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## G = (C4×C20)⋊C6order 480 = 25·3·5

### 1st semidirect product of C4×C20 and C6 acting faithfully

Aliases: (C4×C20)⋊1C6, C422D5⋊C3, C42⋊C31D5, C5⋊(C42⋊C6), C421(C3×D5), C22.2(D5×A4), (C22×D5).2A4, (C5×C42⋊C3)⋊1C2, (C2×C10).2(C2×A4), SmallGroup(480,263)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4×C20 — (C4×C20)⋊C6
 Chief series C1 — C22 — C2×C10 — C4×C20 — C5×C42⋊C3 — (C4×C20)⋊C6
 Lower central C4×C20 — (C4×C20)⋊C6
 Upper central C1

Generators and relations for (C4×C20)⋊C6
G = < a,b,c | a4=b20=c6=1, ab=ba, cac-1=a-1b5, cbc-1=a-1b14 >

Character table of (C4×C20)⋊C6

 class 1 2A 2B 3A 3B 4A 4B 4C 5A 5B 6A 6B 10A 10B 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 20G 20H size 1 3 20 16 16 6 6 60 2 2 80 80 6 6 32 32 32 32 6 6 6 6 6 6 6 6 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 ζ32 ζ3 1 1 -1 1 1 ζ65 ζ6 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 1 1 linear of order 6 ρ4 1 1 1 ζ32 ζ3 1 1 1 1 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 1 1 linear of order 3 ρ5 1 1 -1 ζ3 ζ32 1 1 -1 1 1 ζ6 ζ65 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 1 linear of order 6 ρ6 1 1 1 ζ3 ζ32 1 1 1 1 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 1 linear of order 3 ρ7 2 2 0 2 2 2 2 0 -1-√5/2 -1+√5/2 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 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ8 2 2 0 2 2 2 2 0 -1+√5/2 -1-√5/2 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 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ9 2 2 0 -1-√-3 -1+√-3 2 2 0 -1-√5/2 -1+√5/2 0 0 -1+√5/2 -1-√5/2 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 complex lifted from C3×D5 ρ10 2 2 0 -1+√-3 -1-√-3 2 2 0 -1-√5/2 -1+√5/2 0 0 -1+√5/2 -1-√5/2 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 complex lifted from C3×D5 ρ11 2 2 0 -1-√-3 -1+√-3 2 2 0 -1+√5/2 -1-√5/2 0 0 -1-√5/2 -1+√5/2 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 complex lifted from C3×D5 ρ12 2 2 0 -1+√-3 -1-√-3 2 2 0 -1+√5/2 -1-√5/2 0 0 -1-√5/2 -1+√5/2 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 complex lifted from C3×D5 ρ13 3 3 3 0 0 -1 -1 -1 3 3 0 0 3 3 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from