metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8.2F5, C40.2C4, D10.1Q8, Dic5.10D4, C5⋊2C8.3C4, (C8×D5).6C2, C2.6(C4⋊F5), C10.3(C4⋊C4), C4.10(C2×F5), C5⋊1(C8.C4), C4.F5.1C2, C20.10(C2×C4), (C4×D5).27C22, SmallGroup(160,70)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.C4
G = < a,b | a40=1, b4=a20, bab-1=a3 >
Character table of C40.C4
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10 | 20A | 20B | 40A | 40B | 40C | 40D | |
| size | 1 | 1 | 10 | 2 | 5 | 5 | 4 | 2 | 2 | 10 | 10 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | i | -i | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | i | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | -i | i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ11 | 2 | -2 | 0 | 0 | -2i | 2i | 2 | √-2 | -√-2 | -√2 | √2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | complex lifted from C8.C4 |
| ρ12 | 2 | -2 | 0 | 0 | 2i | -2i | 2 | √-2 | -√-2 | √2 | -√2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | complex lifted from C8.C4 |
| ρ13 | 2 | -2 | 0 | 0 | 2i | -2i | 2 | -√-2 | √-2 | -√2 | √2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | complex lifted from C8.C4 |
| ρ14 | 2 | -2 | 0 | 0 | -2i | 2i | 2 | -√-2 | √-2 | √2 | -√2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | complex lifted from C8.C4 |
| ρ15 | 4 | 4 | 0 | 4 | 0 | 0 | -1 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ16 | 4 | 4 | 0 | 4 | 0 | 0 | -1 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ17 | 4 | 4 | 0 | -4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√-5 | -√-5 | √-5 | √-5 | complex lifted from C4⋊F5 |
| ρ18 | 4 | 4 | 0 | -4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √-5 | √-5 | -√-5 | -√-5 | complex lifted from C4⋊F5 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | -2√-2 | 2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -√-5 | √-5 | -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5+ζ85 | ζ83ζ53+ζ83ζ52+ζ83-ζ8ζ53-ζ8ζ52 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ87ζ54+ζ87ζ5+ζ87-ζ85ζ54-ζ85ζ5 | complex faithful |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | 2√-2 | -2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | √-5 | -√-5 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ87ζ54+ζ87ζ5+ζ87-ζ85ζ54-ζ85ζ5 | -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5+ζ85 | ζ83ζ53+ζ83ζ52+ζ83-ζ8ζ53-ζ8ζ52 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | -2√-2 | 2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | √-5 | -√-5 | ζ87ζ54+ζ87ζ5+ζ87-ζ85ζ54-ζ85ζ5 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ83ζ53+ζ83ζ52+ζ83-ζ8ζ53-ζ8ζ52 | -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5+ζ85 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | 2√-2 | -2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -√-5 | √-5 | ζ83ζ53+ζ83ζ52+ζ83-ζ8ζ53-ζ8ζ52 | -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5+ζ85 | ζ87ζ54+ζ87ζ5+ζ87-ζ85ζ54-ζ85ζ5 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | complex faithful |
►On 80 points
Generators in S80
(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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 76 31 46 21 56 11 66)(2 63 40 49 22 43 20 69)(3 50 9 52 23 70 29 72)(4 77 18 55 24 57 38 75)(5 64 27 58 25 44 7 78)(6 51 36 61 26 71 16 41)(8 65 14 67 28 45 34 47)(10 79 32 73 30 59 12 53)(13 80 19 42 33 60 39 62)(15 54 37 48 35 74 17 68)
G:=sub<Sym(80)| (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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,76,31,46,21,56,11,66)(2,63,40,49,22,43,20,69)(3,50,9,52,23,70,29,72)(4,77,18,55,24,57,38,75)(5,64,27,58,25,44,7,78)(6,51,36,61,26,71,16,41)(8,65,14,67,28,45,34,47)(10,79,32,73,30,59,12,53)(13,80,19,42,33,60,39,62)(15,54,37,48,35,74,17,68)>;
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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,76,31,46,21,56,11,66)(2,63,40,49,22,43,20,69)(3,50,9,52,23,70,29,72)(4,77,18,55,24,57,38,75)(5,64,27,58,25,44,7,78)(6,51,36,61,26,71,16,41)(8,65,14,67,28,45,34,47)(10,79,32,73,30,59,12,53)(13,80,19,42,33,60,39,62)(15,54,37,48,35,74,17,68) );
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),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,76,31,46,21,56,11,66),(2,63,40,49,22,43,20,69),(3,50,9,52,23,70,29,72),(4,77,18,55,24,57,38,75),(5,64,27,58,25,44,7,78),(6,51,36,61,26,71,16,41),(8,65,14,67,28,45,34,47),(10,79,32,73,30,59,12,53),(13,80,19,42,33,60,39,62),(15,54,37,48,35,74,17,68)])
C40.C4 is a maximal subgroup of
C80⋊4C4 C80⋊5C4 D40.C4 Dic20.C4 (C8×D5).C4 M4(2).1F5 D8⋊F5 SD16⋊3F5 Q16⋊F5 D10.Dic6 C40.Dic3
C40.C4 is a maximal quotient of
C40⋊2C8 Dic5.13D8 C20.10C42 D10.Dic6 C40.Dic3
Matrix representation of C40.C4 ►in GL4(𝔽3) generated by
| 0 | 0 | 2 | 2 |
| 1 | 0 | 1 | 1 |
| 2 | 1 | 2 | 1 |
| 0 | 0 | 2 | 1 |
| 0 | 1 | 1 | 0 |
| 2 | 0 | 2 | 1 |
| 1 | 0 | 2 | 1 |
| 1 | 1 | 1 | 1 |
G:=sub<GL(4,GF(3))| [0,1,2,0,0,0,1,0,2,1,2,2,2,1,1,1],[0,2,1,1,1,0,0,1,1,2,2,1,0,1,1,1] >;
C40.C4 in GAP, Magma, Sage, TeX
C_{40}.C_4 % in TeX
G:=Group("C40.C4"); // GroupNames label
G:=SmallGroup(160,70);
// by ID
G=gap.SmallGroup(160,70);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,55,86,579,69,2309,1169]);
// Polycyclic
G:=Group<a,b|a^40=1,b^4=a^20,b*a*b^-1=a^3>;
// generators/relations
Export
Subgroup lattice of C40.C4 in TeX
Character table of C40.C4 in TeX