metacyclic, supersoluble, monomial, Z-group
Aliases: C41⋊C10, D41⋊C5, C41⋊C5⋊C2, SmallGroup(410,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C41 — C41⋊C10 |
Generators and relations for C41⋊C10
G = < a,b | a41=b10=1, bab-1=a23 >
Character table of C41⋊C10
| class | 1 | 2 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 41A | 41B | 41C | 41D | |
| size | 1 | 41 | 41 | 41 | 41 | 41 | 41 | 41 | 41 | 41 | 10 | 10 | 10 | 10 | |
| 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 | ζ52 | ζ54 | ζ5 | ζ53 | ζ5 | ζ54 | ζ53 | ζ52 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ4 | 1 | 1 | ζ5 | ζ52 | ζ53 | ζ54 | ζ53 | ζ52 | ζ54 | ζ5 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ5 | 1 | -1 | ζ54 | ζ53 | ζ52 | ζ5 | -ζ52 | -ζ53 | -ζ5 | -ζ54 | 1 | 1 | 1 | 1 | linear of order 10 |
| ρ6 | 1 | -1 | ζ5 | ζ52 | ζ53 | ζ54 | -ζ53 | -ζ52 | -ζ54 | -ζ5 | 1 | 1 | 1 | 1 | linear of order 10 |
| ρ7 | 1 | -1 | ζ53 | ζ5 | ζ54 | ζ52 | -ζ54 | -ζ5 | -ζ52 | -ζ53 | 1 | 1 | 1 | 1 | linear of order 10 |
| ρ8 | 1 | 1 | ζ53 | ζ5 | ζ54 | ζ52 | ζ54 | ζ5 | ζ52 | ζ53 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ9 | 1 | -1 | ζ52 | ζ54 | ζ5 | ζ53 | -ζ5 | -ζ54 | -ζ53 | -ζ52 | 1 | 1 | 1 | 1 | linear of order 10 |
| ρ10 | 1 | 1 | ζ54 | ζ53 | ζ52 | ζ5 | ζ52 | ζ53 | ζ5 | ζ54 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ11 | 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4135+ζ4127+ζ4126+ζ4124+ζ4122+ζ4119+ζ4117+ζ4115+ζ4114+ζ416 | ζ4138+ζ4134+ζ4130+ζ4129+ζ4128+ζ4113+ζ4112+ζ4111+ζ417+ζ413 | ζ4139+ζ4136+ζ4133+ζ4132+ζ4121+ζ4120+ζ419+ζ418+ζ415+ζ412 | ζ4140+ζ4137+ζ4131+ζ4125+ζ4123+ζ4118+ζ4116+ζ4110+ζ414+ζ41 | orthogonal faithful |
| ρ12 | 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4139+ζ4136+ζ4133+ζ4132+ζ4121+ζ4120+ζ419+ζ418+ζ415+ζ412 | ζ4140+ζ4137+ζ4131+ζ4125+ζ4123+ζ4118+ζ4116+ζ4110+ζ414+ζ41 | ζ4138+ζ4134+ζ4130+ζ4129+ζ4128+ζ4113+ζ4112+ζ4111+ζ417+ζ413 | ζ4135+ζ4127+ζ4126+ζ4124+ζ4122+ζ4119+ζ4117+ζ4115+ζ4114+ζ416 | orthogonal faithful |
| ρ13 | 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4138+ζ4134+ζ4130+ζ4129+ζ4128+ζ4113+ζ4112+ζ4111+ζ417+ζ413 | ζ4135+ζ4127+ζ4126+ζ4124+ζ4122+ζ4119+ζ4117+ζ4115+ζ4114+ζ416 | ζ4140+ζ4137+ζ4131+ζ4125+ζ4123+ζ4118+ζ4116+ζ4110+ζ414+ζ41 | ζ4139+ζ4136+ζ4133+ζ4132+ζ4121+ζ4120+ζ419+ζ418+ζ415+ζ412 | orthogonal faithful |
| ρ14 | 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4140+ζ4137+ζ4131+ζ4125+ζ4123+ζ4118+ζ4116+ζ4110+ζ414+ζ41 | ζ4139+ζ4136+ζ4133+ζ4132+ζ4121+ζ4120+ζ419+ζ418+ζ415+ζ412 | ζ4135+ζ4127+ζ4126+ζ4124+ζ4122+ζ4119+ζ4117+ζ4115+ζ4114+ζ416 | ζ4138+ζ4134+ζ4130+ζ4129+ζ4128+ζ4113+ζ4112+ζ4111+ζ417+ζ413 | 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 26 11 5 19 41 17 32 38 24)(3 10 21 9 37 40 33 22 34 6)(4 35 31 13 14 39 8 12 30 29)(7 28 20 25 27 36 15 23 18 16)
