Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #3567⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | #3567⋊C3 |
| Fitting, Sylow | #3567 ⋊ C3 |
| derived | #564 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #564 | C2×C3 |
| Frattini, socle | C22 | C2×C22⋊A4 |
| Fitting | #3567 | C3 |
| LCS | G ▷ #564 | |
| UCS | C1 | |
185 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 185 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) = 48 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,7)(3,12,35)(4,28,22,16,17,42)(5,19,24)(6,11,25)(8,23,36,26,27,13)(9,30,15)(10,33,21)(14,32,46)(18,41,48,20,40,43)(29,45,37,34,44,38)(31,47,39);(1,5,6,21)(2,11)(3,39,15,14)(4,20,16,18)(7,24,25,33)(8,26)(10,19)(13,37,36,38)(22,48,42,43)(23,44,27,45)(29,34)(30,47,35,46);(1,13,6,36)(2,28)(3,16,15,4)(5,38,21,37)(7,35,25,30)(8,12)(9,26)(10,41)(11,17)(14,18,39,20)(19,40)(22,23,42,27)(24,46,33,47)(29,32)(31,34)(43,45,48,44)Polycyclic presentation:
2,3,2^6|c^2=h,d^2=gh,e^2=h,f^2=gh,c^a=cgh,d^a=dg,c^b=dgh,d^b=cdgh,e^b=fgh,f^b=efgh,g^b=h,h^b=gh,e^c=egh,f^c=fh,e^d=eh,f^d=fg18 conjugacy classes — view the full character table.
| maximal subgroup of | #8515⋊C3, #19096⋊C3, C2×#3567⋊C3, … |
| maximal normal subgroup of | #8515⋊C3, #19096⋊C3, C2×#3567⋊C3, … |
| maximal quotient of | #34530⋊C3, #37564⋊C3, #37604⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18238);