Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #67774 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #1349 |
114 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 114 conjugacy classes of subgroups — too many to draw. Only its 37 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 — 2 generators on 27 points:
(1,2,3)(4,6)(5,8)(7,11)(9,10)(13,18)(14,24)(15,16)(19,20)(21,27)(22,23)(25,26);(2,3)(4,5)(6,10,11,8)(7,9)(12,13,15,20,17,22,16,21)(14,23,25,19,26,18,24,27)Polycyclic presentation:
2^7,3|a^2=d,c^2=fg,d^2=g,e^2=g,f^2=g,b^a=bc,c^a=ce,e^a=eg,f^a=fg,h^a=h^2,c^b=cf,d^b=defg,e^b=eg,f^b=fg51 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1349, C2×#67775, #10705438, … |
| maximal normal subgroup of | D3×#1349, C2×#67775, #10705438, … |
| maximal quotient of | #10681872, #10681881, #10681884, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 351);