Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ C4×SD32 |
162 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 59 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)(4,6,7,5);(2,3)(8,9,12,15)(10,17,19,23)(11,20,18,14)(13,21,22,16);(2,3)(4,5,7,6)(9,13)(10,16)(11,18)(14,23)(15,22)(17,20)(19,21)Polycyclic presentation:
2^7,3|a^2=e,b^2=g,c^2=e,d^2=fg,f^2=g,b^a=bd,d^a=df,f^a=fg,h^a=h^2,d^b=df,f^b=fg,h^b=h^2,h^c=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D3×SD32, C2×#69175, C4×#4912, … |
| maximal normal subgroup of | C4×D3×SD32, C2×#69175, C4×#4912, … |
| maximal quotient of | #10706917, #10707086, #10707099, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1751);