Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #2237 |
132 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(13,19)(15,26)(16,27)(18,30)(20,21)(22,35)(23,37)(24,38)(28,36)(29,34)(31,32)(33,42)(39,43)(40,41);(5,8)(9,11)(12,14,17,25)(13,18,22,33)(15,23,28,39)(16,24,27,38)(19,30,35,42)(20,31,34,41)(21,32,29,40)(26,37,36,43);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13,17,22)(14,18,25,33)(15,19,28,35)(16,29,27,21)(20,36,34,26)(23,30,39,42)(24,40,38,32)(31,43,41,37)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,c^2=d,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,h^a=h^2,e^c=ef,f^c=fg57 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2237, C2×#70655, #10949985, … |
| maximal normal subgroup of | D3×#2237, C2×#70655, #10949985, … |
| maximal quotient of | #10711484, #10711697, #10712140, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3231);