Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22×C4⋊C4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×S4 |
| commutator | C2×A4 | C22×C4 |
| Fitting | C22×C4⋊C4 | D3 |
| socle | C24 | C22×D3 |
| prime radical | C2×C4⋊S4 | C2 |
| cosocle | C22×A4 | C23 |
| hypercenter | C4⋊C4 | S4 |
| hypocenter | A4 | C4×D4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
427 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 427 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|b^2=e,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 | #11713057, #11713060, #11713068, … |
| maximal normal subgroup of | #11713057, #11713060, #11713068, … |
| maximal quotient of | C4⋊C4×SL(2,ℤ/4), #11711692, #11711719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17884);