Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×#346 ⋊ C2 |
| Sylow | C3 ⋊ C4×M32 |
196 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 113 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) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(9,13)(14,19)(15,20)(21,23);(2,3)(4,5,6,7);(8,9,10,14,11,15,16,21,12,13,17,19,18,20,22,23)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,d^2=f,f^2=g,c^a=cg,h^b=h^2,h^c=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #10704186, #10704190, #10704194, … |
| maximal normal subgroup of | #10704186, #10704190, #10704194, … |
| maximal quotient of | #10704165, #10704179, #10704212, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1647);