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 ⋊ #3567 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#564 | C2 |
| hypercenter | #564 | D3 |
| hypocenter | C3 | #3567 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #564 | |
950 conjugacy classes of subgroups; only the 449 normal subgroups are shown.
This group has 950 conjugacy classes of subgroups — too many to draw. Only its 449 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)(5,11)(9,12)(10,15)(18,19)(20,24,25,23)(21,28,27,32)(22,31,29,30)(26,33,34,35);(5,11)(6,14)(10,15)(13,16)(20,23,25,24)(21,28,27,32)(22,30,29,31)(26,33,34,35);(2,3)(4,7,8,17)(5,10,11,15)(6,13,14,16)(9,18,12,19)(21,27)(22,29)(28,32)(30,31);(2,3)(4,6)(5,9)(7,16)(8,14)(10,19)(11,12)(13,17)(15,18)(20,22)(21,26)(23,30)(24,31)(25,29)(27,34)(28,33)(32,35);(2,3)(4,5)(6,12)(7,15)(8,11)(9,14)(10,17)(13,18)(16,19)(20,21)(22,26)(23,32)(24,28)(25,27)(29,34)(30,35)(31,33)Polycyclic presentation:
2^7,3|c^2=f,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,h^a=h^2,c^b=cg,d^b=df,h^b=h^2,h^c=h^2,h^e=h^257 conjugacy classes — view the full character table.
| maximal subgroup of | #10776870, #10776878, #10776885, … |
| maximal normal subgroup of | #10776870, #10776878, #10776885, … |
| maximal quotient of | #10958527, #10959493, #10959548, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19717);