Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1556 |
228 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 228 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) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,10)(9,11);(2,3)(5,9)(6,7)(10,11)(12,14,16,22)(13,17,18,20)(15,23,24,21)(19,27,26,25);(4,5,7,9,8,10,6,11)(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,d^2=g,e^2=fg,b^a=bd,c^a=cg,d^a=df,d^b=dg,e^b=efg,h^b=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10687583, #10687589, #10687708, … |
| maximal normal subgroup of | #10687583, #10687589, #10687708, … |
| maximal quotient of | #10683266, #10683310, #10683412, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3398);