Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3083 |
376 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 141 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)(12,14,16,21)(13,18,20,25)(15,22,23,27)(17,24,19,26);(13,17)(15,23)(18,24)(19,20)(22,27)(25,26);(2,3)(4,6,7,11)(5,8,9,10)(14,21)(18,25)(22,27)(24,26);(2,3)(4,5,7,9)(6,10,11,8)(12,13)(14,18)(15,17)(16,20)(19,23)(21,25)(22,24)(26,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,d^b=dg,h^b=h^2,f^c=fg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10852723, #10852725, #10852744, … |
| maximal normal subgroup of | #10852723, #10852725, #10852744, … |
| maximal quotient of | #10794718, #10796195, #10796218, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11448);