metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary
Aliases: C41⋊C5, SmallGroup(205,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C41 — C41⋊C5 |
Generators and relations for C41⋊C5
G = < a,b | a41=b5=1, bab-1=a37 >
Character table of C41⋊C5
| class | 1 | 5A | 5B | 5C | 5D | 41A | 41B | 41C | 41D | 41E | 41F | 41G | 41H | |
| size | 1 | 41 | 41 | 41 | 41 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | ζ5 | ζ53 | ζ52 | ζ54 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ3 | 1 | ζ54 | ζ52 | ζ53 | ζ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ4 | 1 | ζ52 | ζ5 | ζ54 | ζ53 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ5 | 1 | ζ53 | ζ54 | ζ5 | ζ52 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ6 | 5 | 0 | 0 | 0 | 0 | ζ4137+ζ4118+ζ4116+ζ4110+ζ41 | ζ4138+ζ4134+ζ4128+ζ4112+ζ4111 | ζ4135+ζ4127+ζ4124+ζ4122+ζ4115 | ζ4130+ζ4129+ζ4113+ζ417+ζ413 | ζ4136+ζ4133+ζ4132+ζ4120+ζ412 | ζ4140+ζ4131+ζ4125+ζ4123+ζ414 | ζ4126+ζ4119+ζ4117+ζ4114+ζ416 | ζ4139+ζ4121+ζ419+ζ418+ζ415 | complex faithful |
| ρ7 | 5 | 0 | 0 | 0 | 0 | ζ4135+ζ4127+ζ4124+ζ4122+ζ4115 | ζ4137+ζ4118+ζ4116+ζ4110+ζ41 | ζ4136+ζ4133+ζ4132+ζ4120+ζ412 | ζ4140+ζ4131+ζ4125+ζ4123+ζ414 | ζ4130+ζ4129+ζ4113+ζ417+ζ413 | ζ4126+ζ4119+ζ4117+ζ4114+ζ416 | ζ4139+ζ4121+ζ419+ζ418+ζ415 | ζ4138+ζ4134+ζ4128+ζ4112+ζ4111 | complex faithful |
| ρ8 | 5 | 0 | 0 | 0 | 0 | ζ4138+ζ4134+ζ4128+ζ4112+ζ4111 | ζ4139+ζ4121+ζ419+ζ418+ζ415 | ζ4137+ζ4118+ζ4116+ζ4110+ζ41 | ζ4136+ζ4133+ζ4132+ζ4120+ζ412 | ζ4135+ζ4127+ζ4124+ζ4122+ζ4115 | ζ4130+ζ4129+ζ4113+ζ417+ζ413 | ζ4140+ζ4131+ζ4125+ζ4123+ζ414 | ζ4126+ζ4119+ζ4117+ζ4114+ζ416 | complex faithful |
| ρ9 | 5 | 0 | 0 | 0 | 0 | ζ4140+ζ4131+ζ4125+ζ4123+ζ414 | ζ4130+ζ4129+ζ4113+ζ417+ζ413 | ζ4126+ζ4119+ζ4117+ζ4114+ζ416 | ζ4138+ζ4134+ζ4128+ζ4112+ζ4111 | ζ4139+ζ4121+ζ419+ζ418+ζ415 | ζ4137+ζ4118+ζ4116+ζ4110+ζ41 | ζ4135+ζ4127+ζ4124+ζ4122+ζ4115 | ζ4136+ζ4133+ζ4132+ζ4120+ζ412 | complex faithful |
| ρ10 | 5 | 0 | 0 | 0 | 0 | ζ4126+ζ4119+ζ4117+ζ4114+ζ416 | ζ4140+ζ4131+ζ4125+ζ4123+ζ414 | ζ4139+ζ4121+ζ419+ζ418+ζ415 | ζ4137+ζ4118+ζ4116+ζ4110+ζ41 | ζ4138+ζ4134+ζ4128+ζ4112+ζ4111 | ζ4135+ζ4127+ζ4124+ζ4122+ζ4115 | ζ4136+ζ4133+ζ4132+ζ4120+ζ412 | ζ4130+ζ4129+ζ4113+ζ417+ζ413 | complex faithful |
