Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2164 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C22×D3 |
| commutator, socle | C22×C3 | C22×C8 |
| Fitting | C3×#405 | C2 |
| prime radical | C23×C3 | C2×C8 |
| cosocle | C2×C3×C8 | C23 |
| hypercenter | #405 | D3 |
| hypocenter | C3 | #2164 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C8 ◁ #405 | |
138 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(14,22)(18,28)(23,34)(24,35)(29,38)(30,39)(36,42)(40,43);(5,8)(9,11)(12,14)(13,18)(15,23)(16,24)(17,22)(19,29)(20,30)(21,28)(25,36)(26,34)(27,35)(31,40)(32,38)(33,39)(37,42)(41,43);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13,15,19,16,20,25,31,17,21,26,32,27,33,37,41)(14,18,23,29,24,30,36,40,22,28,34,38,35,39,42,43)Polycyclic presentation:
2^7,3|a^2=e,c^2=d,e^2=f,f^2=g,b^a=bd,h^a=h^2,c^b=cg96 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2164, C2×#69883, #10948270, … |
| maximal normal subgroup of | D3×#2164, C2×#69883, #10948270, … |
| maximal quotient of | #10710192, #10710222, #10710500, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2459);