Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2020 |
342 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2)(8,11,14,29)(9,16,19,35)(10,21,24,38)(12,28,13,27)(15,30,32,36)(17,34,18,25)(20,33,31,39)(22,26,23,37);(5,7)(8,10)(9,15)(11,26)(12,23)(13,22)(14,24)(16,33)(17,31)(18,20)(19,32)(21,28)(25,36)(27,38)(29,37)(30,34)(35,39);(2,3)(4,5)(6,7)(8,9,14,19)(10,20,24,31)(11,25,29,34)(12,18,13,17)(15,23,32,22)(16,28,35,27)(21,36,38,30)(26,33,37,39)Polycyclic presentation:
2^7,3|a^2=g,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,h^a=h^2,c^b=cf,d^b=dg,f^b=fg,d^c=dg,e^c=eg,f^c=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10682684, #10682686, #10682690, … |
| maximal normal subgroup of | #10682684, #10682686, #10682690, … |
| maximal quotient of | #10695799, #10695816, #10695874, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2584);