Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #204 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1735 |
424 conjugacy classes of subgroups; only the 205 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 205 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 — 3 generators on 27 points:
(1,2,3)(12,15,17,16)(13,19,21,20)(14,23,25,24)(18,26,27,22);(8,9,10,11)(12,14,16,24,17,25,15,23)(13,18,19,26,21,27,20,22);(2,3)(4,5,6,7)(12,13)(14,22)(15,19)(16,20)(17,21)(18,23)(24,27)(25,26)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=g,d^2=g,f^2=g,b^a=bd,d^a=dg,f^a=fg,h^a=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #10682336, #10682338, #10682342, … |
| maximal normal subgroup of | #10682336, #10682338, #10682342, … |
| maximal quotient of | #10684646, #10684656, #10684667, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4545);