Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #3500⋊C3 |
| Fitting, Sylow | #3500 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Fitting | #3500 | C3 |
| hypercenter | #124 | A4 |
| hypocenter | Q8 | C2×C3×D4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ #124 | |
365 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 365 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,7,8,16)(6,12,11,15)(9,33,10,31,21,32,19,34,20,17,22,18)(13,35,14,29,27,30,23,36,24,25,28,26);(1,2,3,4)(5,22,8,21)(6,27,11,28)(7,34,16,33)(9,20,19,10)(12,35,15,36)(13,14,23,24)(17,32,31,18)(25,26,29,30);(2,4)(5,6,8,11)(7,15,16,12)(9,13,19,23)(10,14,20,24)(17,29,31,25)(18,30,32,26)(21,27,22,28)(33,36,34,35)Polycyclic presentation:
2^3,3,2^4|a^2=h,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 | #17688⋊C3, #33362⋊C3, #33371⋊C3, … |
| maximal normal subgroup of | #17688⋊C3, #33362⋊C3, #33371⋊C3, … |
| maximal quotient of | #34184⋊C3, #34258⋊C3, #34259⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18188);