Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×#328 ⋊ C2 |
| Sylow | C3 ⋊ #1512 |
222 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 61 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)(4,6,8,7)(5,9,10,11)(12,15,16,23)(13,18,19,25)(14,21,22,27)(17,24,20,26);(2,3)(4,5)(6,9)(7,11)(8,10)(12,14)(13,17)(15,21)(16,22)(18,24)(19,20)(23,27)(25,26);(5,9)(6,7)(10,11)(12,13)(14,20,22,17)(15,18)(16,19)(21,26,27,24)(23,25)Polycyclic presentation:
2^7,3|a^2=e,c^2=f,d^2=g,b^a=bd,c^a=cg,d^a=df,d^b=dg,e^b=efg,h^b=h^2,h^c=h^245 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1512, C2×#68312, #10762432, … |
| maximal normal subgroup of | D3×#1512, C2×#68312, #10762432, … |
| maximal quotient of | #10683271, #10683277, #10683301, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 888);