Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3177 |
338 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(14,23)(18,31)(22,37)(24,36)(30,42)(32,39)(35,43)(38,41);(2,3)(12,15,26,33,17,28,16,27)(13,19,29,40,21,34,20,25)(14,22,36,41,23,37,24,38)(18,30,39,43,31,42,32,35);(5,8)(9,11)(12,14,17,23)(13,18,21,31)(15,22,28,37)(16,24,26,36)(19,30,34,42)(20,32,29,39)(25,35,40,43)(27,38,33,41);(4,5)(6,9)(7,8)(10,11)(12,13,17,21)(14,18,23,31)(15,25,28,40)(16,29,26,20)(19,33,34,27)(22,35,37,43)(24,39,36,32)(30,41,42,38)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,c^2=fg,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=dg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3177, C2×#83444, #11665035, … |
| maximal normal subgroup of | D3×#3177, C2×#83444, #11665035, … |
| maximal quotient of | #10795753, #10799455, #10799535, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16020);