Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×C3×C4⋊C8 ⋊ C2 |
| Sylow | C3 ⋊ #1551 |
252 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,8)(9,11)(12,14,16,15)(13,17,18,19);(2,3)(5,8)(6,10)(12,13)(14,17)(15,19)(16,18);(4,5,6,9,7,8,10,11)(13,17)(14,15)(18,19)Polycyclic presentation:
2^7,3|a^2=e,c^2=g,d^2=g,e^2=fg,b^a=bd,c^a=cf,d^a=df,d^b=dg,e^b=efg,h^b=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10687327, #10687329, #10687337, … |
| maximal normal subgroup of | #10687327, #10687329, #10687337, … |
| maximal quotient of | #10683197, #10683227, #10683346, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3009);