Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #68807 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2097 |
238 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2)(5,7)(16,18,20,27)(17,22,24,30)(19,29,28,26)(21,25,23,31);(4,5)(6,7)(8,10,11,15)(9,12,13,14)(17,21)(19,28)(22,25)(23,24)(26,29)(30,31);(2,3)(8,9,11,13)(10,14,15,12)(16,17,19,23,20,24,28,21)(18,25,26,30,27,31,29,22)Polycyclic presentation:
2^7,3|a^2=d,b^2=deg,c^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,h^a=h^2,c^b=cf,d^b=dg,e^b=eg,d^c=dg,e^c=eg,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2097, C2×#70591, #10824981, … |
| maximal normal subgroup of | D3×#2097, C2×#70591, #10824981, … |
| maximal quotient of | #10700186, #10700211, #10700243, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3167);