Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3504⋊C3 |
| Fitting, Sylow | #3504 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Fitting | #3504 | C3 |
| prime radical | C2×#41⋊C3 | C22 |
| hypercenter | #125 | A4 |
| hypocenter | Q8 | C2×C3×Q8 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ #125 | |
243 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(9,11,12,20)(10,16,15,19)(13,37,14,35,25,36,23,38,24,21,26,22)(17,39,18,33,31,34,27,40,28,29,32,30);(1,3,4,8)(2,5,6,7)(9,26,12,25)(10,31,15,32)(11,38,20,37)(13,24,23,14)(16,39,19,40)(17,18,27,28)(21,36,35,22)(29,30,33,34);(1,2,4,6)(3,7,8,5)(9,10)(11,19)(12,15)(13,17)(14,18)(16,20)(21,33)(22,34)(23,27)(24,28)(25,31)(26,32)(29,35)(30,36)(37,40)(38,39)Polycyclic presentation:
2^3,3,2^4|a^2=eh,b^2=eh,c^2=h,f^2=h,g^2=h,b^a=be,c^a=ch,f^d=fgh,g^d=fh,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #17692⋊C3, #60496⋊C3, #60499⋊C3, … |
| maximal normal subgroup of | #17692⋊C3, #60496⋊C3, #60499⋊C3, … |
| maximal quotient of | #34184⋊C3, #34250⋊C3, #34336⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18191);