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 ⋊ #1536 |
238 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 238 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)(12,15,17,16)(13,19,20,24)(14,22,23,25)(18,26,27,21);(5,8)(6,10)(12,14)(13,18)(15,22)(16,25)(17,23)(19,26)(20,27)(21,24);(2,3)(4,5,6,9,7,8,10,11)(12,13)(14,21,23,26)(15,24)(16,19)(17,20)(18,22,27,25)Polycyclic presentation:
2^7,3|a^2=e,c^2=g,d^2=g,e^2=f,b^a=bd,c^a=cg,d^a=df,h^a=h^2,d^b=dg,e^b=efg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10682922, #10682923, #10682952, … |
| maximal normal subgroup of | #10682922, #10682923, #10682952, … |
| maximal quotient of | #10683285, #10683308, #10683315, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4375);