Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1290 |
136 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(2,3)(5,8)(6,10)(12,14,16,24,17,23,15,22)(13,18,19,21,20,26,25,27);(1,2,3)(4,5,6,9,7,8,10,11)(12,13)(14,21,23,27)(15,19)(16,25)(17,20)(18,22,26,24)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,c^2=fg,d^2=f,e^2=fg,b^a=bc,c^a=cf,e^a=efg,d^b=dg,h^b=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1290, C2×#67755, #10685080, … |
| maximal normal subgroup of | D3×#1290, C2×#67755, #10685080, … |
| maximal quotient of | #10681724, #10681732, #10681734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 331);