Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1922 |
186 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(4,6,7,10)(5,9,8,11)(12,15)(13,17)(14,21)(16,19)(18,23)(20,24)(22,25)(26,27);(5,8)(9,11)(12,14,16,22,17,23,24,27)(13,18,20,26,15,21,19,25);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13,16,20,17,15,24,19)(14,18,22,26,23,21,27,25)Polycyclic presentation:
2^7,3|a^2=f,b^2=f,c^2=d,e^2=g,f^2=g,b^a=bd,c^a=ce,e^a=eg,h^a=h^2,e^c=eg,h^c=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1922, C2×#68645, #10797435, … |
| maximal normal subgroup of | D3×#1922, C2×#68645, #10797435, … |
| maximal quotient of | #10693305, #10693312, #10693319, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1221);