Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #1574 |
158 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 77 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)(4,7,9,8)(5,6,10,11)(14,20)(17,23)(21,26)(24,27);(2,3)(4,6,8,10,9,11,7,5)(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(4,5,9,10)(6,8,11,7)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=d,c^2=f,d^2=f,e^2=g,b^a=bd,d^a=df,c^b=cfg,h^b=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10689459, #10689462, #10689466, … |
| maximal normal subgroup of | #10689459, #10689462, #10689466, … |
| maximal quotient of | #10684674, #10686406, #10688041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3324);