Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #857⋊C2 |
| commutator, prime radical, cosocle | C2×C42⋊C3 | C22 |
| Frattini, socle | C23 | C2×S4 |
| Fitting | #384 | D3 |
| hypocenter | C42⋊C3 | Q8 |
| LCS | G ▷ C2×C42⋊C3 ▷ C42⋊C3 | |
| UCS | C1 ◁ C2 | |
113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,3,4,8)(2,5,6,7)(9,10,17,12,15,19)(11,20,13)(14,18,16);(1,2,4,6)(3,7,8,5)(10,13)(14,19)(15,18)(17,20)Polycyclic presentation:
2^3,3,2^4|a^2=c,b^2=c,e^2=g,f^2=gh,b^a=bc,d^a=d^2,e^a=fh,f^a=eh,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 | #11711434, #11711439, C2×#72992, … |
| maximal normal subgroup of | #11711434, #11711439, C2×#72992, … |
| maximal quotient of | #11711281, #11711282, #11711283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5568);