Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1394 |
172 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 172 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) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3)(4,5,6,7)(9,13,16,14)(15,22,23,21);(2,3)(8,9,11,15,12,16,20,23)(10,14,17,22,18,13,19,21)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,c^2=g,d^2=g,b^a=bc,c^a=cf,e^a=eg,f^a=fg,h^a=h^2,d^b=dfg,d^c=dg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10682700, #10682701, #10682703, … |
| maximal normal subgroup of | #10682700, #10682701, #10682703, … |
| maximal quotient of | #10682150, #10682191, #10682218, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 220);