Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | SL(2,𝔽3) ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C8 |
| Frattini | C2×C4 | C2×S4 |
| Fitting | C4×#41 | D3 |
| socle | C22 | C2×SL(2,ℤ/4) |
| prime radical | C2×#41⋊C3 | C4 |
| cosocle | C4×SL(2,𝔽3) | C22 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C42 | |
127 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(9,23,18,16,19,24,13,14,20,10,11,22)(12,15,17,21);(1,2,3,5,4,6,7,8)(11,17)(12,19)(14,21)(15,23)(18,20)(22,24)Polycyclic presentation:
2^4,3,2^3|a^2=c,b^2=h,c^2=d,f^2=h,g^2=h,e^a=e^2,f^a=gh,g^a=fh,f^e=g,g^e=fgh,g^f=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #11710990, #11710991, #11710992, … |
| maximal normal subgroup of | #11710990, #11710991, #11710992, … |
| maximal quotient of | C4×SL(2,𝔽3)⋊C8, #11711014, #11711065, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5644);