Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2141 |
220 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 85 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:
(1,2,3)(8,11,14,13)(9,16,18,29)(10,20,23,22)(12,24,28,27)(15,30,31,32)(17,25,34,35)(19,33,36,39)(21,37,38,26);(2,3)(8,10,14,23)(9,15,18,31)(11,22,13,20)(12,26,28,37)(16,32,29,30)(17,33,34,39)(19,35,36,25)(21,24,38,27);(4,5,6,7)(8,9)(10,19)(11,16)(12,25)(13,29)(14,18)(15,21)(17,27)(20,33)(22,39)(23,36)(24,34)(26,32)(28,35)(30,37)(31,38)Polycyclic presentation:
2^7,3|a^2=e,b^2=g,c^2=g,d^2=fg,f^2=g,b^a=bd,d^a=df,f^a=fg,c^b=cg,d^b=df,f^b=fg,h^b=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10710465, #10710468, #10710877, … |
| maximal normal subgroup of | #10710465, #10710468, #10710877, … |
| maximal quotient of | #10704939, #10705183, #10705235, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3230);