Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4625 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | C22⋊A4 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4625 |
| commutator, hypocenter | C22⋊A4 | C8 |
| Frattini | C22×C4 | S4 |
| Fitting | #379 | D3 |
| socle | C23 | SL(2,ℤ/4) |
| prime radical | C2×C22⋊A4 | C4 |
| cosocle | #379⋊C3 | C2 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C2 | |
81 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,8,4,2)(3,15)(5,23,10,21,11,20,14,12)(6,7)(9,18)(13,16,22,19)(17,24);(1,5,13)(2,9,12)(3,10,18)(4,11,22)(6,14,24)(7,16,20)(8,17,21)(15,19,23)Polycyclic presentation:
2^3,3,2^4|a^2=b,b^2=c,d^a=d^2,e^a=fh,f^a=eg,g^a=gh,e^b=egh,f^b=fg,e^d=f,f^d=ef,g^d=h,h^d=gh22 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#67998, #11711340, #11711342, … |
| maximal normal subgroup of | C2×#67998, #11711340, #11711342, … |
| maximal quotient of | #11710944, #11710945, #11710951, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 574);