Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1451 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C22×D3 |
| commutator, socle | C22×C3 | C2×C42 |
| Fitting | C3×C4×M16 | C2 |
| prime radical | C23×C3 | C42 |
| cosocle | C3×C42 | C23 |
| hypercenter | C4×M16 | D3 |
| hypocenter | C3 | #1451 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C42 ◁ C4×M16 | |
208 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,8)(9,11);(2,3)(12,13,14,17,15,16,18,19);(4,5,6,9,7,8,10,11)(13,16)(17,19)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,d^2=f,e^2=g,b^a=bg,c^a=cf,h^b=h^296 conjugacy classes — view the full character table.
| maximal subgroup of | #10685754, #10685757, #10685759, … |
| maximal normal subgroup of | #10685754, #10685757, #10685759, … |
| maximal quotient of | #10684547, #10684568, #10685612, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2423);