Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2194 |
216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,6)(5,11)(7,16)(8,18)(9,19)(10,12)(13,30)(14,31)(15,25)(17,22)(20,28)(21,23)(24,26)(27,35)(29,33)(32,34);(4,6)(5,10)(7,21)(8,17)(9,19)(11,12)(13,27)(14,29)(15,20)(16,23)(18,22)(24,34)(25,28)(26,32)(30,35)(31,33);(2,3)(4,5,9,14)(6,15,19,30)(7,20,24,35)(8,25,22,13)(10,26,29,16)(11,18,31,17)(12,32,33,23)(21,27,34,28)Polycyclic presentation:
2^7,3|a^2=g,d^2=fg,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fg,h^a=h^2,d^b=df,f^b=fg,e^c=eg45 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2194, C2×#72123, #10949138, … |
| maximal normal subgroup of | D3×#2194, C2×#72123, #10949138, … |
| maximal quotient of | #10707335, #10707346, #10707679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4699);