Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #1291 |
138 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 57 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 — 2 generators on 27 points:
(1,2)(5,8,10,9)(12,14,16,23,17,24,15,25)(13,18,21,26,19,22,20,27);(2,3)(4,5)(6,8)(7,10)(9,11)(12,13,15,20,17,19,16,21)(14,22,23,27,24,18,25,26)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,c^2=fg,d^2=g,e^2=g,b^a=bc,c^a=cf,e^a=efg,h^a=h^2,d^b=dg,h^b=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1291, C2×#67754, #10685101, … |
| maximal normal subgroup of | D3×#1291, C2×#67754, #10685101, … |
| maximal quotient of | #10681722, #10681728, #10681729, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 330);