| ρ11 | 5 | 0 | 0 | 0 | 0 | ζ4130+ζ4129+ζ4113+ζ417+ζ413 | ζ4136+ζ4133+ζ4132+ζ4120+ζ412 | ζ4140+ζ4131+ζ4125+ζ4123+ζ414 | ζ4139+ζ4121+ζ419+ζ418+ζ415 | ζ4126+ζ4119+ζ4117+ζ4114+ζ416 | ζ4138+ζ4134+ζ4128+ζ4112+ζ4111 | ζ4137+ζ4118+ζ4116+ζ4110+ζ41 | ζ4135+ζ4127+ζ4124+ζ4122+ζ4115 | complex faithful |
| ρ12 | 5 | 0 | 0 | 0 | 0 | ζ4139+ζ4121+ζ419+ζ418+ζ415 | ζ4126+ζ4119+ζ4117+ζ4114+ζ416 | ζ4138+ζ4134+ζ4128+ζ4112+ζ4111 | ζ4135+ζ4127+ζ4124+ζ4122+ζ4115 | ζ4137+ζ4118+ζ4116+ζ4110+ζ41 | ζ4136+ζ4133+ζ4132+ζ4120+ζ412 | ζ4130+ζ4129+ζ4113+ζ417+ζ413 | ζ4140+ζ4131+ζ4125+ζ4123+ζ414 | complex faithful |
| ρ13 | 5 | 0 | 0 | 0 | 0 | ζ4136+ζ4133+ζ4132+ζ4120+ζ412 | ζ4135+ζ4127+ζ4124+ζ4122+ζ4115 | ζ4130+ζ4129+ζ4113+ζ417+ζ413 | ζ4126+ζ4119+ζ4117+ζ4114+ζ416 | ζ4140+ζ4131+ζ4125+ζ4123+ζ414 | ζ4139+ζ4121+ζ419+ζ418+ζ415 | ζ4138+ζ4134+ζ4128+ζ4112+ζ4111 | ζ4137+ζ4118+ζ4116+ζ4110+ζ41 | complex faithful |
►On 41 points: primitive
Generators in S41
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41)
(2 11 19 17 38)(3 21 37 33 34)(4 31 14 8 30)(5 41 32 24 26)(6 10 9 40 22)(7 20 27 15 18)(12 29 35 13 39)(16 28 25 36 23)
G:=sub<Sym(41)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41), (2,11,19,17,38)(3,21,37,33,34)(4,31,14,8,30)(5,41,32,24,26)(6,10,9,40,22)(7,20,27,15,18)(12,29,35,13,39)(16,28,25,36,23)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41), (2,11,19,17,38)(3,21,37,33,34)(4,31,14,8,30)(5,41,32,24,26)(6,10,9,40,22)(7,20,27,15,18)(12,29,35,13,39)(16,28,25,36,23) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)], [(2,11,19,17,38),(3,21,37,33,34),(4,31,14,8,30),(5,41,32,24,26),(6,10,9,40,22),(7,20,27,15,18),(12,29,35,13,39),(16,28,25,36,23)]])
C41⋊C5 is a maximal subgroup of
C41⋊C10
Matrix representation of C41⋊C5 ►in GL5(𝔽821)
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 408 | 787 | 471 | 550 |
| 1 | 0 | 0 | 0 | 0 |
| 313 | 753 | 518 | 596 | 433 |
| 577 | 780 | 503 | 322 | 39 |
| 239 | 807 | 32 | 619 | 802 |
| 545 | 752 | 453 | 594 | 587 |
G:=sub<GL(5,GF(821))| [0,0,0,0,1,1,0,0,0,408,0,1,0,0,787,0,0,1,0,471,0,0,0,1,550],[1,313,577,239,545,0,753,780,807,752,0,518,503,32,453,0,596,322,619,594,0,433,39,802,587] >;
C41⋊C5 in GAP, Magma, Sage, TeX
C_{41}\rtimes C_5 % in TeX
G:=Group("C41:C5"); // GroupNames label
G:=SmallGroup(205,1);
// by ID
G=gap.SmallGroup(205,1);
# by ID
G:=PCGroup([2,-5,-41,201]);
// Polycyclic
G:=Group<a,b|a^41=b^5=1,b*a*b^-1=a^37>;
// generators/relations
Export
Subgroup lattice of C41⋊C5 in TeX
Character table of C41⋊C5 in TeX