A4 ρ14 3 3 -3 0 0 -1 -1 1 3 3 0 0 3 3 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from C2×A4 ρ15 6 6 0 0 0 -2 -2 0 -3+3√5/2 -3-3√5/2 0 0 -3-3√5/2 -3+3√5/2 0 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 orthogonal lifted from D5×A4 ρ16 6 6 0 0 0 -2 -2 0 -3-3√5/2 -3+3√5/2 0 0 -3+3√5/2 -3-3√5/2 0 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 orthogonal lifted from D5×A4 ρ17 6 -2 0 0 0 -2i 2i 0 6 6 0 0 -2 -2 0 0 0 0 2i -2i -2i 2i 2i -2i -2i 2i complex lifted from C42⋊C6 ρ18 6 -2 0 0 0 2i -2i 0 6 6 0 0 -2 -2 0 0 0 0 -2i 2i 2i -2i -2i 2i 2i -2i complex lifted from C42⋊C6 ρ19 6 -2 0 0 0 -2i 2i 0 -3-3√5/2 -3+3√5/2 0 0 1-√5/2 1+√5/2 0 0 0 0 2ζ4ζ54-ζ54+ζ5 2ζ43ζ5+ζ54-ζ5 2ζ43ζ54-ζ54+ζ5 2ζ4ζ5+ζ54-ζ5 2ζ4ζ52+ζ53-ζ52 2ζ43ζ53-ζ53+ζ52 2ζ43ζ52+ζ53-ζ52 2ζ4ζ53-ζ53+ζ52 complex faithful ρ20 6 -2 0 0 0 -2i 2i 0 -3+3√5/2 -3-3√5/2 0 0 1+√5/2 1-√5/2 0 0 0 0 2ζ4ζ52+ζ53-ζ52 2ζ43ζ53-ζ53+ζ52 2ζ43ζ52+ζ53-ζ52 2ζ4ζ53-ζ53+ζ52 2ζ4ζ5+ζ54-ζ5 2ζ43ζ54-ζ54+ζ5 2ζ43ζ5+ζ54-ζ5 2ζ4ζ54-ζ54+ζ5 complex faithful ρ21 6 -2 0 0 0 2i -2i 0 -3+3√5/2 -3-3√5/2 0 0 1+√5/2 1-√5/2 0 0 0 0 2ζ43ζ53-ζ53+ζ52 2ζ4ζ52+ζ53-ζ52 2ζ4ζ53-ζ53+ζ52 2ζ43ζ52+ζ53-ζ52 2ζ43ζ54-ζ54+ζ5 2ζ4ζ5+ζ54-ζ5 2ζ4ζ54-ζ54+ζ5 2ζ43ζ5+ζ54-ζ5 complex faithful ρ22 6 -2 0 0 0 -2i 2i 0 -3-3√5/2 -3+3√5/2 0 0 1-√5/2 1+√5/2 0 0 0 0 2ζ4ζ5+ζ54-ζ5 2ζ43ζ54-ζ54+ζ5 2ζ43ζ5+ζ54-ζ5 2ζ4ζ54-ζ54+ζ5 2ζ4ζ53-ζ53+ζ52 2ζ43ζ52+ζ53-ζ52 2ζ43ζ53-ζ53+ζ52 2ζ4ζ52+ζ53-ζ52 complex faithful ρ23 6 -2 0 0 0 2i -2i 0 -3-3√5/2 -3+3√5/2 0 0 1-√5/2 1+√5/2 0 0 0 0 2ζ43ζ5+ζ54-ζ5 2ζ4ζ54-ζ54+ζ5 2ζ4ζ5+ζ54-ζ5 2ζ43ζ54-ζ54+ζ5 2ζ43ζ53-ζ53+ζ52 2ζ4ζ52+ζ53-ζ52 2ζ4ζ53-ζ53+ζ52 2ζ43ζ52+ζ53-ζ52 complex faithful ρ24 6 -2 0 0 0 2i -2i 0 -3-3√5/2 -3+3√5/2 0 0 1-√5/2 1+√5/2 0 0 0 0 2ζ43ζ54-ζ54+ζ5 2ζ4ζ5+ζ54-ζ5 2ζ4ζ54-ζ54+ζ5 2ζ43ζ5+ζ54-ζ5 2ζ43ζ52+ζ53-ζ52 2ζ4ζ53-ζ53+ζ52 2ζ4ζ52+ζ53-ζ52 2ζ43ζ53-ζ53+ζ52 complex faithful ρ25 6 -2 0 0 0 -2i 2i 0 -3+3√5/2 -3-3√5/2 0 0 1+√5/2 1-√5/2 0 0 0 0 2ζ4ζ53-ζ53+ζ52 2ζ43ζ52+ζ53-ζ52 2ζ43ζ53-ζ53+ζ52 2ζ4ζ52+ζ53-ζ52 2ζ4ζ54-ζ54+ζ5 2ζ43ζ5+ζ54-ζ5 2ζ43ζ54-ζ54+ζ5 2ζ4ζ5+ζ54-ζ5 complex faithful ρ26 6 -2 0 0 0 2i -2i 0 -3+3√5/2 -3-3√5/2 0 0 1+√5/2 1-√5/2 0 0 0 0 2ζ43ζ52+ζ53-ζ52 2ζ4ζ53-ζ53+ζ52 2ζ4ζ52+ζ53-ζ52 2ζ43ζ53-ζ53+ζ52 2ζ43ζ5+ζ54-ζ5 2ζ4ζ54-ζ54+ζ5 2ζ4ζ5+ζ54-ζ5 2ζ43ζ54-ζ54+ζ5 complex faithful

