Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #2089 |
176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,7,8,6)(5,9,10,11)(13,19)(14,22)(15,16)(18,25)(20,21)(23,24);(4,6,8,7)(5,9,10,11)(12,14)(13,18)(15,23)(16,22)(17,24)(19,27)(20,26)(21,25);(2,3)(4,5)(6,11)(7,9)(8,10)(12,13,15,20,17,21,16,19)(14,18,23,26,24,25,22,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=d,c^2=d,e^2=g,f^2=g,b^a=bd,c^a=ce,h^a=h^2,c^b=cf,f^b=fg,e^c=eg,f^c=fg48 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2089, C2×#70670, #10824310, … |
| maximal normal subgroup of | D3×#2089, C2×#70670, #10824310, … |
| maximal quotient of | #10700096, #10700101, #10700106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3246);