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## G = C52⋊SD16order 400 = 24·52

### The semidirect product of C52 and SD16 acting via SD16/C2=D4

Aliases: C52⋊SD16, C2.4D5≀C2, (C5×C10).4D4, C522D4.C2, C525C84C2, C522Q81C2, C526C4.7C22, SmallGroup(400,132)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊6C4 — C52⋊SD16
 Chief series C1 — C52 — C5×C10 — C52⋊6C4 — C52⋊2D4 — C52⋊SD16
 Lower central C52 — C5×C10 — C52⋊6C4 — C52⋊SD16
 Upper central C1 — C2

Generators and relations for C52⋊SD16
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=a3, dad=b3, cbc-1=b2, dbd=a2, dcd=c3 >

20C2
2C5
2C5
2C5
10C22
10C4
25C4
2C10
2C10
2C10
4D5
20C10
25C8
25Q8
25D4
2D10
2Dic5
10Dic5
10C20
10Dic5
10Dic5
10C2×C10
25SD16
10Dic10
10C5⋊D4
10C5⋊C8

Character table of C52⋊SD16

 class 1 2A 2B 4A 4B 5A 5B 5C 5D 5E 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 20A 20B 20C 20D size 1 1 20 20 50 4 4 4 4 8 50 50 4 4 4 4 8 20 20 20 20 20 20 20 20 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 2 2 0 0 -2 2 2 2 2 2 0 0 2 2 2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ6 2 -2 0 0 0 2 2 2 2 2 √-2 -√-2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ7 2 -2 0 0 0 2 2 2 2 2 -√-2 √-2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ8 4 4 2 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 0 0 0 0 orthogonal lifted from D5≀C2 ρ9 4 4 -2 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 1+√5/2 1+√5/2 1-√5/2 1-√5/2 0 0 0 0 orthogonal lifted from D5≀C2 ρ10 4 4 0 -2 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 0 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 0 0 0 0 1-√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from D5≀C2 ρ11 4 4 0 2 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 0 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 0 0 0 0 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5≀C2 ρ12 4 4 0 2 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 0 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 0 0 0 0 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5≀C2 ρ13 4 4 0 -2 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 0 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 0 0 0 0 1+√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from D5≀C2 ρ14 4 4 -2 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 1-√5/2 1-√5/2 1+√5/2 1+√5/2 0 0 0 0 orthogonal lifted from D5≀C2 ρ15 4 4 2 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 0 0 0 0 orthogonal lifted from D5≀C2 ρ16 4 -4 0 0 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 0 0 -3+√5/2 -3-√5/2 1-√5 1+√5 1 0 0 0 0 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 symplectic faithful, Schur index 2 ρ17 4 -4 0 0 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 0 0 -3-√5/2 -3+√5/2 1+√5 1-√5 1 0 0 0 0 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 symplectic faithful, Schur index 2 ρ18 4 -4 0 0 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 0 0 -3+√5/2 -3-√5/2 1-√5 1+√5 1 0 0 0 0 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 symplectic faithful, Schur index 2 ρ19 4 -4 0 0 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 0 0 -3-√5/2 -3+√5/2 1+√5 1-√5 1 0 0 0 0 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 symplectic faithful, Schur index 2 ρ20 4 -4 0 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 0 0 1-√5 1+√5 -3-√5/2 -3+√5/2 1 -ζ54+ζ5 ζ54-ζ5 -ζ53+ζ52 ζ53-ζ52 0 0 0 0 complex faithful ρ21 4 -4 0 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 0 0 1+√5 1-√5 -3+√5/2 -3-√5/2 1 ζ53-ζ52 -ζ53+ζ52 -ζ54+ζ5 ζ54-ζ5 0 0 0 0 complex faithful ρ22 4 -4 0 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 0 0 1+√5 1-√5 -3+√5/2 -3-√5/2 1 -ζ53+ζ52 ζ53-ζ52 ζ54-ζ5 -ζ54+ζ5 0 0 0 0 complex faithful ρ23 4 -4 0 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 0 0 1-√5 1+√5 -3-√5/2 -3+√5/2 1 ζ54-ζ5 -ζ54+ζ5 ζ53-ζ52 -ζ53+ζ52 0 0 0 0 complex faithful ρ24 8 8 0 0 0 -2 -2 -2 -2 3 0 0 -2 -2 -2 -2 3 0 0 0 0 0 0 0 0 orthogonal lifted from D5≀C2 ρ25 8 -8 0 0 0 -2 -2 -2 -2 3 0 0 2 2 2 2 -3 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C52⋊SD16
On 40 points
Generators in S40
(1 18 35 29 12)(2 30 19 13 36)(3 14 31 37 20)(4 38 15 21 32)(5 22 39 25 16)(6 26 23 9 40)(7 10 27 33 24)(8 34 11 17 28)
(1 35 12 18 29)(2 13 30 36 19)(3 31 20 14 37)(4 21 38 32 15)(5 39 16 22 25)(6 9 26 40 23)(7 27 24 10 33)(8 17 34 28 11)
(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 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 23)(19 21)(20 24)(26 28)(27 31)(30 32)(33 37)(34 40)(36 38)

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

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

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

Matrix representation of C52⋊SD16 in GL4(𝔽41) generated by

 18 0 0 0 0 16 0 0 0 0 37 0 0 0 0 10
,
 37 0 0 0 0 10 0 0 0 0 18 0 0 0 0 16
,
 0 0 1 0 0 0 0 40 0 40 0 0 40 0 0 0
,
 1 0 0 0 0 40 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(41))| [18,0,0,0,0,16,0,0,0,0,37,0,0,0,0,10],[37,0,0,0,0,10,0,0,0,0,18,0,0,0,0,16],[0,0,0,40,0,0,40,0,1,0,0,0,0,40,0,0],[1,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0] >;

C52⋊SD16 in GAP, Magma, Sage, TeX

C_5^2\rtimes {\rm SD}_{16}
% in TeX

G:=Group("C5^2:SD16");
// GroupNames label

G:=SmallGroup(400,132);
// by ID

G=gap.SmallGroup(400,132);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,5,73,55,218,116,50,7204,1210,496,1157,299,2897]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=a^3,d*a*d=b^3,c*b*c^-1=b^2,d*b*d=a^2,d*c*d=c^3>;
// generators/relations

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