Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #557 ⋊ D3 |
209 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 36 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)(2,5,6,7)(9,21,18,14,19,24,13,15,20,10,11,23)(12,16,17,22);(1,2,4,6)(3,7,8,5);(10,14)(11,17)(12,19)(15,16)(18,20)(21,22)Polycyclic presentation:
2^4,3,2^3|b^2=d,c^2=dh,f^2=h,g^2=h,c^a=ch,e^a=e^2,f^a=gh,g^a=fh,c^b=cd,f^e=g,g^e=fgh,g^f=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #11712276, #11712278, #11712314, … |
| maximal normal subgroup of | #11712276, #11712278, #11712314, … |
| maximal quotient of | #11712975, #11713011, #11713015, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18000);