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## G = C5×D4order 40 = 23·5

### Direct product of C5 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×D4, C4⋊C10, C203C2, C22⋊C10, C10.6C22, (C2×C10)⋊1C2, C2.1(C2×C10), SmallGroup(40,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×D4
 Chief series C1 — C2 — C10 — C2×C10 — C5×D4
 Lower central C1 — C2 — C5×D4
 Upper central C1 — C10 — C5×D4

Generators and relations for C5×D4
G = < a,b,c | a5=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Character table of C5×D4

 class 1 2A 2B 2C 4 5A 5B 5C 5D 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 20A 20B 20C 20D size 1 1 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 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 -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 -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 1 1 1 linear of order 2 ρ5 1 1 -1 1 -1 ζ5 ζ54 ζ52 ζ53 ζ52 ζ54 ζ53 ζ5 -ζ5 -ζ52 -ζ53 ζ53 ζ5 ζ54 ζ52 -ζ54 -ζ53 -ζ5 -ζ54 -ζ52 linear of order 10 ρ6 1 1 1 -1 -1 ζ5 ζ54 ζ52 ζ53 ζ52 ζ54 ζ53 ζ5 ζ5 ζ52 ζ53 -ζ53 -ζ5 -ζ54 -ζ52 ζ54 -ζ53 -ζ5 -ζ54 -ζ52 linear of order 10 ρ7 1 1 1 1 1 ζ54 ζ5 ζ53 ζ52 ζ53 ζ5 ζ52 ζ54 ζ54 ζ53 ζ52 ζ52 ζ54 ζ5 ζ53 ζ5 ζ52 ζ54 ζ5 ζ53 linear of order 5 ρ8 1 1 -1 -1 1 ζ54 ζ5 ζ53 ζ52 ζ53 ζ5 ζ52 ζ54 -ζ54 -ζ53 -ζ52 -ζ52 -ζ54 -ζ5 -ζ53 -ζ5 ζ52 ζ54 ζ5 ζ53 linear of order 10 ρ9 1 1 1 -1 -1 ζ53 ζ52 ζ5 ζ54 ζ5 ζ52 ζ54 ζ53 ζ53 ζ5 ζ54 -ζ54 -ζ53 -ζ52 -ζ5 ζ52 -ζ54 -ζ53 -ζ52 -ζ5 linear of order 10 ρ10 1 1 -1 -1 1 ζ53 ζ52 ζ5 ζ54 ζ5 ζ52 ζ54 ζ53 -ζ53 -ζ5 -ζ54 -ζ54 -ζ53 -ζ52 -ζ5 -ζ52 ζ54 ζ53 ζ52 ζ5 linear of order 10 ρ11 1 1 1 1 1 ζ53 ζ52 ζ5 ζ54 ζ5 ζ52 ζ54 ζ53 ζ53 ζ5 ζ54 ζ54 ζ53 ζ52 ζ5 ζ52 ζ54 ζ53 ζ52 ζ5 linear of order 5 ρ12 1 1 -1 -1 1 ζ5 ζ54 ζ52 ζ53 ζ52 ζ54 ζ53 ζ5 -ζ5 -ζ52 -ζ53 -ζ53 -ζ5 -ζ54 -ζ52 -ζ54 ζ53 ζ5 ζ54 ζ52 linear of order 10 ρ13 1 1 -1 1 -1 ζ53 ζ52 ζ5 ζ54 ζ5 ζ52 ζ54 ζ53 -ζ53 -ζ5 -ζ54 ζ54 ζ53 ζ52 ζ5 -ζ52 -ζ54 -ζ53 -ζ52 -ζ5 linear of order 10 ρ14 1 1 1 1 1 ζ52 ζ53 ζ54 ζ5 ζ54 ζ53 ζ5 ζ52 ζ52 ζ54 ζ5 ζ5 ζ52 ζ53 ζ54 ζ53 ζ5 ζ52 ζ53 ζ54 linear of order 5 ρ15 1 1 1 1 1 ζ5 ζ54 ζ52 ζ53 ζ52 ζ54 ζ53 ζ5 ζ5 ζ52 ζ53 ζ53 ζ5 ζ54 ζ52 ζ54 ζ53 ζ5 ζ54 ζ52 linear of order 5 ρ16 1 1 1 -1 -1 ζ52 ζ53 ζ54 ζ5 ζ54 ζ53 ζ5 ζ52 ζ52 ζ54 ζ5 -ζ5 -ζ52 -ζ53 -ζ54 ζ53 -ζ5 -ζ52 -ζ53 -ζ54 linear of order 10 ρ17 1 1 1 -1 -1 ζ54 ζ5 ζ53 ζ52 ζ53 ζ5 ζ52 ζ54 ζ54 ζ53 ζ52 -ζ52 -ζ54 -ζ5 -ζ53 ζ5 -ζ52 -ζ54 -ζ5 -ζ53 linear of order 10 ρ18 1 1 -1 1 -1 ζ52 ζ53 ζ54 ζ5 ζ54 ζ53 ζ5 ζ52 -ζ52 -ζ54 -ζ5 ζ5 ζ52 ζ53 ζ54 -ζ53 -ζ5 -ζ52 -ζ53 -ζ54 linear of order 10 ρ19 1 1 -1 1 -1 ζ54 ζ5 ζ53 ζ52 ζ53 ζ5 ζ52 ζ54 -ζ54 -ζ53 -ζ52 ζ52 ζ54 ζ5 ζ53 -ζ5 -ζ52 -ζ54 -ζ5 -ζ53 linear of order 10 ρ20 1 1 -1 -1 1 ζ52 ζ53 ζ54 ζ5 ζ54 ζ53 ζ5 ζ52 -ζ52 -ζ54 -ζ5 -ζ5 -ζ52 -ζ53 -ζ54 -ζ53 ζ5 ζ52 ζ53 ζ54 linear of order 10 ρ21 2 -2 0 0 0 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 -2 0 0 0 2ζ54 2ζ5 2ζ53 2ζ52 -2ζ53 -2ζ5 -2ζ52 -2ζ54 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ23 2 -2 0 0 0 2ζ5 2ζ54 2ζ52 2ζ53 -2ζ52 -2ζ54 -2ζ53 -2ζ5 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ24 2 -2 0 0 0 2ζ52 2ζ53 2ζ54 2ζ5 -2ζ54 -2ζ53 -2ζ5 -2ζ52 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ25 2 -2 0 0 0 2ζ53 2ζ52 2ζ5 2ζ54 -2ζ5 -2ζ52 -2ζ54 -2ζ53 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C5×D4
On 20 points - transitive group 20T12
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 8 13 17)(2 9 14 18)(3 10 15 19)(4 6 11 20)(5 7 12 16)
(1 17)(2 18)(3 19)(4 20)(5 16)(6 11)(7 12)(8 13)(9 14)(10 15)

