Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#428 ⋊ C2 |
| Sylow | C3 ⋊ #3097 |
386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 386 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)(14,21)(17,25)(22,27)(24,26);(4,6,7,10)(5,9,8,11)(13,18)(15,23)(17,24)(19,20)(22,27)(25,26);(2,3)(5,8)(9,11)(12,14)(13,17)(15,22)(16,21)(18,24)(19,26)(20,25)(23,27);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13,16,20)(14,17,21,25)(15,18,23,19)(22,24,27,26)Polycyclic presentation:
2^7,3|a^2=g,c^2=e,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 | #10855675, #10855678, #10855690, … |
| maximal normal subgroup of | #10855675, #10855678, #10855690, … |
| maximal quotient of | #10795009, #10795960, #10796444, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11796);