Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2123 |
234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,7,9,20)(5,11,13,27)(6,15,17,18)(8,19,22,35)(10,23,24,25)(12,26,21,34)(14,28,29,32)(16,30,31,33);(2,3)(4,6,9,17)(5,10,13,24)(7,18,20,15)(8,16,22,31)(11,25,27,23)(12,14,21,29)(19,33,35,30)(26,32,34,28);(2,3)(4,5)(6,14)(7,11)(8,21,22,12)(9,13)(10,16,24,31)(15,28)(17,29)(18,32)(19,34,35,26)(20,27)(23,30,25,33)Polycyclic presentation:
2^7,3|a^2=e,b^2=g,c^2=g,b^a=bd,d^a=df,f^a=fg,h^a=h^2,c^b=cg,e^b=ef,h^b=h^2,e^d=eg45 conjugacy classes — view the full character table.
| maximal subgroup of | #10710757, #10710760, #10710762, … |
| maximal normal subgroup of | #10710757, #10710760, #10710762, … |
| maximal quotient of | #10704282, #10704367, #10704450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2723);