Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1306 |
192 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 57 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 — 2 generators on 27 points:
(2,3)(4,6)(5,8,9,10)(7,11)(12,14,15,22,16,21,23,27)(13,17,19,24,18,20,25,26);(1,2,3)(4,5)(6,10)(7,9)(8,11)(12,13)(14,20,21,17)(15,19)(16,18)(22,26,27,24)(23,25)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,e^2=f,b^a=bc,c^a=cf,e^a=eg,c^b=cg,d^b=df,h^b=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10682673, #10682674, #10682769, … |
| maximal normal subgroup of | #10682673, #10682674, #10682769, … |
| maximal quotient of | #10681758, #10681759, #10681769, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 205);