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## G = C32⋊Dic10order 360 = 23·32·5

### The semidirect product of C32 and Dic10 acting via Dic10/C5=Q8

Aliases: C32⋊Dic10, C5⋊PSU3(𝔽2), (C3×C15)⋊2Q8, C3⋊S3.3D10, C32⋊C4.2D5, C32⋊Dic5.2C2, (C5×C32⋊C4).2C2, (C5×C3⋊S3).6C22, SmallGroup(360,136)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C5×C3⋊S3 — C32⋊Dic10
 Chief series C1 — C5 — C3×C15 — C5×C3⋊S3 — C32⋊Dic5 — C32⋊Dic10
 Lower central C3×C15 — C5×C3⋊S3 — C32⋊Dic10
 Upper central C1

Generators and relations for C32⋊Dic10
G = < a,b,c,d | a3=b3=c20=1, d2=c10, ab=ba, cac-1=b, dad-1=ab-1, cbc-1=a-1, dbd-1=a-1b-1, dcd-1=c-1 >

Character table of C32⋊Dic10

 class 1 2 3 4A 4B 4C 5A 5B 10A 10B 15A 15B 15C 15D 20A 20B 20C 20D size 1 9 8 18 90 90 2 2 18 18 8 8 8 8 18 18 18 18 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 2 -2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 orthogonal lifted from D10 ρ6 2 2 2 2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ7 2 2 2 -2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 orthogonal lifted from D10 ρ8 2 2 2 2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ9 2 -2 2 0 0 0 2 2 -2 -2 2 2 2 2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ10 2 -2 2 0 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 -ζ43ζ54+ζ43ζ5 symplectic lifted from Dic10, Schur index 2 ρ11 2 -2 2 0 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 symplectic lifted from Dic10, Schur index 2 ρ12 2 -2 2 0 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 ζ43ζ54-ζ43ζ5 symplectic lifted from Dic10, Schur index 2 ρ13 2 -2 2 0 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 symplectic lifted from Dic10, Schur index 2 ρ14 8 0 -1 0 0 0 8 8 0 0 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from PSU3(𝔽2) ρ15 8 0 -1 0 0 0 -2+2√5 -2-2√5 0 0 -2ζ53+ζ52 -2ζ54+ζ5 ζ54-2ζ5 ζ53-2ζ52 0 0 0 0 complex faithful ρ16 8 0 -1 0 0 0 -2-2√5 -2+2√5 0 0 -2ζ54+ζ5 ζ53-2ζ52 -2ζ53+ζ52 ζ54-2ζ5 0 0 0 0 complex faithful ρ17 8 0 -1 0 0 0 -2+2√5 -2-2√5 0 0 ζ53-2ζ52 ζ54-2ζ5 -2ζ54+ζ5 -2ζ53+ζ52 0 0 0 0 complex faithful ρ18 8 0 -1 0 0 0 -2-2√5 -2+2√5 0 0 ζ54-2ζ5 -2ζ53+ζ52 ζ53-2ζ52 -2ζ54+ζ5 0 0 0 0 complex faithful

Smallest permutation representation of C32⋊Dic10
On 45 points
Generators in S45
(1 42 32)(2 38 28)(3 34 44)(4 30 40)(5 26 36)(6 39 21)(7 45 12)(8 23 41)(9 14 27)(10 43 25)(11 29 16)(13 18 31)(15 33 20)(17 22 35)(19 37 24)
(1 37 27)(2 33 43)(3 29 39)(4 45 35)(5 41 31)(6 44 11)(7 22 40)(8 13 26)(9 42 24)(10 28 15)(12 17 30)(14 32 19)(16 21 34)(18 36 23)(20 25 38)
(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)
(2 5)(3 4)(6 45 16 35)(7 44 17 34)(8 43 18 33)(9 42 19 32)(10 41 20 31)(11 40 21 30)(12 39 22 29)(13 38 23 28)(14 37 24 27)(15 36 25 26)

G:=sub<Sym(45)| (1,42,32)(2,38,28)(3,34,44)(4,30,40)(5,26,36)(6,39,21)(7,45,12)(8,23,41)(9,14,27)(10,43,25)(11,29,16)(13,18,31)(15,33,20)(17,22,35)(19,37,24), (1,37,27)(2,33,43)(3,29,39)(4,45,35)(5,41,31)(6,44,11)(7,22,40)(8,13,26)(9,42,24)(10,28,15)(12,17,30)(14,32,19)(16,21,34)(18,36,23)(20,25,38), (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), (2,5)(3,4)(6,45,16,35)(7,44,17,34)(8,43,18,33)(9,42,19,32)(10,41,20,31)(11,40,21,30)(12,39,22,29)(13,38,23,28)(14,37,24,27)(15,36,25,26)>;

