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 | #3362⋊C3 |
| Fitting, Sylow | #3362 ⋊ C3 |
117 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 117 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:
(1,2,4,6)(3,8,7,5)(11,17,20)(12,19,18)(14,21,24)(15,23,22);(2,5)(3,7)(6,8)(9,22)(10,17)(11,24)(12,14)(13,21)(15,19)(16,18)(20,23)Polycyclic presentation:
2^2,3,2^5|a^2=h,b^2=e,d^2=e,f^2=h,g^2=h,b^a=bd,d^a=de,d^b=de,f^c=fgh,g^c=fh,g^f=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3362⋊C3, #61303⋊C3, #61319⋊C3, … |
| maximal normal subgroup of | C2×#3362⋊C3, #61303⋊C3, #61319⋊C3, … |
| maximal quotient of | SL(2,𝔽3)×#103, #22540⋊C3, #22738⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5841);