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## G = C32⋊Dic5order 180 = 22·32·5

### The semidirect product of C32 and Dic5 acting via Dic5/C5=C4

Aliases: C32⋊Dic5, C3⋊S3.D5, (C3×C15)⋊6C4, C52(C32⋊C4), (C5×C3⋊S3).2C2, SmallGroup(180,24)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C32⋊Dic5
G = < a,b,c,d | a3=b3=c10=1, d2=c5, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=b-1, dbd-1=a-1b-1, dcd-1=c-1 >

Character table of C32⋊Dic5

 class 1 2 3A 3B 4A 4B 5A 5B 10A 10B 15A 15B 15C 15D 15E 15F 15G 15H size 1 9 4 4 45 45 2 2 18 18 4 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 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 i -i 1 1 -1 -1 1 1 1 1 1 1 1 1 linear of order 4 ρ4 1 -1 1 1 -i i 1 1 -1 -1 1 1 1 1 1 1 1 1 linear of order 4 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 D5 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 symplectic lifted from Dic5, Schur index 2 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 symplectic lifted from Dic5, Schur index 2 ρ9 4 0 -2 1 0 0 4 4 0 0 1 -2 -2 1 -2 -2 1 1 orthogonal lifted from C32⋊C4 ρ10 4 0 1 -2 0 0 4 4 0 0 -2 1 1 -2 1 1 -2 -2 orthogonal lifted from C32⋊C4 ρ11 4 0 1 -2 0 0 -1-√5 -1+√5 0 0 1+√5/2 -ζ53+2ζ52 -ζ54+2ζ5 1-√5/2 2ζ54-ζ5 2ζ53-ζ52 1-√5/2 1+√5/2 complex faithful ρ12 4 0 -2 1 0 0 -1-√5 -1+√5 0 0 -ζ53+2ζ52 1+√5/2 1-√5/2 -ζ54+2ζ5 1-√5/2 1+√5/2 2ζ54-ζ5 2ζ53-ζ52 complex faithful ρ13 4 0 1 -2 0 0 -1+√5 -1-√5 0 0 1-√5/2 -ζ54+2ζ5 2ζ53-ζ52 1+√5/2 -ζ53+2ζ52 2ζ54-ζ5 1+√5/2 1-√5/2 complex faithful ρ14 4 0 -2 1 0 0 -1-√5 -1+√5 0 0 2ζ53-ζ52 1+√5/2 1-√5/2 2ζ54-ζ5 1-√5/2 1+√5/2 -ζ54+2ζ5 -ζ53+2ζ52 complex faithful ρ15 4 0 1 -2 0 0 -1+√5 -1-√5 0 0 1-√5/2 2ζ54-ζ5 -ζ53+2ζ52 1+√5/2 2ζ53-ζ52 -ζ54+2ζ5 1+√5/2 1-√5/2 complex faithful ρ16 4 0 1 -2 0 0 -1-√5 -1+√5 0 0 1+√5/2 2ζ53-ζ52 2ζ54-ζ5 1-√5/2 -ζ54+2ζ5 -ζ53+2ζ52 1-√5/2 1+√5/2 complex faithful ρ17 4 0 -2 1 0 0 -1+√5 -1-√5 0 0 -ζ54+2ζ5 1-√5/2 1+√5/2 2ζ53-ζ52 1+√5/2 1-√5/2 -ζ53+2ζ52 2ζ54-ζ5 complex faithful ρ18 4 0 -2 1 0 0 -1+√5 -1-√5 0 0 2ζ54-ζ5 1-√5/2 1+√5/2 -ζ53+2ζ52 1+√5/2 1-√5/2 2ζ53-ζ52 -ζ54+2ζ5 complex faithful

Permutation representations of C32⋊Dic5
On 30 points - transitive group 30T48
Generators in S30
(1 21 26)(2 27 22)(3 23 28)(4 29 24)(5 25 30)(6 20 15)(7 16 11)(8 12 17)(9 18 13)(10 14 19)
(6 15 20)(7 11 16)(8 17 12)(9 13 18)(10 19 14)
(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)
(1 6)(2 10)(3 9)(4 8)(5 7)(11 25 16 30)(12 24 17 29)(13 23 18 28)(14 22 19 27)(15 21 20 26)

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

G:=Group( (1,21,26)(2,27,22)(3,23,28)(4,29,24)(5,25,30)(6,20,15)(7,16,11)(8,12,17)(9,18,13)(10,14,19), (6,15,20)(7,11,16)(8,17,12)(9,13,18)(10,19,14), (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), (1,6)(2,10)(3,9)(4,8)(5,7)(11,25,16,30)(12,24,17,29)(13,23,18,28)(14,22,19,27)(15,21,20,26) );

G=PermutationGroup([[(1,21,26),(2,27,22),(3,23,28),(4,29,24),(5,25,30),(6,20,15),(7,16,11),(8,12,17),(9,18,13),(10,14,19)], [(6,15,20),(7,11,16),(8,17,12),(9,13,18),(10,19,14)], [(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)], [(1,6),(2,10),(3,9),(4,8),(5,7),(11,25,16,30),(12,24,17,29),(13,23,18,28),(14,22,19,27),(15,21,20,26)]])

G:=TransitiveGroup(30,48);

C32⋊Dic5 is a maximal subgroup of   C5⋊F9  D5×C32⋊C4  S32⋊D5  C32⋊Dic10
C32⋊Dic5 is a maximal quotient of   (C3×C15)⋊9C8

Matrix representation of C32⋊Dic5 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 60 60
,
 0 1 0 0 60 60 0 0 0 0 0 1 0 0 60 60
,
 34 0 0 0 27 27 0 0 0 0 9 0 0 0 52 52
,
 0 0 1 0 0 0 0 1 1 0 0 0 60 60 0 0
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,60],[0,60,0,0,1,60,0,0,0,0,0,60,0,0,1,60],[34,27,0,0,0,27,0,0,0,0,9,52,0,0,0,52],[0,0,1,60,0,0,0,60,1,0,0,0,0,1,0,0] >;

C32⋊Dic5 in GAP, Magma, Sage, TeX

C_3^2\rtimes {\rm Dic}_5
% in TeX

G:=Group("C3^2:Dic5");
// GroupNames label

G:=SmallGroup(180,24);
// by ID

G=gap.SmallGroup(180,24);
# by ID

G:=PCGroup([5,-2,-2,-3,3,-5,10,302,67,323,248,3604]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^10=1,d^2=c^5,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations

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