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 | C2 | C2×#857 |
| commutator, prime radical, cosocle | C2×C42⋊C3 | C22 |
| Frattini, socle | C23 | C2×S4 |
| Fitting | C43 | D3 |
| hypercenter | C4 | #857 |
| hypocenter | C42⋊C3 | Q8 |
| LCS | G ▷ C2×C42⋊C3 ▷ C42⋊C3 | |
| UCS | C1 ◁ C2 ◁ C4 | |
120 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)(11,14,13)(12,15,19)(16,20,18);(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^d=fg,f^d=ef,g^d=gh,h^d=g28 conjugacy classes — view the full character table.
| maximal subgroup of | #11711184, #11711185, #11711230, … |
| maximal normal subgroup of | #11711184, #11711185, #11711230, … |
| maximal quotient of | #11711233, #11711234, #11711235, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5556);