Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #67526 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C3×C4⋊C16 ⋊ C2 |
| Sylow | C3 ⋊ #1360 |
102 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 102 conjugacy classes of subgroups — too many to draw. Only its 39 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 — 2 generators on 27 points:
(1,2)(5,8)(6,10)(12,13)(14,23)(15,21)(16,20)(17,19)(18,24)(22,26)(25,27);(2,3)(4,5,6,9,7,8,10,11)(13,18,19,26)(14,22)(15,16)(20,23,21,27)(24,25)Polycyclic presentation:
2^7,3|a^2=d,c^2=fg,d^2=ef,e^2=g,f^2=g,b^a=bc,c^a=ce,e^a=eg,f^a=fg,h^a=h^2,c^b=cf,d^b=defg,e^b=eg,f^b=fg,h^b=h^269 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1360, C2×#67562, #10706409, … |
| maximal normal subgroup of | D3×#1360, C2×#67562, #10706409, … |
| maximal quotient of | #10681907, #10681909, #10681911, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 138);