Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Sylow | C3 ⋊ #1782 |
612 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 125 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)(5,7)(8,11,12,19)(9,14,15,21)(10,17,18,23)(13,20,16,22);(8,10)(9,13)(11,17)(12,18)(14,20)(15,16)(19,23)(21,22);(2,3)(4,5)(6,7)(8,9)(10,16,18,13)(11,14)(12,15)(17,22,23,20)(19,21)Polycyclic presentation:
2^7,3|a^2=f,c^2=g,b^a=bd,c^a=ce,d^a=dg,h^a=h^2,f^b=fg72 conjugacy classes — view the full character table.
| maximal subgroup of | #10682461, #10682462, #10682470, … |
| maximal normal subgroup of | #10682461, #10682462, #10682470, … |
| maximal quotient of | #10686521, #10686528, #10686531, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4715);