Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | GL(2,ℤ/4)⋊C4 |
| Fitting | C22×C22⋊C4 ⋊ D3 |
457 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 457 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,3,4)(6,8)(9,10,12);(5,6)(7,8);(6,8)(10,11)Polycyclic presentation:
2^5,3,2^2|c^2=d,b^a=be,f^a=f^2,g^a=h,h^a=g,c^b=ce,g^f=h,h^f=gh50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×GL(2,ℤ/4)⋊C4, #11714306, C4×D4×S4, … |
| maximal normal subgroup of | C2×GL(2,ℤ/4)⋊C4, #11714306, C4×D4×S4, … |
| maximal quotient of | C22⋊C4×SL(2,ℤ/4), #11711692, #11711693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17872);