metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C41⋊C4, D41.C2, SmallGroup(164,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C41 — C41⋊C4 |
Generators and relations for C41⋊C4
G = < a,b | a41=b4=1, bab-1=a9 >
Character table of C41⋊C4
| class | 1 | 2 | 4A | 4B | 41A | 41B | 41C | 41D | 41E | 41F | 41G | 41H | 41I | 41J | |
| size | 1 | 41 | 41 | 41 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 4 | 0 | 0 | 0 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4137+ζ4136+ζ415+ζ414 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4140+ζ4132+ζ419+ζ41 | ζ4134+ζ4122+ζ4119+ζ417 | orthogonal faithful |
| ρ6 | 4 | 0 | 0 | 0 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4137+ζ4136+ζ415+ζ414 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4140+ζ4132+ζ419+ζ41 | orthogonal faithful |
| ρ7 | 4 | 0 | 0 | 0 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4137+ζ4136+ζ415+ζ414 | ζ4140+ζ4132+ζ419+ζ41 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4139+ζ4123+ζ4118+ζ412 | orthogonal faithful |
| ρ8 | 4 | 0 | 0 | 0 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4137+ζ4136+ζ415+ζ414 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4140+ζ4132+ζ419+ζ41 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4133+ζ4131+ζ4110+ζ418 | orthogonal faithful |
| ρ9 | 4 | 0 | 0 | 0 | ζ4140+ζ4132+ζ419+ζ41 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4137+ζ4136+ζ415+ζ414 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4130+ζ4124+ζ4117+ζ4111 | orthogonal faithful |
| ρ10 | 4 | 0 | 0 | 0 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4140+ζ4132+ζ419+ζ41 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4137+ζ4136+ζ415+ζ414 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4125+ζ4121+ζ4120+ζ4116 | orthogonal faithful |
| ρ11 | 4 | 0 | 0 | 0 | ζ4137+ζ4136+ζ415+ζ414 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4140+ζ4132+ζ419+ζ41 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4138+ζ4127+ζ4114+ζ413 | orthogonal faithful |
| ρ12 | 4 | 0 | 0 | 0 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4140+ζ4132+ζ419+ζ41 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4137+ζ4136+ζ415+ζ414 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4129+ζ4126+ζ4115+ζ4112 | orthogonal faithful |
| ρ13 | 4 | 0 | 0 | 0 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4140+ζ4132+ζ419+ζ41 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4137+ζ4136+ζ415+ζ414 | ζ4135+ζ4128+ζ4113+ζ416 | orthogonal faithful |
| ρ14 | 4 | 0 | 0 | 0 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4140+ζ4132+ζ419+ζ41 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4137+ζ4136+ζ415+ζ414 | orthogonal 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 33 41 10)(3 24 40 19)(4 15 39 28)(5 6 38 37)(7 29 36 14)(8 20 35 23)(9 11 34 32)(12 25 31 18)(13 16 30 27)(17 21 26 22)
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), (2,33,41,10)(3,24,40,19)(4,15,39,28)(5,6,38,37)(7,29,36,14)(8,20,35,23)(9,11,34,32)(12,25,31,18)(13,16,30,27)(17,21,26,22)>;
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), (2,33,41,10)(3,24,40,19)(4,15,39,28)(5,6,38,37)(7,29,36,14)(8,20,35,23)(9,11,34,32)(12,25,31,18)(13,16,30,27)(17,21,26,22) );
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)], [(2,33,41,10),(3,24,40,19),(4,15,39,28),(5,6,38,37),(7,29,36,14),(8,20,35,23),(9,11,34,32),(12,25,31,18),(13,16,30,27),(17,21,26,22)]])
C41⋊C4 is a maximal subgroup of
C41⋊C8 C41⋊Dic3
C41⋊C4 is a maximal quotient of C41⋊2C8 C41⋊Dic3
Matrix representation of C41⋊C4 ►in GL4(𝔽821) generated by
| 817 | 1 | 0 | 0 |
| 267 | 0 | 1 | 0 |
| 554 | 0 | 0 | 1 |
| 109 | 756 | 401 | 789 |
| 787 | 474 | 393 | 437 |
| 47 | 277 | 249 | 307 |
| 652 | 653 | 448 | 756 |
| 446 | 396 | 609 | 130 |
G:=sub<GL(4,GF(821))| [817,267,554,109,1,0,0,756,0,1,0,401,0,0,1,789],[787,47,652,446,474,277,653,396,393,249,448,609,437,307,756,130] >;
C41⋊C4 in GAP, Magma, Sage, TeX
C_{41}\rtimes C_4 % in TeX
G:=Group("C41:C4"); // GroupNames label
G:=SmallGroup(164,3);
// by ID
G=gap.SmallGroup(164,3);
# by ID
G:=PCGroup([3,-2,-2,-41,6,1154,725]);
// Polycyclic
G:=Group<a,b|a^41=b^4=1,b*a*b^-1=a^9>;
// generators/relations
Export
Subgroup lattice of C41⋊C4 in TeX
Character table of C41⋊C4 in TeX