Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3252 |
348 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 121 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)(4,6,7,10)(5,9,8,11);(5,8)(9,11)(13,18)(14,21)(15,23)(17,24)(19,20)(25,26);(4,5)(6,9)(7,8)(10,11)(12,14)(13,17)(15,22)(16,21)(18,26)(19,24)(20,25)(23,27);(2,3)(5,8)(6,10)(12,13,16,20)(14,17,21,25)(15,18,23,19)(22,26,27,24)Polycyclic presentation:
2^7,3|a^2=g,d^2=e,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,h^a=h^2,c^b=ceg,f^c=fg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10772002, #10772008, #10772032, … |
| maximal normal subgroup of | #10772002, #10772008, #10772032, … |
| maximal quotient of | #10805115, #10805125, #10805583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15732);