Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #3137 |
372 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 125 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)(14,21)(18,26)(20,27)(22,25);(2,3)(5,9)(8,11)(12,15,16,19,17,23,24,13)(14,20,22,26,21,27,25,18);(4,6,7,10)(5,8,9,11)(12,14,16,22,17,21,24,25)(13,18,15,20,19,26,23,27);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,17,19)(14,20,21,27)(15,16,23,24)(18,25,26,22)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,c^2=e,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,h^a=h^2,d^b=dg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10865146, #10865150, #10865176, … |
| maximal normal subgroup of | #10865146, #10865150, #10865176, … |
| maximal quotient of | #10795391, #10797831, #10797847, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11453);