Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #548⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊(#131⋊C3) |
| unambiguous semidirect product | #2906⋊C3 |
| Fitting, Sylow | #2906 ⋊ C3 |
134 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3,4)(8,18,21)(9,19,20)(12,25,28)(13,26,27)(15,29,32)(16,30,31)(22,33,36)(23,34,35);(2,4)(5,28,17,34,10,26,7,36)(6,32,24,21,14,30,11,19)(8,13,31,33,20,25,15,23)(9,27,29,22,18,12,16,35)Polycyclic presentation:
2^2,3,2^5|a^2=e,b^2=d,e^2=h,f^2=h,g^2=h,b^a=bd,f^c=gh,g^c=fg,g^f=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | #17507⋊C3, #17509⋊C3, #17720⋊C3, … |
| maximal normal subgroup of | #17507⋊C3, #17509⋊C3, #17720⋊C3, … |
| maximal quotient of | #17826⋊C3, #17828⋊C3, #17835⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5818);