Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#358 ⋊ C2 |
| Sylow | C3 ⋊ #2138 |
228 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2)(8,11,14,13)(9,16,18,28)(10,20,22,29)(12,23,27,26)(15,30,32,31)(17,24,34,38)(19,35,36,33)(21,25,37,39);(2,3)(8,10)(9,15)(11,20)(12,25)(13,29)(14,22)(16,30)(17,33)(18,32)(19,24)(21,26)(23,37)(27,39)(28,31)(34,35)(36,38);(4,5,6,7)(8,9)(10,19)(11,16)(12,24)(13,28)(14,18)(15,21)(17,26)(20,35)(22,36)(23,34)(25,30)(27,38)(29,33)(31,39)(32,37)Polycyclic presentation:
2^7,3|a^2=e,c^2=g,d^2=fg,f^2=g,b^a=bd,d^a=df,f^a=fg,d^b=df,f^b=fg,h^b=h^2,h^c=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2138, C2×#69220, #10948089, … |
| maximal normal subgroup of | D3×#2138, C2×#69220, #10948089, … |
| maximal quotient of | #10704926, #10705170, #10705222, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1796);