7,10)(8,11)(12,14)(13,17)(15,23)(16,22)(18,25)(19,26)(20,21)(24,27);(2,3)(4,5,8,7)(6,10,11,9)(12,13)(14,21)(15,18)(16,20)(17,22)(19,24)(23,26)(25,27)Polycyclic presentation:
2^7,3|a^2=e,c^2=e,f^2=g,b^a=beg,c^a=cf,f^a=fg,h^a=h^2,d^b=deg,f^c=fg60 conjugacy classes — view the full character table.
| maximal subgroup of | #10785412, #10785413, #10785419, … |
| maximal normal subgroup of | #10785412, #10785413, #10785419, … |
| maximal quotient of | #10811421, #10811427, #10812840, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16199);