Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1531 |
212 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 81 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)(13,18)(17,20)(19,26)(25,27);(4,5)(6,9)(7,11)(8,10)(12,14,16,22)(13,17,18,20)(15,23,24,21)(19,27,26,25);(2,3)(5,9,10,11)(12,13,15,19,16,18,24,26)(14,20,21,27,22,17,23,25)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=g,d^2=g,e^2=f,b^a=bd,c^a=cf,d^a=df,h^a=h^2,d^b=dg,e^b=efg60 conjugacy classes — view the full character table.
| maximal subgroup of | #10687076, #10687079, #10687083, … |
| maximal normal subgroup of | #10687076, #10687079, #10687083, … |
| maximal quotient of | #10683204, #10683247, #10683260, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4398);