Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#436 ⋊ C2 |
| Sylow | C3 ⋊ #3357 |
354 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 125 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 — 4 generators on 27 points:
(1,2,3)(6,11)(9,10)(13,19)(14,22)(18,20)(23,27);(2,3)(4,7,8,5)(6,10,11,9)(13,18)(14,22)(15,24)(17,25)(19,20)(21,26);(4,6)(5,9)(7,10)(8,11)(12,14,16,22)(13,17,19,21)(15,23,24,27)(18,26,20,25);(2,3)(4,5,8,7)(6,10,11,9)(12,13)(14,21)(15,18)(16,19)(17,22)(20,24)(23,25)(26,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=g,c^2=e,f^2=g,b^a=beg,c^a=cf,d^a=dg,f^a=fg,h^a=h^2,c^b=cg,d^b=deg,f^c=fg,h^c=h^257 conjugacy classes — view the full character table.
| maximal subgroup of | #10786129, #10786131, #10786174, … |
| maximal normal subgroup of | #10786129, #10786131, #10786174, … |
| maximal quotient of | #10811637, #10811648, #10812915, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12111);