Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | GL(2,𝔽3)⋊C8 |
| Fitting | C8×Q8 ⋊ D3 |
| derived | SL(2,𝔽3) ⋊ C2×C8 |
112 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 24 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) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,19,21)(10,15,24,14,20,23)(11,18,13)(16,17);(9,10)(11,17)(12,20)(13,14)(15,22)(16,19)(18,24)(21,23)Polycyclic presentation:
2^4,3,2^3|b^2=c,c^2=d,f^2=h,g^2=h,b^a=bh,e^a=e^2,f^a=gh,g^a=fh,f^e=g,g^e=fgh,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×GL(2,𝔽3)⋊C8, #11711829, C8×#986, … |
| maximal normal subgroup of | C2×GL(2,𝔽3)⋊C8, #11711829, C8×#986, … |
| maximal quotient of | #11711000, #11711086, #11711107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5595);