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## G = C5×C32⋊C4order 180 = 22·32·5

### Direct product of C5 and C32⋊C4

Aliases: C5×C32⋊C4, C32⋊C20, C3⋊S3.C10, (C3×C15)⋊5C4, (C5×C3⋊S3).1C2, SmallGroup(180,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C5×C32⋊C4
 Chief series C1 — C32 — C3⋊S3 — C5×C3⋊S3 — C5×C32⋊C4
 Lower central C32 — C5×C32⋊C4
 Upper central C1 — C5

Generators and relations for C5×C32⋊C4
G = < a,b,c,d | a5=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

Character table of C5×C32⋊C4

 class 1 2 3A 3B 4A 4B 5A 5B 5C 5D 10A 10B 10C 10D 15A 15B 15C 15D 15E 15F 15G 15H 20A 20B 20C 20D 20E 20F 20G 20H size 1 9 4 4 9 9 1 1 1 1 9 9 9 9 4 4 4 4 4 4 4 4 9 9 9 9 9 9 9 9 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 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 -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 1 1 1 1 i i -i i -i i -i -i linear of order 4 ρ4 1 -1 1 1 -i i 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -i -i i -i i -i i i linear of order 4 ρ5 1 1 1 1 1 1 ζ52 ζ54 ζ5 ζ53 ζ52 ζ5 ζ53 ζ54 ζ52 ζ53 ζ53 ζ5 ζ5 ζ54 ζ54 ζ52 ζ5 ζ54 ζ53 ζ53 ζ52 ζ52 ζ5 ζ54 linear of order 5 ρ6 1 1 1 1 1 1 ζ53 ζ5 ζ54 ζ52 ζ53 ζ54 ζ52 ζ5 ζ53 ζ52 ζ52 ζ54 ζ54 ζ5 ζ5 ζ53 ζ54 ζ5 ζ52 ζ52 ζ53 ζ53 ζ54 ζ5 linear of order 5 ρ7 1 1 1 1 1 1 ζ54 ζ53 ζ52 ζ5 ζ54 ζ52 ζ5 ζ53 ζ54 ζ5 ζ5 ζ52 ζ52 ζ53 ζ53 ζ54 ζ52 ζ53 ζ5 ζ5 ζ54 ζ54 ζ52 ζ53 linear of order 5 ρ8 1 1 1 1 -1 -1 ζ53 ζ5 ζ54 ζ52 ζ53 ζ54 ζ52 ζ5 ζ53 ζ52 ζ52 ζ54 ζ54 ζ5 ζ5 ζ53 -ζ54 -ζ5 -ζ52 -ζ52 -ζ53 -ζ53 -ζ54 -ζ5 linear of order 10 ρ9 1 1 1 1 -1 -1 ζ54 ζ53 ζ52 ζ5 ζ54 ζ52 ζ5 ζ53 ζ54 ζ5 ζ5 ζ52 ζ52 ζ53 ζ53 ζ54 -ζ52 -ζ53 -ζ5 -ζ5 -ζ54 -ζ54 -ζ52 -ζ53 linear of order 10 ρ10 1 1 1 1 -1 -1 ζ52 ζ54 ζ5 ζ53 ζ52 ζ5 ζ53 ζ54 ζ52 ζ53 ζ53 ζ5 ζ5 ζ54 ζ54 ζ52 -ζ5 -ζ54 -ζ53 -ζ53 -ζ52 -ζ52 -ζ5 -ζ54 linear of order 10 ρ11 1 1 1 1 -1 -1 ζ5 ζ52 ζ53 ζ54 ζ5 