Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3095 |
410 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 137 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)(12,15,17,16)(13,19,21,20)(14,23,24,25)(18,22,26,27);(2,3)(4,6,7,10)(5,9,8,11)(12,14)(13,18)(15,23)(16,25)(17,24)(19,22)(20,27)(21,26);(5,8)(9,11);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13,17,21)(14,22,24,27)(15,20,16,19)(18,25,26,23)Polycyclic presentation:
2^7,3|a^2=g,c^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=dg,f^a=fg,h^a=h^2,f^c=fg,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10692242, #10692244, #10692354, … |
| maximal normal subgroup of | #10692242, #10692244, #10692354, … |
| maximal quotient of | #10794999, #10795937, #10796427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11201);