Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #68807 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2101 |
198 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,9)(8,11)(12,14,15,19)(13,17,16,18)(21,24)(22,26)(25,27);(2,3)(4,6,7,10)(5,8,9,11)(13,16)(14,19)(20,22,23,27)(21,25,24,26);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,15,16)(14,18,19,17)(20,21,23,24)(22,25,27,26)Polycyclic presentation:
2^7,3|a^2=d,b^2=de,c^2=dg,f^2=g,b^a=bdg,c^a=ceg,h^a=h^2,c^b=cf,f^b=fg,h^b=h^2,f^c=fg60 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2101, C2×#68844, #10825102, … |
| maximal normal subgroup of | D3×#2101, C2×#68844, #10825102, … |
| maximal quotient of | #10700326, #10703100, #10703449, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1420);