Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #1472 |
270 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 77 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)(12,15,17,16)(13,19,21,20)(14,22,23,24)(18,25,26,27);(2,3)(4,6,7,11)(5,8,9,10)(12,14)(13,18)(15,24)(16,22)(17,23)(19,27)(20,25)(21,26);(4,5)(6,10,11,8)(7,9)(12,13,16,20,17,21,15,19)(14,18,22,25,23,26,24,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=g,e^2=g,b^a=bd,d^a=df,c^b=cg,e^b=ef,h^b=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10687387, #10687393, #10687394, … |
| maximal normal subgroup of | #10687387, #10687393, #10687394, … |
| maximal quotient of | #10683567, #10683580, #10683594, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2499);