Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | Aut(G) = GO+(4,𝔽3) (order 1152) · Inn(G) = #5934 · Out(G) = D3 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q82 ⋊ D3 |
| derived | Q8⋊SL(2,𝔽3) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #5934 |
| commutator, cosocle, hypocenter | Q8⋊SL(2,𝔽3) | C2 |
| Frattini, hypercenter | C22 | #1020 |
| Fitting | Q82 | D3 |
| LCS | G ▷ Q8⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C22 | |
109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,11,13,2,6,8,3,10)(4,16,12,14,15,9,5,7);(3,4,5)(7,8,9)(11,16,14)(12,13,15)Polycyclic presentation:
2,3,2^6|c^2=h,d^2=h,e^2=h,f^2=h,b^a=b^2,c^a=dh,d^a=ch,e^a=fh,f^a=eh,g^a=gh,c^b=d,d^b=cdh,e^b=ef,f^b=e,d^c=dh,e^c=eg,f^c=fh,e^d=eh,f^d=fgh,f^e=fh17 conjugacy classes — view the full character table.
| maximal subgroup of | #11711673, #11711674, C2×#85554, … |
| maximal normal subgroup of | #11711673, #11711674, C2×#85554, … |
| maximal quotient of | #11714251, #11714259, #11714277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18130);