Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3390 |
320 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 129 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) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,7,8,5)(6,9,11,10)(12,15,17,16)(13,19,20,24)(14,23,22,25)(18,27,26,21);(12,14,17,22)(13,18,20,26)(15,23,16,25)(19,27,24,21);(2,3)(4,6,8,11)(5,9,7,10)(14,22)(18,26)(21,27)(23,25);(4,5,8,7)(6,10,11,9)(12,13)(14,21)(15,19)(16,24)(17,20)(18,23)(22,27)(25,26)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,c^2=g,d^2=eg,f^2=g,b^a=be,c^a=cf,f^a=fg,c^b=cg,d^b=de,h^b=h^2,f^c=fg57 conjugacy classes — view the full character table.
| maximal subgroup of | #10708925, #10708929, #10708939, … |
| maximal normal subgroup of | #10708925, #10708929, #10708939, … |
| maximal quotient of | #10811869, #10813513, #10813569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12285);