Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C2×C3×AGL(1,ℤ/8) ⋊ C2 |
| Sylow | C3 ⋊ #3153 |
572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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 — 4 generators on 23 points:
(1,2,3)(10,17)(13,21)(16,22)(18,23);(2,3)(4,5,6,7)(9,14)(13,21)(15,19)(16,22);(8,10,11,18,12,17,20,23)(9,13,15,16,14,21,19,22);(5,7)(8,9)(10,16)(11,19)(12,14)(13,18)(15,20)(17,22)(21,23)Polycyclic presentation:
2^7,3|b^2=e,c^2=fg,e^2=g,b^a=be,c^a=cf,e^a=eg,d^b=dg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10821767, #10821772, #10846213, … |
| maximal normal subgroup of | #10821767, #10821772, #10846213, … |
| maximal quotient of | #10783046, #10783050, #10783064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16514);