G:=Group( (1,42,32)(2,38,28)(3,34,44)(4,30,40)(5,26,36)(6,39,21)(7,45,12)(8,23,41)(9,14,27)(10,43,25)(11,29,16)(13,18,31)(15,33,20)(17,22,35)(19,37,24), (1,37,27)(2,33,43)(3,29,39)(4,45,35)(5,41,31)(6,44,11)(7,22,40)(8,13,26)(9,42,24)(10,28,15)(12,17,30)(14,32,19)(16,21,34)(18,36,23)(20,25,38), (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), (2,5)(3,4)(6,45,16,35)(7,44,17,34)(8,43,18,33)(9,42,19,32)(10,41,20,31)(11,40,21,30)(12,39,22,29)(13,38,23,28)(14,37,24,27)(15,36,25,26) );

G=PermutationGroup([[(1,42,32),(2,38,28),(3,34,44),(4,30,40),(5,26,36),(6,39,21),(7,45,12),(8,23,41),(9,14,27),(10,43,25),(11,29,16),(13,18,31),(15,33,20),(17,22,35),(19,37,24)], [(1,37,27),(2,33,43),(3,29,39),(4,45,35),(5,41,31),(6,44,11),(7,22,40),(8,13,26),(9,42,24),(10,28,15),(12,17,30),(14,32,19),(16,21,34),(18,36,23),(20,25,38)], [(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)], [(2,5),(3,4),(6,45,16,35),(7,44,17,34),(8,43,18,33),(9,42,19,32),(10,41,20,31),(11,40,21,30),(12,39,22,29),(13,38,23,28),(14,37,24,27),(15,36,25,26)]])

Matrix representation of C32⋊Dic10 in GL8(𝔽61)

 1 0 0 0 0 26 11 0 0 1 0 0 0 35 26 0 0 0 0 0 0 60 0 1 0 0 41 1 0 41 0 20 0 0 1 0 1 51 37 60 27 51 60 0 60 59 0 0 11 59 3 0 3 0 59 0 27 51 51 37 60 60 0 60
,
 1 0 26 11 0 0 0 0 0 1 35 26 0 0 0 0 0 0 60 0 1 0 0 0 31 29 58 59 58 0 0 0 0 0 60 0 0 0 0 0 0 0 60 0 0 0 0 1 0 0 41 0 0 41 1 20 0 0 0 0 1 52 37 60
,
 17 1 0 0 0 0 0 0 60 0 0 0 0 0 0 0 11 59 3 0 3 0 60 0 38 37 52 0 52 1 44 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 60 17 0 0 0 0 11 59 47 9 3 20 60 41
,
 17 44 0 0 0 0 0 0 60 44 0 0 0 0 0 0 15 26 9 24 0 9 24 1 26 0 14 51 0 14 51 58 0 0 0 0 0 0 0 1 0 0 1 0 1 52 37 60 30 32 0 1 20 47 10 3 0 0 1 0 0 0 0 0

G:=sub<GL(8,GF(61))| [1,0,0,0,0,27,11,27,0,1,0,0,0,51,59,51,0,0,0,41,1,60,3,51,0,0,0,1,0,0,0,37,0,0,0,0,1,60,3,60,26,35,60,41,51,59,0,60,11,26,0,0,37,0,59,0,0,0,1,20,60,0,0,60],[1,0,0,31,0,0,0,0,0,1,0,29,0,0,0,0,26,35,60,58,60,60,41,0,11,26,0,59,0,0,0,0,0,0,1,58,0,0,0,1,0,0,0,0,0,0,41,52,0,0,0,0,0,0,1,37,0,0,0,0,0,1,20,60],[17,60,11,38,0,0,0,11,1,0,59,37,0,0,0,59,0,0,3,52,0,0,60,47,0,0,0,0,0,1,17,9,0,0,3,52,0,0,0,3,0,0,0,1,0,0,0,20,0,0,60,44,1,0,0,60,0,0,0,0,0,0,0,41],[17,60,15,26,0,0,30,0,44,44,26,0,0,0,32,0,0,0,9,14,0,1,0,1,0,0,24,51,0,0,1,0,0,0,0,0,0,1,20,0,0,0,9,14,0,52,47,0,0,0,24,51,0,37,10,0,0,0,1,58,1,60,3,0] >;

C32⋊Dic10 in GAP, Magma, Sage, TeX

C_3^2\rtimes {\rm Dic}_{10}
% in TeX

G:=Group("C3^2:Dic10");
// GroupNames label

G:=SmallGroup(360,136);
// by ID

G=gap.SmallGroup(360,136);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,3,-5,24,73,31,963,585,111,964,130,376,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^20=1,d^2=c^10,a*b=b*a,c*a*c^-1=b,d*a*d^-1=a*b^-1,c*b*c^-1=a^-1,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations

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