Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #3568⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | #3568⋊C3 |
| Fitting, Sylow | #3568 ⋊ C3 |
| derived | H(3,𝔽4) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | H(3,𝔽4) | C2×C3 |
| Frattini, socle | C22 | C2×C22⋊A4 |
| Fitting | #3568 | C3 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 | |
193 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 193 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,9,7,13,8,2)(3,24,15,17,18,30)(4,31,21,11,6,29)(5,25,22,27,32,26)(10,20,28,16,12,19)(14,23);(1,6,23,32)(2,12,9,24)(3,15,5,21)(4,19,16,22)(7,20,8,18)(10,26,11,28)(13,31,14,25)(17,30,27,29);(1,15,23,21)(2,29,9,30)(3,32,5,6)(4,18,16,20)(7,19,8,22)(10,25,11,31)(12,27,24,17)(13,26,14,28)Polycyclic presentation:
2,3,2^6|c^2=g,e^2=h,f^2=gh,b^a=bgh,c^a=cgh,d^a=dg,c^b=def,d^b=cdfgh,e^b=fg,f^b=efh,g^b=h,h^b=gh,d^c=dg,e^c=egh,f^c=fh,e^d=eh,f^d=fg18 conjugacy classes — view the full character table.
| maximal subgroup of | #8514⋊C3, #19094⋊C3, C2×#3568⋊C3, … |
| maximal normal subgroup of | #8514⋊C3, #19094⋊C3, C2×#3568⋊C3, … |
| maximal quotient of | #34529⋊C3, #37563⋊C3, #37603⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18239);