Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×#336 ⋊ C2 |
| Sylow | C3 ⋊ #1929 |
224 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 224 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) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(4,6,7,5)(8,11)(9,13)(10,17)(12,15)(14,19)(16,20)(18,21)(22,23);(4,5,7,6)(8,10,12,18,13,19,20,23)(9,14,16,22,11,17,15,21);(2,3)(5,6)(8,9,12,16,13,11,20,15)(10,14,18,22,19,17,23,21)Polycyclic presentation:
2^7,3|a^2=f,b^2=df,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×#1929, C2×#68646, #10797690, … |
| maximal normal subgroup of | D3×#1929, C2×#68646, #10797690, … |
| maximal quotient of | #10693361, #10693368, #10693375, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1222);