Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×C4×M16 ⋊ C2 |
| Sylow | C3 ⋊ #3183 |
370 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 370 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)(15,24)(19,27)(21,23)(22,26);(12,15,16,23,17,24,14,21)(13,19,18,22,20,27,25,26);(5,8)(9,11)(12,14,17,16)(13,18,20,25)(15,23,24,21)(19,26,27,22);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,18)(15,22)(16,25)(17,20)(19,23)(21,27)(24,26)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,c^b=cg,d^c=dg72 conjugacy classes — view the full character table.
| maximal subgroup of | #10706026, #10706030, #10950435, … |
| maximal normal subgroup of | #10706026, #10706030, #10950435, … |
| maximal quotient of | #10795792, #10799504, #10799515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11110);