Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3081 |
398 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(14,21)(17,25)(22,27)(24,26);(13,18)(15,23)(17,24)(19,20)(22,27)(25,26);(4,6,7,11)(5,8,9,10)(12,14)(13,17)(15,22)(16,21)(18,24)(19,26)(20,25)(23,27);(2,3)(4,5,7,9)(6,10,11,8)(12,13)(14,17)(15,18)(16,20)(19,23)(21,25)(22,24)(26,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,d^b=dg,f^c=fg66 conjugacy classes — view the full character table.
| maximal subgroup of | #10858612, #10858617, #10858642, … |
| maximal normal subgroup of | #10858612, #10858617, #10858642, … |
| maximal quotient of | #10794696, #10796173, #10796202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15115);