metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊2F5, C40⋊2C4, D10.8D4, Dic5.2Q8, D5.1SD16, C5⋊(C4.Q8), C5⋊2C8⋊5C4, C4⋊F5.3C2, C4.8(C2×F5), C20.8(C2×C4), (C8×D5).5C2, C2.4(C4⋊F5), C10.1(C4⋊C4), (C4×D5).25C22, SmallGroup(160,68)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊C4
G = < a,b | a40=b4=1, bab-1=a3 >
Character table of C40⋊C4
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 10 | 20A | 20B | 40A | 40B | 40C | 40D | |
| size | 1 | 1 | 5 | 5 | 2 | 10 | 20 | 20 | 20 | 20 | 4 | 2 | 2 | 10 | 10 | 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 | i | i | -i | -i | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | -i | i | i | -i | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | -i | -i | i | i | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | i | -i | -i | i | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √-2 | -√-2 | -√-2 | √-2 | -2 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | complex lifted from SD16 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√-2 | √-2 | √-2 | -√-2 | -2 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | complex lifted from SD16 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √-2 | -√-2 | √-2 | -√-2 | -2 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | complex lifted from SD16 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√-2 | √-2 | -√-2 | √-2 | -2 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | complex lifted from SD16 |
| ρ15 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | -4 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ16 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 4 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ17 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√-5 | √-5 | √-5 | -√-5 | complex lifted from C4⋊F5 |
| ρ18 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √-5 | -√-5 | -√-5 | √-5 | complex lifted from C4⋊F5 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -2√-2 | 2√-2 | 0 | 0 | 1 | -√-5 | √-5 | ζ87ζ54+ζ87ζ5+ζ87-ζ85ζ54-ζ85ζ5 | -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5+ζ85 | ζ83ζ53+ζ83ζ52+ζ83-ζ8ζ53-ζ8ζ52 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | complex faithful |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2√-2 | -2√-2 | 0 | 0 | 1 | √-5 | -√-5 | ζ83ζ53+ζ83ζ52+ζ83-ζ8ζ53-ζ8ζ52 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ87ζ54+ζ87ζ5+ζ87-ζ85ζ54-ζ85ζ5 | -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5+ζ85 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2√-2 | -2√-2 | 0 | 0 | 1 | -√-5 | √-5 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ83ζ53+ζ83ζ52+ζ83-ζ8ζ53-ζ8ζ52 | -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5+ζ85 | ζ87ζ54+ζ87ζ5+ζ87-ζ85ζ54-ζ85ζ5 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -2√-2 | 2√-2 | 0 | 0 | 1 | √-5 | -√-5 | -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5+ζ85 | ζ87ζ54+ζ87ζ5+ζ87-ζ85ζ54-ζ85ζ5 | -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ83ζ53+ζ83ζ52+ζ83-ζ8ζ53-ζ8ζ52 | complex faithful |
►On 40 points
Generators in S40
(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)
(2 28 10 4)(3 15 19 7)(5 29 37 13)(6 16)(8 30 24 22)(9 17 33 25)(11 31)(12 18 20 34)(14 32 38 40)(23 35 39 27)(26 36)
G:=sub<Sym(40)| (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), (2,28,10,4)(3,15,19,7)(5,29,37,13)(6,16)(8,30,24,22)(9,17,33,25)(11,31)(12,18,20,34)(14,32,38,40)(23,35,39,27)(26,36)>;
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), (2,28,10,4)(3,15,19,7)(5,29,37,13)(6,16)(8,30,24,22)(9,17,33,25)(11,31)(12,18,20,34)(14,32,38,40)(23,35,39,27)(26,36) );
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)], [(2,28,10,4),(3,15,19,7),(5,29,37,13),(6,16),(8,30,24,22),(9,17,33,25),(11,31),(12,18,20,34),(14,32,38,40),(23,35,39,27),(26,36)]])
C40⋊C4 is a maximal subgroup of
C80⋊4C4 C80⋊5C4 D10.D8 D40⋊1C4 (C2×C8)⋊6F5 M4(2)⋊1F5 D40⋊C4 SD16×F5 Dic20⋊C4 Dic5.Dic6 C120⋊C4
C40⋊C4 is a maximal quotient of
C80⋊4C4 C80⋊5C4 C40⋊2C8 C20.26M4(2) D10.10D8 Dic5.Dic6 C120⋊C4
Matrix representation of C40⋊C4 ►in GL4(𝔽3) generated by
| 0 | 0 | 2 | 2 |
| 1 | 0 | 0 | 0 |
| 2 | 1 | 0 | 2 |
| 2 | 2 | 0 | 0 |
| 2 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(3))| [0,1,2,2,0,0,1,2,2,0,0,0,2,0,2,0],[2,0,0,0,0,0,2,0,1,1,0,0,0,0,0,1] >;
C40⋊C4 in GAP, Magma, Sage, TeX
C_{40}\rtimes C_4 % in TeX
G:=Group("C40:C4"); // GroupNames label
G:=SmallGroup(160,68);
// by ID
G=gap.SmallGroup(160,68);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,55,579,69,2309,1169]);
// Polycyclic
G:=Group<a,b|a^40=b^4=1,b*a*b^-1=a^3>;
// generators/relations
Export
Subgroup lattice of C40⋊C4 in TeX
Character table of C40⋊C4 in TeX