Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4626 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| derived | C42⋊C3 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4626 |
| commutator, hypocenter | C42⋊C3 | C8 |
| Frattini | C22×C4 | S4 |
| Fitting | #384 | D3 |
| socle | C23 | SL(2,ℤ/4) |
| prime radical | C2×C42⋊C3 | C4 |
| cosocle | #384⋊C3 | C2 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C2 | |
51 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,14,24,18)(10,21,20,19)(11,12,17,23)(13,22,16,15);(10,15,23)(11,16,17)(12,19,13)(14,20,21)(18,24,22)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=f,f^a=eg,g^a=gh,e^b=eg,f^b=fgh,e^d=fg,f^d=ef,g^d=gh,h^d=g22 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#67999, #11711341, #11711343, … |
| maximal normal subgroup of | C2×#67999, #11711341, #11711343, … |
| maximal quotient of | #11710946, #11710947, #11710950, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 575);