ζ53 ζ54 ζ52 ζ5 ζ54 ζ54 ζ53 ζ53 ζ52 ζ52 ζ5 -ζ53 -ζ52 -ζ54 -ζ54 -ζ5 -ζ5 -ζ53 -ζ52 linear of order 10 ρ12 1 1 1 1 1 1 ζ5 ζ52 ζ53 ζ54 ζ5 ζ53 ζ54 ζ52 ζ5 ζ54 ζ54 ζ53 ζ53 ζ52 ζ52 ζ5 ζ53 ζ52 ζ54 ζ54 ζ5 ζ5 ζ53 ζ52 linear of order 5 ρ13 1 -1 1 1 -i i ζ53 ζ5 ζ54 ζ52 -ζ53 -ζ54 -ζ52 -ζ5 ζ53 ζ52 ζ52 ζ54 ζ54 ζ5 ζ5 ζ53 ζ43ζ54 ζ43ζ5 ζ4ζ52 ζ43ζ52 ζ4ζ53 ζ43ζ53 ζ4ζ54 ζ4ζ5 linear of order 20 ρ14 1 -1 1 1 -i i ζ52 ζ54 ζ5 ζ53 -ζ52 -ζ5 -ζ53 -ζ54 ζ52 ζ53 ζ53 ζ5 ζ5 ζ54 ζ54 ζ52 ζ43ζ5 ζ43ζ54 ζ4ζ53 ζ43ζ53 ζ4ζ52 ζ43ζ52 ζ4ζ5 ζ4ζ54 linear of order 20 ρ15 1 -1 1 1 i -i ζ53 ζ5 ζ54 ζ52 -ζ53 -ζ54 -ζ52 -ζ5 ζ53 ζ52 ζ52 ζ54 ζ54 ζ5 ζ5 ζ53 ζ4ζ54 ζ4ζ5 ζ43ζ52 ζ4ζ52 ζ43ζ53 ζ4ζ53 ζ43ζ54 ζ43ζ5 linear of order 20 ρ16 1 -1 1 1 i -i ζ52 ζ54 ζ5 ζ53 -ζ52 -ζ5 -ζ53 -ζ54 ζ52 ζ53 ζ53 ζ5 ζ5 ζ54 ζ54 ζ52 ζ4ζ5 ζ4ζ54 ζ43ζ53 ζ4ζ53 ζ43ζ52 ζ4ζ52 ζ43ζ5 ζ43ζ54 linear of order 20 ρ17 1 -1 1 1 i -i ζ5 ζ52 ζ53 ζ54 -ζ5 -ζ53 -ζ54 -ζ52 ζ5 ζ54 ζ54 ζ53 ζ53 ζ52 ζ52 ζ5 ζ4ζ53 ζ4ζ52 ζ43ζ54 ζ4ζ54 ζ43ζ5 ζ4ζ5 ζ43ζ53 ζ43ζ52 linear of order 20 ρ18 1 -1 1 1 -i i ζ54 ζ53 ζ52 ζ5 -ζ54 -ζ52 -ζ5 -ζ53 ζ54 ζ5 ζ5 ζ52 ζ52 ζ53 ζ53 ζ54 ζ43ζ52 ζ43ζ53 ζ4ζ5 ζ43ζ5 ζ4ζ54 ζ43ζ54 ζ4ζ52 ζ4ζ53 linear of order 20 ρ19 1 -1 1 1 i -i ζ54 ζ53 ζ52 ζ5 -ζ54 -ζ52 -ζ5 -ζ53 ζ54 ζ5 ζ5 ζ52 ζ52 ζ53 ζ53 ζ54 ζ4ζ52 ζ4ζ53 ζ43ζ5 ζ4ζ5 ζ43ζ54 ζ4ζ54 ζ43ζ52 ζ43ζ53 linear of order 20 ρ20 1 -1 1 1 -i i ζ5 ζ52 ζ53 ζ54 -ζ5 -ζ53 -ζ54 -ζ52 ζ5 ζ54 ζ54 ζ53 ζ53 ζ52 ζ52 ζ5 ζ43ζ53 ζ43ζ52 ζ4ζ54 ζ43ζ54 ζ4ζ5 ζ43ζ5 ζ4ζ53 ζ4ζ52 linear of order 20 ρ21 4 0 1 -2 0 0 4 4 4 4 0 0 0 0 1 -2 1 -2 1 -2 1 -2 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C4 ρ22 4 0 -2 1 0 0 4 4 4 4 0 0 0 0 -2 1 -2 1 -2 1 -2 1 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C4 ρ23 4 0 -2 1 0 0 4ζ54 4ζ53 4ζ52 4ζ5 0 0 0 0 -2ζ54 ζ5 -2ζ5 ζ52 -2ζ52 ζ53 -2ζ53 ζ54 0 0 0 0 0 0 0 0 complex faithful ρ24 4 0 -2 1 0 0 4ζ52 4ζ54 4ζ5 4ζ53 0 0 0 0 -2ζ52 ζ53 -2ζ53 ζ5 -2ζ5 ζ54 -2ζ54 ζ52 0 0 0 0 0 0 0 0 complex faithful ρ25 4 0 1 -2 0 0 4ζ53 4ζ5 4ζ54 4ζ52 0 0 0 0 ζ53 -2ζ52 ζ52 -2ζ54 ζ54 -2ζ5 ζ5 -2ζ53 0 0 0 0 0 0 0 0 complex faithful ρ26 4 0 1 -2 0 0 4ζ54 4ζ53 4ζ52 4ζ5 0 0 0 0 ζ54 -2ζ5 ζ5 -2ζ52 ζ52 -2ζ53 ζ53 -2ζ54 0 0 0 0 0 0 0 0 complex faithful ρ27 4 0 1 -2 0 0 4ζ52 4ζ54 4ζ5 4ζ53 0 0 0 0 ζ52 -2ζ53 ζ53 -2ζ5 ζ5 -2ζ54 ζ54 -2ζ52 0 0 0 0 0 0 0 0 complex faithful ρ28 4 0 -2 1 0 0 4ζ5 4ζ52 4ζ53 4ζ54 0 0 0 0 -2ζ5 ζ54 -2ζ54 ζ53 -2ζ53 ζ52 -2ζ52 ζ5 0 0 0 0 0 0 0 0 complex faithful ρ29 4 0 -2 1 0 0 4ζ53 4ζ5 4ζ54 4ζ52 0 0 0 0 -2ζ53 ζ52 -2ζ52 ζ54 -2ζ54 ζ5 -2ζ5 ζ53 0 0 0 0 0 0 0 0 complex faithful ρ30 4 0 1 -2 0 0 4ζ5 4ζ52 4ζ53 4ζ54 0 0 0 0 ζ5 -2ζ54 ζ54 -2ζ53 ζ53 -2ζ52 ζ52 -2ζ5 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C5×C32⋊C4
On 30 points - transitive group 30T49
Generators in S30
(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 17 23)(2 18 24)(3 19 25)(4 20 21)(5 16 22)(6 29 15)(7 30 11)(8 26 12)(9 27 13)(10 28 14)
(6 15 29)(7 11 30)(8 12 26)(9 13 27)(10 14 28)
(1 8)(2 9)(3 10)(4 6)(5 7)(11 22 30 16)(12 23 26 17)(13 24 27 18)(14 25 28 19)(15 21 29 20)

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

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

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

G:=TransitiveGroup(30,49);

C5×C32⋊C4 is a maximal subgroup of   C52F9  C32⋊D20  C32⋊Dic10

Matrix representation of C5×C32⋊C4 in GL4(𝔽61) generated by

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

C5×C32⋊C4 in GAP, Magma, Sage, TeX

C_5\times C_3^2\rtimes C_4
% in TeX

G:=Group("C5xC3^2:C4");
// GroupNames label

G:=SmallGroup(180,23);
// by ID

G=gap.SmallGroup(180,23);
# by ID

G:=PCGroup([5,-2,-5,-2,-3,3,50,2803,93,4004,314]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations

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