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## G = C52⋊3C8order 200 = 23·52

### 2nd semidirect product of C52 and C8 acting via C8/C2=C4

Aliases: C523C8, C10.Dic5, C10.5F5, Dic5.2D5, C5⋊(C52C8), C53(C5⋊C8), (C5×C10).2C4, C2.(D5.D5), (C5×Dic5).3C2, SmallGroup(200,19)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊3C8
 Chief series C1 — C5 — C52 — C5×C10 — C5×Dic5 — C52⋊3C8
 Lower central C52 — C52⋊3C8
 Upper central C1 — C2

Generators and relations for C523C8
G = < a,b,c | a5=b5=c8=1, ab=ba, cac-1=a-1, cbc-1=b2 >

Character table of C523C8

 class 1 2 4A 4B 5A 5B 5C 5D 5E 5F 5G 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 20A 20B 20C 20D size 1 1 5 5 2 2 4 4 4 4 4 25 25 25 25 2 2 4 4 4 4 4 10 10 10 10 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 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 1 linear of order 2 ρ3 1 1 -1 -1 1 1 1 1 1 1 1 i -i i -i 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ4 1 1 -1 -1 1 1 1 1 1 1 1 -i i -i i 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ5 1 -1 -i i 1 1 1 1 1 1 1 ζ87 ζ85 ζ83 ζ8 -1 -1 -1 -1 -1 -1 -1 i i -i -i linear of order 8 ρ6 1 -1 -i i 1 1 1 1 1 1 1 ζ83 ζ8 ζ87 ζ85 -1 -1 -1 -1 -1 -1 -1 i i -i -i linear of order 8 ρ7 1 -1 i -i 1 1 1 1 1 1 1 ζ8 ζ83 ζ85 ζ87 -1 -1 -1 -1 -1 -1 -1 -i -i i i linear of order 8 ρ8 1 -1 i -i 1 1 1 1 1 1 1 ζ85 ζ87 ζ8 ζ83 -1 -1 -1 -1 -1 -1 -1 -i -i i i linear of order 8 ρ9 2 2 2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 0 0 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ10 2 2 2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 0 0 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ11 2 2 -2 -2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 0 0 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 symplectic lifted from Dic5, Schur index 2 ρ12 2 2 -2 -2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 0 0 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 symplectic lifted from Dic5, Schur index 2 ρ13 2 -2 -2i 2i -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 0 0 0 0 1+√5/2 1-√5/2 1+√5/2 1-√5/2 1-√5/2 -2 1+√5/2 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 complex lifted from C5⋊2C8, Schur index 2 ρ14 2 -2 2i -2i -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 0 0 0 0 1-√5/2 1+√5/2 1-√5/2 1+√5/2 1+√5/2 -2 1-√5/2 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 complex lifted from C5⋊2C8, Schur index 2 ρ15 2 -2 2i -2i -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 0 0 0 0 1+√5/2 1-√5/2 1+√5/2 1-√5/2 1-√5/2 -2 1+√5/2 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 complex lifted from C5⋊2C8, Schur index 2 ρ16 2 -2 -2i 2i -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 0 0 0 0 1-√5/2 1+√5/2 1-√5/2 1+√5/2 1+√5/2 -2 1-√5/2 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 complex lifted from C5⋊2C8, Schur index 2 ρ17 4 4 0 0 4 4 -1 -1 -1 -1 -1 0 0 0 0 4 4 -1 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from F5 ρ18 4 -4 0 0 4 4 -1 -1 -1 -1 -1 0 0 0 0 -4 -4 1 1 1 1 1 0 0 0 0 symplectic lifted from C5⋊C8, Schur index 2 ρ19 4 4 0 0 -1-√5 -1+√5 2ζ54+ζ52+1 2ζ52+ζ5+1 -1 ζ54+2ζ53+1 ζ53+2ζ5+1 0 0 0 0 -1+√5 -1-√5 2ζ52+ζ5+1 ζ53+2ζ5+1 2ζ54+ζ52+1 -1 ζ54+2ζ53+1 0 0 0 0 complex lifted from D5.D5 ρ20 4 -4 0 0 -1+√5 -1-√5 2ζ52+ζ5+1 ζ53+2ζ5+1 -1 2ζ54+ζ52+1 ζ54+2ζ53+1 0 0 0 0 1+√5 1-√5 ζ54+ζ52-ζ5 -ζ53+ζ52+ζ5 ζ54+ζ53-ζ52 1 -ζ54+ζ53+ζ5 0 0 0 0 complex faithful ρ21 4 -4 0 0 -1-√5 -1+√5 ζ53+2ζ5+1 ζ54+2ζ53+1 -1 2ζ52+ζ5+1 2ζ54+ζ52+1 0 0 0 0 1-√5 1+√5 -ζ53+ζ52+ζ5 -ζ54+ζ53+ζ5 ζ54+ζ52-ζ5 1 ζ54+ζ53-ζ52 0 0 0 0 complex faithful ρ22 4 -4 0 0 -1-√5 -1+√5 2ζ54+ζ52+1 2ζ52+ζ5+1 -1 ζ54+2ζ53+1 ζ53+2ζ5+1 0 0 0 0 1-√5 1+√5 ζ54+ζ53-ζ52 ζ54+ζ52-ζ5 -ζ54+ζ53+ζ5 1 -ζ53+ζ52+ζ5 0 0 0 0 complex faithful ρ23 4 -4 0 0 -1+√5 -1-√5 ζ54+2ζ53+1 2ζ54+ζ52+1 -1 ζ53+2ζ5+1 2ζ52+ζ5+1 0 0 0 0 1+√5 1-√5 -ζ54+ζ53+ζ5 ζ54+ζ53-ζ52 -ζ53+ζ52+ζ5 1 ζ54+ζ52-ζ5 0 0 0 0 complex faithful ρ24 4 4 0 0 -1+√5 -1-√5 2ζ52+ζ5+1 ζ53+2ζ5+1 -1 2ζ54+ζ52+1 ζ54+2ζ53+1 0 0 0 0 -1-√5 -1+√5 ζ53+2ζ5+1 ζ54+2ζ53+1 2ζ52+ζ5+1 -1 2ζ54+ζ52+1 0 0 0 0 complex lifted from D5.D5 ρ25 4 4 0 0 -1-√5 -1+√5 ζ53+2ζ5+1 ζ54+2ζ53+1 -1 2ζ52+ζ5+1 2ζ54+ζ52+1 0 0 0 0 -1+√5 -1-√5 ζ54+2ζ53+1 2ζ54+ζ52+1 ζ53+2ζ5+1 -1 2ζ52+ζ5+1 0 0 0 0 complex lifted from D5.D5 ρ26 4 4 0 0 -1+√5 -1-√5 ζ54+2ζ53+1 2ζ54+ζ52+1 -1 ζ53+2ζ5+1 2ζ52+ζ5+1 0 0 0 0 -1-√5 -1+√5 2ζ54+ζ52+1 2ζ52+ζ5+1 ζ54+2ζ53+1 -1 ζ53+2ζ5+1 0 0 0 0 complex lifted from D5.D5

