Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #857 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | C42⋊C3 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #857 |
| commutator, hypocenter | C42⋊C3 | C8 |
| Frattini | C22×C4 | S4 |
| Fitting | C43 | D3 |
| socle | C23 | SL(2,ℤ/4) |
| prime radical | C2×C42⋊C3 | C4 |
| cosocle | C4×C42⋊C3 | C2 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C4 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,5,4,6,7,8)(9,11,12,16)(10,17,20,18,15,19,14,13);(9,10,13)(11,14,19)(12,15,18)(16,20,17)Polycyclic presentation:
2^3,3,2^4|a^2=b,b^2=c,e^2=g,f^2=gh,d^a=d^2,e^a=fh,f^a=eh,g^a=gh,e^d=fg,f^d=ef,g^d=gh,h^d=g40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#67995, #11710976, C8×#857, … |
| maximal normal subgroup of | C2×#67995, #11710976, C8×#857, … |
| maximal quotient of | #11710943, #11710952, #11710953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 571);