metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C41⋊C8, D41.C4, C41⋊C4.C2, SmallGroup(328,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C41 — C41⋊C8 |
Generators and relations for C41⋊C8
G = < a,b | a41=b8=1, bab-1=a38 >
Character table of C41⋊C8
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 41A | 41B | 41C | 41D | 41E | |
| size | 1 | 41 | 41 | 41 | 41 | 41 | 41 | 41 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | i | -i | -i | i | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | 1 | -1 | -1 | -i | i | i | -i | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 1 | -1 | i | -i | ζ85 | ζ83 | ζ87 | ζ8 | 1 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ6 | 1 | -1 | -i | i | ζ87 | ζ8 | ζ85 | ζ83 | 1 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ7 | 1 | -1 | -i | i | ζ83 | ζ85 | ζ8 | ζ87 | 1 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ8 | 1 | -1 | i | -i | ζ8 | ζ87 | ζ83 | ζ85 | 1 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ9 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4134+ζ4125+ζ4122+ζ4121+ζ4120+ζ4119+ζ4116+ζ417 | ζ4133+ζ4131+ζ4130+ζ4124+ζ4117+ζ4111+ζ4110+ζ418 | ζ4140+ζ4138+ζ4132+ζ4127+ζ4114+ζ419+ζ413+ζ41 | ζ4137+ζ4136+ζ4129+ζ4126+ζ4115+ζ4112+ζ415+ζ414 | ζ4139+ζ4135+ζ4128+ζ4123+ζ4118+ζ4113+ζ416+ζ412 | orthogonal faithful |
| ρ10 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4139+ζ4135+ζ4128+ζ4123+ζ4118+ζ4113+ζ416+ζ412 | ζ4140+ζ4138+ζ4132+ζ4127+ζ4114+ζ419+ζ413+ζ41 | ζ4137+ζ4136+ζ4129+ζ4126+ζ4115+ζ4112+ζ415+ζ414 | ζ4134+ζ4125+ζ4122+ζ4121+ζ4120+ζ4119+ζ4116+ζ417 | ζ4133+ζ4131+ζ4130+ζ4124+ζ4117+ζ4111+ζ4110+ζ418 | orthogonal faithful |
| ρ11 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4140+ζ4138+ζ4132+ζ4127+ζ4114+ζ419+ζ413+ζ41 | ζ4134+ζ4125+ζ4122+ζ4121+ζ4120+ζ4119+ζ4116+ζ417 | ζ4139+ζ4135+ζ4128+ζ4123+ζ4118+ζ4113+ζ416+ζ412 | ζ4133+ζ4131+ζ4130+ζ4124+ζ4117+ζ4111+ζ4110+ζ418 | ζ4137+ζ4136+ζ4129+ζ4126+ζ4115+ζ4112+ζ415+ζ414 | orthogonal faithful |
| ρ12 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4137+ζ4136+ζ4129+ζ4126+ζ4115+ζ4112+ζ415+ζ414 | ζ4139+ζ4135+ζ4128+ζ4123+ζ4118+ζ4113+ζ416+ζ412 | ζ4133+ζ4131+ζ4130+ζ4124+ζ4117+ζ4111+ζ4110+ζ418 | ζ4140+ζ4138+ζ4132+ζ4127+ζ4114+ζ419+ζ413+ζ41 | ζ4134+ζ4125+ζ4122+ζ4121+ζ4120+ζ4119+ζ4116+ζ417 | orthogonal faithful |
| ρ13 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4133+ζ4131+ζ4130+ζ4124+ζ4117+ζ4111+ζ4110+ζ418 | ζ4137+ζ4136+ζ4129+ζ4126+ζ4115+ζ4112+ζ415+ζ414 | ζ4134+ζ4125+ζ4122+ζ4121+ζ4120+ζ4119+ζ4116+ζ417 | ζ4139+ζ4135+ζ4128+ζ4123+ζ4118+ζ4113+ζ416+ζ412 | ζ4140+ζ4138+ζ4132+ζ4127+ζ4114+ζ419+ζ413+ζ41 | 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 28 33 4 41 15 10 39)(3 14 24 7 40 29 19 36)(5 27 6 13 38 16 37 30)(8 26 20 22 35 17 23 21)(9 12 11 25 34 31 32 18)
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), (2,28,33,4,41,15,10,39)(3,14,24,7,40,29,19,36)(5,27,6,13,38,16,37,30)(8,26,20,22,35,17,23,21)(9,12,11,25,34,31,32,18)>;
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), (2,28,33,4,41,15,10,39)(3,14,24,7,40,29,19,36)(5,27,6,13,38,16,37,30)(8,26,20,22,35,17,23,21)(9,12,11,25,34,31,32,18) );
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)], [(2,28,33,4,41,15,10,39),(3,14,24,7,40,29,19,36),(5,27,6,13,38,16,37,30),(8,26,20,22,35,17,23,21),(9,12,11,25,34,31,32,18)]])
Matrix representation of C41⋊C8 ►in GL8(𝔽2297)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 2296 | 2270 | 1712 | 2242 | 1711 | 2242 | 1712 | 2270 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 367 | 1648 | 956 | 1505 | 957 | 1477 | 733 | 2153 |
| 847 | 72 | 1126 | 95 | 1520 | 878 | 427 | 2177 |
| 1488 | 2188 | 270 | 236 | 191 | 261 | 2002 | 591 |
| 1927 | 2258 | 658 | 1693 | 70 | 1831 | 273 | 730 |
| 1014 | 1717 | 1805 | 288 | 295 | 1806 | 1915 | 1035 |
| 1283 | 1221 | 1351 | 1164 | 1014 | 1943 | 1241 | 1903 |
| 370 | 1532 | 805 | 1508 | 972 | 1370 | 1190 | 763 |
G:=sub<GL(8,GF(2297))| [0,0,0,0,0,0,0,2296,1,0,0,0,0,0,0,2270,0,1,0,0,0,0,0,1712,0,0,1,0,0,0,0,2242,0,0,0,1,0,0,0,1711,0,0,0,0,1,0,0,2242,0,0,0,0,0,1,0,1712,0,0,0,0,0,0,1,2270],[1,367,847,1488,1927,1014,1283,370,0,1648,72,2188,2258,1717,1221,1532,0,956,1126,270,658,1805,1351,805,0,1505,95,236,1693,288,1164,1508,0,957,1520,191,70,295,1014,972,0,1477,878,261,1831,1806,1943,1370,0,733,427,2002,273,1915,1241,1190,0,2153,2177,591,730,1035,1903,763] >;
C41⋊C8 in GAP, Magma, Sage, TeX
C_{41}\rtimes C_8 % in TeX
G:=Group("C41:C8"); // GroupNames label
G:=SmallGroup(328,12);
// by ID
G=gap.SmallGroup(328,12);
# by ID
G:=PCGroup([4,-2,-2,-2,-41,8,21,3459,2055,1291]);
// Polycyclic
G:=Group<a,b|a^41=b^8=1,b*a*b^-1=a^38>;
// generators/relations
Export
Subgroup lattice of C41⋊C8 in TeX
Character table of C41⋊C8 in TeX