Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2171 |
144 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 79 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)(4,7,9,17,10,22,27,12,11,23,28,30,8,21,26,31)(5,13,15,32,14,25,33,35,16,6,18,20,24,19,34,29);(4,6)(5,12)(7,18)(8,25)(9,20)(10,19)(11,13)(14,30)(15,23)(16,31)(17,24)(21,33)(22,34)(26,35)(27,29)(28,32);(2,3)(4,5,9,15,10,14,27,33,11,16,28,18,8,24,26,34)(6,17,20,22,19,12,29,23,13,30,32,21,25,31,35,7)Polycyclic presentation:
2^7,3|a^2=e,c^2=e,d^2=g,e^2=f,f^2=g,b^a=bd,c^a=cg,d^a=dg,h^a=h^2,d^b=dg,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10707074, #10707077, #10709588, … |
| maximal normal subgroup of | #10707074, #10707077, #10709588, … |
| maximal quotient of | #10706898, #10706961, #10706968, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3500);