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×M32 ⋊ C2 |
| Sylow | C3 ⋊ #2145 |
224 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 81 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)(4,7,9,21)(5,11,14,13)(6,16,18,19)(8,20,24,33)(10,25,26,30)(12,27,29,22)(15,31,32,28)(17,23,34,35);(2,3)(4,6)(5,10)(7,16)(8,23)(9,18)(11,25)(12,28)(13,30)(14,26)(15,27)(17,33)(19,21)(20,34)(22,32)(24,35)(29,31);(2,3)(4,5)(6,15,18,32)(7,11)(8,22)(9,14)(10,17,26,34)(12,20)(13,21)(16,31,19,28)(23,30,35,25)(24,27)(29,33)Polycyclic presentation:
2^7,3|a^2=e,c^2=g,d^2=f,f^2=g,b^a=bd,d^a=df,f^a=fg,h^a=h^2,d^b=dfg,e^b=eg,f^b=fg,h^b=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10710449, #10710452, #10710472, … |
| maximal normal subgroup of | #10710449, #10710452, #10710472, … |
| maximal quotient of | #10704954, #10704963, #10704967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4026);