Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#405 ⋊ C2 |
| Sylow | C3 ⋊ #3174 |
362 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 362 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)(4,6,7,10)(5,9,8,11)(13,18)(17,24)(19,25)(23,27);(12,14,15,20,16,21,22,26)(13,17,19,23,18,24,25,27);(5,8)(9,11)(12,13,16,18)(14,17,21,24)(15,19,22,25)(20,23,26,27);(2,3)(4,5)(6,9)(7,8)(10,11)(14,20)(15,22)(17,23)(19,25)(21,26)(24,27)Polycyclic presentation:
2^7,3|b^2=g,c^2=f,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,d^b=dg72 conjugacy classes — view the full character table.
| maximal subgroup of | #10706367, #10706370, #10706712, … |
| maximal normal subgroup of | #10706367, #10706370, #10706712, … |
| maximal quotient of | #10795750, #10799450, #10799563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11143);