G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,8,13,17)(2,9,14,18)(3,10,15,19)(4,6,11,20)(5,7,12,16), (1,17)(2,18)(3,19)(4,20)(5,16)(6,11)(7,12)(8,13)(9,14)(10,15)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,8,13,17)(2,9,14,18)(3,10,15,19)(4,6,11,20)(5,7,12,16), (1,17)(2,18)(3,19)(4,20)(5,16)(6,11)(7,12)(8,13)(9,14)(10,15) );

G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,8,13,17),(2,9,14,18),(3,10,15,19),(4,6,11,20),(5,7,12,16)], [(1,17),(2,18),(3,19),(4,20),(5,16),(6,11),(7,12),(8,13),(9,14),(10,15)]])

G:=TransitiveGroup(20,12);

C5×D4 is a maximal subgroup of   D4⋊D5  D4.D5  D42D5  D44⋊C5  C22⋊F11
C5×D4 is a maximal quotient of   D44⋊C5  C22⋊F11

Matrix representation of C5×D4 in GL2(𝔽11) generated by

 9 0 0 9
,
 0 7 3 0
,
 0 4 3 0
G:=sub<GL(2,GF(11))| [9,0,0,9],[0,3,7,0],[0,3,4,0] >;

C5×D4 in GAP, Magma, Sage, TeX

C_5\times D_4
% in TeX

G:=Group("C5xD4");
// GroupNames label

G:=SmallGroup(40,10);
// by ID

G=gap.SmallGroup(40,10);
# by ID

G:=PCGroup([4,-2,-2,-5,-2,177]);
// Polycyclic

G:=Group<a,b,c|a^5=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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