Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #3499⋊C3 |
| Fitting, Sylow | #3499 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Fitting | #3499 | C3 |
| hypercenter | #127 | A4 |
| hypocenter | Q8 | C2×C3×D4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ #127 | |
381 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 381 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,9,10)(7,12,13,17,8,20,14,18,15,11,16,19);(1,2,3,4)(5,13,9,15)(6,19,10,20)(7,16,14,8)(11,18,17,12);(2,4)(6,10)(11,17)(12,18)(19,20)Polycyclic presentation:
2^3,3,2^4|b^2=e,c^2=h,f^2=h,g^2=h,b^a=be,c^a=ch,f^d=g,g^d=fgh,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #17721⋊C3, #17773⋊C3, #33360⋊C3, … |
| maximal normal subgroup of | #17721⋊C3, #17773⋊C3, #33360⋊C3, … |
| maximal quotient of | #34183⋊C3, #34259⋊C3, #34311⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18187);