Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #1696 |
140 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2)(6,14)(9,18)(12,19)(15,17)(20,22,24,30)(21,25,27,34)(23,32,33,29)(26,35,31,28);(2,3)(4,6,13,17,8,14,7,15)(5,9,16,19,11,18,10,12);(4,5,8,11)(6,12,14,19)(7,16,13,10)(9,17,18,15)(20,21,24,27)(22,28,30,35)(23,31,33,26)(25,32,34,29)Polycyclic presentation:
2^7,3|a^2=fg,b^2=df,c^2=g,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,c^b=cf,h^b=h^2,e^c=eg,h^c=h^242 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1696, C2×#68513, #10777823, … |
| maximal normal subgroup of | D3×#1696, C2×#68513, #10777823, … |
| maximal quotient of | #10688285, #10691386, #10691390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1089);