Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3364 |
336 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 117 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)(4,6,7,11)(5,9,8,10);(12,15,17,23)(13,19,20,27)(14,21,16,25)(18,26,24,22);(2,3)(4,5,7,8)(6,10,11,9)(12,14,17,16)(13,18,20,24)(15,21,23,25)(19,26,27,22);(5,8)(9,10)(12,13)(14,18)(15,22)(16,24)(17,20)(19,21)(23,26)(25,27)Polycyclic presentation:
2^7,3|b^2=eg,c^2=g,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=de,h^b=h^2,f^c=fg63 conjugacy classes — view the full character table.
| maximal subgroup of | #10785115, #10785124, #10785220, … |
| maximal normal subgroup of | #10785115, #10785124, #10785220, … |
| maximal quotient of | #10811799, #10811981, #10814130, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12199);