Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2986 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×#433 | C2 |
| prime radical | C23×C3 | C22×C4 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #433 | D3 |
| hypocenter | C3 | #2986 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #433 | |
330 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 161 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 — 4 generators on 35 points:
(1,2,3)(5,11)(9,12)(10,19)(17,18)(22,28)(25,31)(29,34)(32,35);(2,3)(4,7,8,16)(5,10,11,19)(6,13,14,15)(9,17,12,18)(22,28)(25,31)(29,34)(32,35);(4,6,8,14)(5,9,11,12)(7,15,16,13)(10,18,19,17)(20,22)(21,25)(23,29)(24,28)(26,32)(27,31)(30,34)(33,35);(4,5)(6,12)(7,10)(8,11)(9,14)(13,18)(15,17)(16,19)(20,21,23,26,24,27,30,33)(22,25,29,32,28,31,34,35)Polycyclic presentation:
2^7,3|a^2=f,b^2=e,c^2=e,f^2=g,b^a=be,d^a=de,c^b=ceg,d^b=dg,h^c=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2986, C2×#79360, #11660166, … |
| maximal normal subgroup of | D3×#2986, C2×#79360, #11660166, … |
| maximal quotient of | #10793944, #10793954, #10793955, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11936);