Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #1704 |
140 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 49 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)(20,22,23,29)(21,26,25,34)(24,33,31,30)(27,35,32,28);(2,3)(4,6,13,17,8,14,7,15)(5,9,16,19,11,18,10,12)(21,25)(26,34)(27,32)(28,35);(2,3)(4,5,8,11)(6,12,14,19)(7,16,13,10)(9,17,18,15)(20,21)(22,28)(23,25)(24,32)(26,30)(27,31)(29,35)(33,34)Polycyclic presentation:
2^7,3|a^2=f,b^2=dfg,c^2=g,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2,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×#1704, C2×#68515, #10777863, … |
| maximal normal subgroup of | D3×#1704, C2×#68515, #10777863, … |
| maximal quotient of | #10688305, #10691386, #10691418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1091);