Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C3×#391 ⋊ C2 |
| Sylow | C3 ⋊ #2326 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#391 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C22×D12 | C22 |
| hypercenter | #391 | D3 |
| hypocenter | C3 | #2326 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #391 | |
612 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 197 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 — 4 generators on 19 points:
(1,2,3)(12,14,15,18)(13,17,16,19);(2,3)(8,9,10,11)(13,16)(17,19);(2,3)(4,6,7,5)(12,13)(14,17)(15,16)(18,19);(2,3)(4,5,7,6)(9,11)(13,16)(14,18)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=ce,h^b=h^2,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10704478, #10704676, #10710376, … |
| maximal normal subgroup of | #10704478, #10704676, #10710376, … |
| maximal quotient of | #10716448, #10716494, #10716499, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9060);