Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C3 ⋊ C8×M16 |
224 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 137 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)(13,16)(17,19);(2,3)(4,5,6,8,7,9,10,11);(2,3)(12,13,14,17,15,16,18,19)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,d^2=f,e^2=g,c^a=cf,h^a=h^2,h^b=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #10685744, #10685748, #10685749, … |
| maximal normal subgroup of | #10685744, #10685748, #10685749, … |
| maximal quotient of | #10684541, #10684570, #10685614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3315);