Smallest permutation representation of (C4×C20)⋊C6
On 80 points
Generators in S80
(1 77 30 44)(2 78 31 45)(3 79 32 46)(4 80 33 47)(5 61 34 48)(6 62 35 49)(7 63 36 50)(8 64 37 51)(9 65 38 52)(10 66 39 53)(11 67 40 54)(12 68 21 55)(13 69 22 56)(14 70 23 57)(15 71 24 58)(16 72 25 59)(17 73 26 60)(18 74 27 41)(19 75 28 42)(20 76 29 43)
(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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(2 53 73 39 55 58)(3 28 22 19 32 38)(4 64 52 37 70 79)(5 17)(6 49 77 35 59 54)(7 24 26 15 36 34)(8 80 56 33 74 75)(9 13)(10 45 61 31 43 50)(11 40 30)(12 76 60 29 78 71)(14 41 65 27 47 46)(16 72 44 25 62 67)(18 57 69 23 51 42)(20 68 48 21 66 63)

G:=sub<Sym(80)| (1,77,30,44)(2,78,31,45)(3,79,32,46)(4,80,33,47)(5,61,34,48)(6,62,35,49)(7,63,36,50)(8,64,37,51)(9,65,38,52)(10,66,39,53)(11,67,40,54)(12,68,21,55)(13,69,22,56)(14,70,23,57)(15,71,24,58)(16,72,25,59)(17,73,26,60)(18,74,27,41)(19,75,28,42)(20,76,29,43), (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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (2,53,73,39,55,58)(3,28,22,19,32,38)(4,64,52,37,70,79)(5,17)(6,49,77,35,59,54)(7,24,26,15,36,34)(8,80,56,33,74,75)(9,13)(10,45,61,31,43,50)(11,40,30)(12,76,60,29,78,71)(14,41,65,27,47,46)(16,72,44,25,62,67)(18,57,69,23,51,42)(20,68,48,21,66,63)>;

G:=Group( (1,77,30,44)(2,78,31,45)(3,79,32,46)(4,80,33,47)(5,61,34,48)(6,62,35,49)(7,63,36,50)(8,64,37,51)(9,65,38,52)(10,66,39,53)(11,67,40,54)(12,68,21,55)(13,69,22,56)(14,70,23,57)(15,71,24,58)(16,72,25,59)(17,73,26,60)(18,74,27,41)(19,75,28,42)(20,76,29,43), (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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (2,53,73,39,55,58)(3,28,22,19,32,38)(4,64,52,37,70,79)(5,17)(6,49,77,35,59,54)(7,24,26,15,36,34)(8,80,56,33,74,75)(9,13)(10,45,61,31,43,50)(11,40,30)(12,76,60,29,78,71)(14,41,65,27,47,46)(16,72,44,25,62,67)(18,57,69,23,51,42)(20,68,48,21,66,63) );

G=PermutationGroup([[(1,77,30,44),(2,78,31,45),(3,79,32,46),(4,80,33,47),(5,61,34,48),(6,62,35,49),(7,63,36,50),(8,64,37,51),(9,65,38,52),(10,66,39,53),(11,67,40,54),(12,68,21,55),(13,69,22,56),(14,70,23,57),(15,71,24,58),(16,72,25,59),(17,73,26,60),(18,74,27,41),(19,75,28,42),(20,76,29,43)], [(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),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(2,53,73,39,55,58),(3,28,22,19,32,38),(4,64,52,37,70,79),(5,17),(6,49,77,35,59,54),(7,24,26,15,36,34),(8,80,56,33,74,75),(9,13),(10,45,61,31,43,50),(11,40,30),(12,76,60,29,78,71),(14,41,65,27,47,46),(16,72,44,25,62,67),(18,57,69,23,51,42),(20,68,48,21,66,63)]])

Matrix representation of (C4×C20)⋊C6 in GL8(𝔽61)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 31 5 35 9 11 38 0 0 32 58 26 52 50 38 0 0 29 53 24 9 11 23 0 0 0 0 0 50 0 0 0 0 25 49 35 9 0 9 0 0 12 12 0 0 0 59
,
 0 60 0 0 0 0 0 0 1 17 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 50 50 0 0 60 0 0 0 11 33 22 46 1 0 0 0 0 12 12 39 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 50
,
 13 38 0 0 0 0 0 0 0 48 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 60 60 0 0 0 0 0 0 1 0 0 0 0 0 0 23 49 19 11 27 0 0 0 30 6 37 52 50 23 0 0 0 45 45 11 0 0

G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,31,32,29,0,25,12,0,0,5,58,53,0,49,12,0,0,35,26,24,0,35,0,0,0,9,52,9,50,9,0,0,0,11,50,11,0,0,0,0,0,38,38,23,0,9,59],[0,1,0,0,0,0,0,0,60,17,0,0,0,0,0,0,0,0,0,50,11,0,0,0,0,0,0,50,33,12,0,0,0,0,0,0,22,12,1,0,0,0,0,0,46,39,0,0,0,0,1,60,1,0,0,0,0,0,0,0,0,0,0,50],[13,0,0,0,0,0,0,0,38,48,0,0,0,0,0,0,0,0,1,60,0,23,30,0,0,0,0,60,1,49,6,45,0,0,0,60,0,19,37,45,0,0,0,0,0,11,52,11,0,0,0,0,0,27,50,0,0,0,0,0,0,0,23,0] >;

(C4×C20)⋊C6 in GAP, Magma, Sage, TeX

(C_4\times C_{20})\rtimes C_6
% in TeX

G:=Group("(C4xC20):C6");
// GroupNames label

G:=SmallGroup(480,263);
// by ID

G=gap.SmallGroup(480,263);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-5,-2,2,7688,198,856,7059,1774,304,3364,5052,8833]);
// Polycyclic

G:=Group<a,b,c|a^4=b^20=c^6=1,a*b=b*a,c*a*c^-1=a^-1*b^5,c*b*c^-1=a^-1*b^14>;
// generators/relations

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