Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #3358⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #3392⋊C3 |
| Fitting, Sylow | #3392 ⋊ C3 |
114 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 114 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,5)(3,8)(6,7)(9,11,14,10)(12,24,22,15,19,18,21,23,13,17,20,16);(1,2,4,7)(3,5,8,6)(9,23)(10,20)(11,19)(12,16)(13,17)(14,24)(15,22)(18,21)Polycyclic presentation:
2^2,3,2^5|a^2=eh,b^2=h,d^2=e,f^2=h,g^2=h,b^a=bd,d^a=de,d^b=de,f^c=fgh,g^c=f,g^f=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3392⋊C3, #61393⋊C3, #61402⋊C3, … |
| maximal normal subgroup of | C2×#3392⋊C3, #61393⋊C3, #61402⋊C3, … |
| maximal quotient of | #22540⋊C3, #22542⋊C3, SL(2,𝔽3)×#106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5847);