Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #1820 |
300 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(12,15)(13,18)(14,20)(16,23)(17,24)(19,22)(21,27)(25,26);(5,8)(9,11)(12,14,16,21)(13,17,19,25)(15,20,23,27)(18,24,22,26);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,17)(15,22,23,18)(16,19)(20,26,27,24)(21,25)Polycyclic presentation:
2^7,3|a^2=f,b^2=g,c^2=d,b^a=bd,c^a=ce,e^a=eg,h^a=h^2,f^c=fg72 conjugacy classes — view the full character table.
| maximal subgroup of | #10682215, #10682216, #10687365, … |
| maximal normal subgroup of | #10682215, #10682216, #10687365, … |
| maximal quotient of | #10688513, #10688519, #10688525, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4170);