Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #204 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #1279 |
208 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 109 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) = 384 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3)(8,10,12,15,13,14,11,9)(17,21,23,22);(2,3)(4,5,6,7)(8,9,11,14,13,15,12,10)(16,17,19,22,20,23,18,21)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,c^2=f,d^2=g,e^2=fg,b^a=bc,c^a=cf,e^a=ef,h^a=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1279, C2×#67655, #10684713, … |
| maximal normal subgroup of | D3×#1279, C2×#67655, #10684713, … |
| maximal quotient of | #10681715, #10681730, #10681736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 231);