Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#564 ⋊ C2 |
| Sylow | C3 ⋊ #3556 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#564 | C2 |
| prime radical | C22×D3 | C24 |
| hypercenter | #564 | D3 |
| hypocenter | C3 | #3556 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #564 | |
890 conjugacy classes of subgroups; only the 455 normal subgroups are shown.
This group has 890 conjugacy classes of subgroups — too many to draw. Only its 455 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2)(20,24,25,23)(21,27,28,33)(22,30,31,32)(26,35,29,34);(5,11)(9,18)(10,14)(17,19)(20,23,25,24)(21,27,28,33)(22,30,31,32)(26,34,29,35);(2,3)(4,7,8,16)(5,10,11,14)(6,12,13,15)(9,17,18,19);(4,6,8,13)(5,9,11,18)(7,15,16,12)(10,19,14,17)(20,22,25,31)(21,26,28,29)(23,32,24,30)(27,35,33,34);(4,5)(6,9)(7,14)(8,11)(10,16)(12,19)(13,18)(15,17)(20,21,25,28)(22,29,31,26)(23,27,24,33)(30,35,32,34)Polycyclic presentation:
2^7,3|a^2=f,b^2=fg,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,h^c=h^2,h^e=h^257 conjugacy classes — view the full character table.
| maximal subgroup of | #10777085, #10777089, #10777118, … |
| maximal normal subgroup of | #10777085, #10777089, #10777118, … |
| maximal quotient of | #10958095, #10958829, #10960458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19716);