Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3385 |
340 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 340 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)(4,7,8,5)(6,9,11,10);(13,18)(14,21)(16,23)(17,24)(19,20)(25,26);(2,3)(4,6,8,11)(5,9,7,10)(12,14)(13,17)(15,21)(16,22)(18,26)(19,25)(20,24)(23,27);(2,3)(4,5,8,7)(6,10,11,9)(12,13,15,19)(14,17,21,25)(16,18,23,20)(22,26,27,24)Polycyclic presentation:
2^7,3|a^2=e,b^2=eg,d^2=eg,f^2=g,b^a=beg,c^a=cf,f^a=fg,h^a=h^2,c^b=cg,d^b=deg,h^b=h^2,f^c=fg48 conjugacy classes — view the full character table.
| maximal subgroup of | #10784688, #10784690, #10784733, … |
| maximal normal subgroup of | #10784688, #10784690, #10784733, … |
| maximal quotient of | #10811816, #10811844, #10813478, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 13632);