Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1559 |
202 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 73 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) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,6,8,7)(5,9,10,11)(12,15,17,16)(13,19,20,21)(14,23,24,25)(18,26,22,27);(2,3)(4,5,8,10)(6,9,7,11)(12,14)(13,18)(15,23)(16,25)(17,24)(19,26)(20,22)(21,27);(2,3)(5,9)(6,7)(10,11)(12,13,16,21,17,20,15,19)(14,22,23,27,24,18,25,26)Polycyclic presentation:
2^7,3|a^2=e,b^2=g,c^2=f,d^2=g,e^2=fg,b^a=bd,c^a=cg,d^a=df,h^a=h^2,d^b=dg,e^b=efg,h^b=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10687490, #10687495, #10687497, … |
| maximal normal subgroup of | #10687490, #10687495, #10687497, … |
| maximal quotient of | #10683281, #10683295, #10683427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3401);