Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #1738 |
372 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(10,19)(14,22)(18,23)(20,21);(4,5,6,7)(8,10,12,21,13,19,11,20)(9,14,16,23,17,22,15,18);(2,3)(8,9,11,16,13,17,12,15)(10,18,20,14,19,23,21,22)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,d^2=g,e^2=g,f^2=g,b^a=bd,d^a=dg,f^a=fg,h^a=h^2,c^b=cg84 conjugacy classes — view the full character table.
| maximal subgroup of | #10700525, #10700526, #10700571, … |
| maximal normal subgroup of | #10700525, #10700526, #10700571, … |
| maximal quotient of | #10684649, #10684659, #10684663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4574);