Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3299 |
388 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 141 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)(6,11)(8,10);(12,14,15,21)(13,17,18,26)(16,25,23,22)(19,27,24,20);(4,6)(5,8)(7,11)(9,10);(2,3)(4,5,7,9)(6,10,11,8)(12,13,15,18)(14,20,21,27)(16,24,23,19)(17,25,26,22)Polycyclic presentation:
2^7,3|a^2=e,c^2=g,f^2=g,b^a=beg,c^a=cf,f^a=fg,h^a=h^2,d^b=deg,f^c=fg60 conjugacy classes — view the full character table.
| maximal subgroup of | #10784920, #10784928, #10784935, … |
| maximal normal subgroup of | #10784920, #10784928, #10784935, … |
| maximal quotient of | D4×#835, #10811421, #10812379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16127);