Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×C8×D4 ⋊ C2 |
| Sylow | C3 ⋊ #1580 |
192 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 73 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)(4,6,7,11,8,5,9,10)(13,17)(16,18);(13,16)(14,19)(17,18);(2,3)(4,5,7,10,8,6,9,11)(12,13,15,17)(14,16,19,18)Polycyclic presentation:
2^7,3|a^2=e,c^2=ef,d^2=f,e^2=g,b^a=bd,c^a=cf,d^a=df,h^a=h^2,d^b=df84 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1580, C2×#70840, #10764588, … |
| maximal normal subgroup of | D3×#1580, C2×#70840, #10764588, … |
| maximal quotient of | #10685963, #10685967, #10686432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3416);