Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×C4×D8 ⋊ C2 |
| Sylow | C3 ⋊ #3168 |
380 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 121 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)(4,6,7,10)(5,9,8,11);(2,3)(12,15,16,23,17,24,14,21)(13,19,18,22,20,26,25,27);(2,3)(5,8)(9,11)(12,14,17,16)(13,18,20,25)(15,21,24,23)(19,22,26,27);(4,5)(6,9)(7,8)(10,11)(12,13)(14,18)(15,22)(16,25)(17,20)(19,23)(21,26)(24,27)Polycyclic presentation:
2^7,3|b^2=g,c^2=f,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,h^b=h^2,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10705168, #10705173, #10705186, … |
| maximal normal subgroup of | #10705168, #10705173, #10705186, … |
| maximal quotient of | #10795732, #10799399, #10799407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12159);