Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×#324 ⋊ C2 |
| Sylow | C3 ⋊ #1473 |
214 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2)(4,7,8,15)(5,10,11,17)(6,13,14,19)(9,16,12,18)(22,29)(25,28)(30,31)(33,35);(2,3)(4,6)(5,9)(7,13)(8,14)(10,16)(11,12)(15,19)(17,18)(20,22)(21,25)(23,31)(24,29)(26,28)(27,35)(30,32)(33,34);(4,5)(6,12,14,9)(7,10)(8,11)(13,18,19,16)(15,17)(20,21,23,27,24,26,32,34)(22,28,30,35,29,25,31,33)Polycyclic presentation:
2^7,3|a^2=e,c^2=fg,e^2=g,b^a=bd,d^a=df,c^b=cg,e^b=ef,h^b=h^2,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1473, C2×#68129, #10761167, … |
| maximal normal subgroup of | D3×#1473, C2×#68129, #10761167, … |
| maximal quotient of | #10683572, #10683585, #10683600, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 705);