Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #2090 |
176 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 71 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)(4,7,8,6)(5,9,10,11)(13,18)(14,20)(15,23)(17,24)(19,27)(21,22);(2,3)(4,6,8,7)(5,9,10,11)(12,14)(13,17)(15,20)(16,22)(18,25)(19,24)(21,23)(26,27);(4,5)(6,11)(7,9)(8,10)(12,13,15,18,16,19,23,27)(14,17,20,25,22,24,21,26)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=ceg,c^b=cf,f^b=fg,h^b=h^2,e^c=eg,f^c=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2090, C2×#68924, #10824309, … |
| maximal normal subgroup of | D3×#2090, C2×#68924, #10824309, … |
| maximal quotient of | #10700115, #10700122, #10700129, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1500);