Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3517⋊C3 |
| Fitting, Sylow | #3517 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Fitting | #3517 | C3 |
| hypercenter | #125 | A4 |
| hypocenter | Q8 | C3×#41 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ #125 | |
267 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 267 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,3,4,7)(2,6,5,8)(9,19,28,15,12,27,13,23,20,11,21,29)(10,22,31,18,16,30,17,26,24,14,25,32);(2,5)(6,8)(9,18)(10,11)(12,21)(13,14)(15,17)(19,23)(20,24)(27,30)(28,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,10,13,17)(11,18,15,14)(12,16,21,25)(19,26,23,22)(20,24,28,31)(27,32,29,30)Polycyclic presentation:
2^3,3,2^4|a^2=h,b^2=h,c^2=e,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 | #17655⋊C3, #61957⋊C3, #61958⋊C3, … |
| maximal normal subgroup of | #17655⋊C3, #61957⋊C3, #61958⋊C3, … |
| maximal quotient of | #34193⋊C3, #34263⋊C3, #34295⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18193);