Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #68001 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | C22⋊SL(2,𝔽3) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4625 |
| commutator, hypocenter | C22⋊SL(2,𝔽3) | C4 |
| Frattini | C24 | S4 |
| Fitting | #393 | D3 |
| socle | C23 | SL(2,ℤ/4) |
| prime radical, cosocle | #393⋊C3 | C2 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,3,12,4,5,11,14,10)(2,15,18,16)(6,19,23,8)(7,22,21,17)(9,20,13,24);(1,2,6)(3,8,22)(4,13,15)(5,9,17)(7,16,14)(10,18,20)(11,21,23)(12,19,24)Polycyclic presentation:
2^2,3,2^5|a^2=b,d^2=h,e^2=h,c^a=c^2,d^a=egh,e^a=dfh,f^a=fg,d^b=dfg,e^b=ef,d^c=e,e^c=deh,f^c=g,g^c=fg,e^d=eh21 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#68001, #11711363, #11711371, … |
| maximal normal subgroup of | C2×#68001, #11711363, #11711371, … |
| maximal quotient of | #11710951, #11710965, #11710966, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 577);