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), (2,26,11,5,19,41,17,32,38,24)(3,10,21,9,37,40,33,22,34,6)(4,35,31,13,14,39,8,12,30,29)(7,28,20,25,27,36,15,23,18,16)>;
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), (2,26,11,5,19,41,17,32,38,24)(3,10,21,9,37,40,33,22,34,6)(4,35,31,13,14,39,8,12,30,29)(7,28,20,25,27,36,15,23,18,16) );
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)], [(2,26,11,5,19,41,17,32,38,24),(3,10,21,9,37,40,33,22,34,6),(4,35,31,13,14,39,8,12,30,29),(7,28,20,25,27,36,15,23,18,16)]])
Matrix representation of C41⋊C10 ►in GL10(𝔽821)
| 678 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 267 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 33 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 407 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 724 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 456 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 455 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 723 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 596 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 699 | 1 | 626 | 192 | 504 | 309 | 189 | 502 | 816 | 505 |
| 71 | 194 | 660 | 530 | 176 | 537 | 605 | 801 | 632 | 435 |
| 472 | 315 | 430 | 625 | 815 | 562 | 309 | 70 | 195 | 436 |
| 769 | 727 | 92 | 793 | 748 | 28 | 738 | 698 | 262 | 610 |
| 150 | 693 | 672 | 326 | 483 | 391 | 644 | 402 | 564 | 308 |
| 201 | 672 | 14 | 810 | 149 | 331 | 23 | 589 | 501 | 727 |
| 352 | 185 | 383 | 379 | 366 | 305 | 373 | 740 | 258 | 18 |
| 177 | 817 | 662 | 234 | 612 | 629 | 389 | 44 | 474 | 759 |
| 238 | 241 | 148 | 817 | 158 | 82 | 247 | 181 | 245 | 782 |
| 816 | 626 | 311 | 153 | 418 | 103 | 545 | 564 | 600 | 793 |
| 37 | 792 | 143 | 113 | 306 | 65 | 203 | 128 | 750 | 35 |
G:=sub<GL(10,GF(821))| [678,267,33,407,724,456,455,723,596,699,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,626,0,0,1,0,0,0,0,0,0,192,0,0,0,1,0,0,0,0,0,504,0,0,0,0,1,0,0,0,0,309,0,0,0,0,0,1,0,0,0,189,0,0,0,0,0,0,1,0,0,502,0,0,0,0,0,0,0,1,0,816,0,0,0,0,0,0,0,0,1,505],[71,472,769,150,201,352,177,238,816,37,194,315,727,693,672,185,817,241,626,792,660,430,92,672,14,383,662,148,311,143,530,625,793,326,810,379,234,817,153,113,176,815,748,483,149,366,612,158,418,306,537,562,28,391,331,305,629,82,103,65,605,309,738,644,23,373,389,247,545,203,801,70,698,402,589,740,44,181,564,128,632,195,262,564,501,258,474,245,600,750,435,436,610,308,727,18,759,782,793,35] >;
C41⋊C10 in GAP, Magma, Sage, TeX
C_{41}\rtimes C_{10} % in TeX
G:=Group("C41:C10"); // GroupNames label
G:=SmallGroup(410,1);
// by ID
G=gap.SmallGroup(410,1);
# by ID
G:=PCGroup([3,-2,-5,-41,3602,455]);
// Polycyclic
G:=Group<a,b|a^41=b^10=1,b*a*b^-1=a^23>;
// generators/relations
Export
Subgroup lattice of C41⋊C10 in TeX
Character table of C41⋊C10 in TeX