Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3164 |
340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 340 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) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(13,18)(17,25)(19,24)(23,27);(12,14,21,26,16,22,15,20)(13,17,24,27,18,25,19,23);(2,3)(4,6,7,11)(5,8,9,10)(12,13)(14,17)(15,19)(16,18)(20,23)(21,24)(22,25)(26,27);(2,3)(4,5,7,9)(6,10,11,8)(14,20)(15,21)(17,23)(19,24)(22,26)(25,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,c^2=fg,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,d^b=dg,h^b=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10714279, #10714280, #10714286, … |
| maximal normal subgroup of | #10714279, #10714280, #10714286, … |
| maximal quotient of | #10795639, #10798555, #10798566, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11749);