Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3402 |
356 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 113 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)(4,7,8,6)(5,10,9,11)(12,15,17,16)(13,19,20,24)(14,23,22,25)(18,27,26,21);(2,3)(4,6,8,7)(5,10,9,11)(12,14)(13,18)(15,23)(16,25)(17,22)(19,27)(20,26)(21,24);(2,3)(4,5,8,9)(6,10,7,11)(14,22)(18,26)(21,27)(23,25);(2,3)(5,9)(10,11)(12,13,17,20)(14,21,22,27)(15,19,16,24)(18,25,26,23)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,c^2=e,d^2=eg,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,c^b=cg,d^b=de,h^b=h^2,f^c=fg,h^c=h^251 conjugacy classes — view the full character table.
| maximal subgroup of | #10708755, #10708759, #10708769, … |
| maximal normal subgroup of | #10708755, #10708759, #10708769, … |
| maximal quotient of | #10812064, #10814075, #10815005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 14393);