Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C4×Q8 ⋊ 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:
(1,3,4,7)(2,6,5,8)(9,19,21)(10,15,24,14,20,23)(11,18,13)(16,17);(2,5)(6,8);(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,20)(13,14)(15,22)(16,19)(18,24)(21,23)Polycyclic presentation:
2^4,3,2^3|c^2=d,f^2=h,g^2=h,b^a=bd,c^a=ch,e^a=e^2,f^a=gh,g^a=fh,f^e=g,g^e=fgh,g^f=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#85330, #11714806, #11714817, … |
| maximal normal subgroup of | C2×#85330, #11714806, #11714817, … |
| maximal quotient of | #11712663, #11712778, #11712779, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17906);