Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S4 |
| commutator, prime radical, cosocle | C2×SL(2,𝔽3) | C23 |
| Fitting | Q82 | D3 |
| hypercenter | C2×Q8 | S4 |
| hypocenter | SL(2,𝔽3) | #41 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,4,5,2)(3,6,7,8)(9,19,21)(10,20,24)(11,18,13)(14,23,16);(1,3,5,7)(2,6,4,8);(1,2,5,4)(3,6,7,8)(9,10,13,16)(11,17,19,15)(12,20,22,14)(18,24,21,23)Polycyclic presentation:
2^4,3,2^3|a^2=dh,b^2=d,c^2=d,f^2=h,g^2=h,e^a=e^2,f^a=gh,g^a=fh,c^b=cd,f^e=g,g^e=fgh,g^f=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #11712313, #11712321, #11712333, … |
| maximal normal subgroup of | #11712313, #11712321, #11712333, … |
| maximal quotient of | Q8×#859, #11713043, #11713116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18004);