Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #557 ⋊ D3 |
214 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 34 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 — 3 generators on 24 points:
(1,3,4,8)(2,6,5,7)(9,17,20)(10,15,24,14,21,23)(11,12,19)(16,22);(1,2,4,5)(3,7,8,6)(9,10,11,14)(12,15,17,21)(13,16,18,22)(19,23,20,24);(2,5)(6,7)(10,14)(12,18)(13,17)(15,16)(19,20)(21,22)Polycyclic presentation:
2^4,3,2^3|b^2=d,c^2=dh,f^2=h,g^2=h,b^a=bd,e^a=e^2,f^a=gh,g^a=fh,c^b=cd,f^e=g,g^e=fgh,g^f=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #11712338, #11712339, #11712342, … |
| maximal normal subgroup of | #11712338, #11712339, #11712342, … |
| maximal quotient of | #11712974, #11713043, #11713044, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17996);