Smallest permutation representation of C523C8
On 40 points
Generators in S40
(1 11 20 27 35)(2 36 28 21 12)(3 13 22 29 37)(4 38 30 23 14)(5 15 24 31 39)(6 40 32 17 16)(7 9 18 25 33)(8 34 26 19 10)
(1 35 27 20 11)(2 28 12 36 21)(3 13 22 29 37)(4 23 38 14 30)(5 39 31 24 15)(6 32 16 40 17)(7 9 18 25 33)(8 19 34 10 26)
(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)

G:=sub<Sym(40)| (1,11,20,27,35)(2,36,28,21,12)(3,13,22,29,37)(4,38,30,23,14)(5,15,24,31,39)(6,40,32,17,16)(7,9,18,25,33)(8,34,26,19,10), (1,35,27,20,11)(2,28,12,36,21)(3,13,22,29,37)(4,23,38,14,30)(5,39,31,24,15)(6,32,16,40,17)(7,9,18,25,33)(8,19,34,10,26), (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)>;

G:=Group( (1,11,20,27,35)(2,36,28,21,12)(3,13,22,29,37)(4,38,30,23,14)(5,15,24,31,39)(6,40,32,17,16)(7,9,18,25,33)(8,34,26,19,10), (1,35,27,20,11)(2,28,12,36,21)(3,13,22,29,37)(4,23,38,14,30)(5,39,31,24,15)(6,32,16,40,17)(7,9,18,25,33)(8,19,34,10,26), (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) );

G=PermutationGroup([[(1,11,20,27,35),(2,36,28,21,12),(3,13,22,29,37),(4,38,30,23,14),(5,15,24,31,39),(6,40,32,17,16),(7,9,18,25,33),(8,34,26,19,10)], [(1,35,27,20,11),(2,28,12,36,21),(3,13,22,29,37),(4,23,38,14,30),(5,39,31,24,15),(6,32,16,40,17),(7,9,18,25,33),(8,19,34,10,26)], [(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)]])

C523C8 is a maximal subgroup of   D5×C5⋊C8  Dic5.4F5  D10.F5  Dic5.F5  C20.14F5  C20.12F5  C102.C4
C523C8 is a maximal quotient of   C523C16

Matrix representation of C523C8 in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 21 0 16 0 16 0 0 16
,
 16 0 0 0 7 18 0 0 5 0 37 0 7 0 0 10
,
 27 0 15 0 0 0 9 1 0 1 14 0 0 0 38 0
G:=sub<GL(4,GF(41))| [18,0,21,16,0,18,0,0,0,0,16,0,0,0,0,16],[16,7,5,7,0,18,0,0,0,0,37,0,0,0,0,10],[27,0,0,0,0,0,1,0,15,9,14,38,0,1,0,0] >;

C523C8 in GAP, Magma, Sage, TeX

C_5^2\rtimes_3C_8
% in TeX

G:=Group("C5^2:3C8");
// GroupNames label

G:=SmallGroup(200,19);
// by ID

G=gap.SmallGroup(200,19);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,10,26,643,3004,2009]);
// Polycyclic

G:=Group<a,b,c|a^5=b^5=c^8=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^2>;
// generators/relations

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