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## G = C2×C20⋊D6order 480 = 25·3·5

### Direct product of C2 and C20⋊D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×C20⋊D6
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — C22×S3×D5 — C2×C20⋊D6
 Lower central C15 — C30 — C2×C20⋊D6
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C20⋊D6
G = < a,b,c,d | a2=b20=c6=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b9, dcd=c-1 >

Subgroups: 2876 in 472 conjugacy classes, 124 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C5×S3, C3×D5, D15, C30, C30, C22×D4, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, Dic15, C60, S3×D5, C6×D5, C6×D5, S3×C10, S3×C10, D30, C2×C30, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, C2×S3×D4, C15⋊D4, C3×D20, C5×D12, C4×D15, C2×Dic15, C2×C60, C2×S3×D5, C2×S3×D5, D5×C2×C6, S3×C2×C10, C22×D15, C2×D4×D5, C20⋊D6, C2×C15⋊D4, C6×D20, C10×D12, C2×C4×D15, C22×S3×D5, C2×C20⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C24, D10, C22×S3, C22×D4, C22×D5, S3×D4, S3×C23, S3×D5, D4×D5, C23×D5, C2×S3×D4, C2×S3×D5, C2×D4×D5, C20⋊D6, C22×S3×D5, C2×C20⋊D6

Smallest permutation representation of C2×C20⋊D6
On 120 points
Generators in S120
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 73)(40 74)(81 119)(82 120)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(91 109)(92 110)(93 111)(94 112)(95 113)(96 114)(97 115)(98 116)(99 117)(100 118)
(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)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 66 109)(2 65 110 20 67 108)(3 64 111 19 68 107)(4 63 112 18 69 106)(5 62 113 17 70 105)(6 61 114 16 71 104)(7 80 115 15 72 103)(8 79 116 14 73 102)(9 78 117 13 74 101)(10 77 118 12 75 120)(11 76 119)(21 82 42 23 100 44)(22 81 43)(24 99 45 40 83 41)(25 98 46 39 84 60)(26 97 47 38 85 59)(27 96 48 37 86 58)(28 95 49 36 87 57)(29 94 50 35 88 56)(30 93 51 34 89 55)(31 92 52 33 90 54)(32 91 53)
(1 81)(2 90)(3 99)(4 88)(5 97)(6 86)(7 95)(8 84)(9 93)(10 82)(11 91)(12 100)(13 89)(14 98)(15 87)(16 96)(17 85)(18 94)(19 83)(20 92)(21 77)(22 66)(23 75)(24 64)(25 73)(26 62)(27 71)(28 80)(29 69)(30 78)(31 67)(32 76)(33 65)(34 74)(35 63)(36 72)(37 61)(38 70)(39 79)(40 68)(41 111)(42 120)(43 109)(44 118)(45 107)(46 116)(47 105)(48 114)(49 103)(50 112)(51 101)(52 110)(53 119)(54 108)(55 117)(56 106)(57 115)(58 104)(59 113)(60 102)

G:=sub<Sym(120)| (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74)(81,119)(82,120)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,113)(96,114)(97,115)(98,116)(99,117)(100,118), (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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,66,109)(2,65,110,20,67,108)(3,64,111,19,68,107)(4,63,112,18,69,106)(5,62,113,17,70,105)(6,61,114,16,71,104)(7,80,115,15,72,103)(8,79,116,14,73,102)(9,78,117,13,74,101)(10,77,118,12,75,120)(11,76,119)(21,82,42,23,100,44)(22,81,43)(24,99,45,40,83,41)(25,98,46,39,84,60)(26,97,47,38,85,59)(27,96,48,37,86,58)(28,95,49,36,87,57)(29,94,50,35,88,56)(30,93,51,34,89,55)(31,92,52,33,90,54)(32,91,53), (1,81)(2,90)(3,99)(4,88)(5,97)(6,86)(7,95)(8,84)(9,93)(10,82)(11,91)(12,100)(13,89)(14,98)(15,87)(16,96)(17,85)(18,94)(19,83)(20,92)(21,77)(22,66)(23,75)(24,64)(25,73)(26,62)(27,71)(28,80)(29,69)(30,78)(31,67)(32,76)(33,65)(34,74)(35,63)(36,72)(37,61)(38,70)(39,79)(40,68)(41,111)(42,120)(43,109)(44,118)(45,107)(46,116)(47,105)(48,114)(49,103)(50,112)(51,101)(52,110)(53,119)(54,108)(55,117)(56,106)(57,115)(58,104)(59,113)(60,102)>;

G:=Group( (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74)(81,119)(82,120)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,113)(96,114)(97,115)(98,116)(99,117)(100,118), (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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,66,109)(2,65,110,20,67,108)(3,64,111,19,68,107)(4,63,112,18,69,106)(5,62,113,17,70,105)(6,61,114,16,71,104)(7,80,115,15,72,103)(8,79,116,14,73,102)(9,78,117,13,74,101)(10,77,118,12,75,120)(11,76,119)(21,82,42,23,100,44)(22,81,43)(24,99,45,40,83,41)(25,98,46,39,84,60)(26,97,47,38,85,59)(27,96,48,37,86,58)(28,95,49,36,87,57)(29,94,50,35,88,56)(30,93,51,34,89,55)(31,92,52,33,90,54)(32,91,53), (1,81)(2,90)(3,99)(4,88)(5,97)(6,86)(7,95)(8,84)(9,93)(10,82)(11,91)(12,100)(13,89)(14,98)(15,87)(16,96)(17,85)(18,94)(19,83)(20,92)(21,77)(22,66)(23,75)(24,64)(25,73)(26,62)(27,71)(28,80)(29,69)(30,78)(31,67)(32,76)(33,65)(34,74)(35,63)(36,72)(37,61)(38,70)(39,79)(40,68)(41,111)(42,120)(43,109)(44,118)(45,107)(46,116)(47,105)(48,114)(49,103)(50,112)(51,101)(52,110)(53,119)(54,108)(55,117)(56,106)(57,115)(58,104)(59,113)(60,102) );

G=PermutationGroup([[(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,49),(18,50),(19,51),(20,52),(21,75),(22,76),(23,77),(24,78),(25,79),(26,80),(27,61),(28,62),(29,63),(30,64),(31,65),(32,66),(33,67),(34,68),(35,69),(36,70),(37,71),(38,72),(39,73),(40,74),(81,119),(82,120),(83,101),(84,102),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108),(91,109),(92,110),(93,111),(94,112),(95,113),(96,114),(97,115),(98,116),(99,117),(100,118)], [(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),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,66,109),(2,65,110,20,67,108),(3,64,111,19,68,107),(4,63,112,18,69,106),(5,62,113,17,70,105),(6,61,114,16,71,104),(7,80,115,15,72,103),(8,79,116,14,73,102),(9,78,117,13,74,101),(10,77,118,12,75,120),(11,76,119),(21,82,42,23,100,44),(22,81,43),(24,99,45,40,83,41),(25,98,46,39,84,60),(26,97,47,38,85,59),(27,96,48,37,86,58),(28,95,49,36,87,57),(29,94,50,35,88,56),(30,93,51,34,89,55),(31,92,52,33,90,54),(32,91,53)], [(1,81),(2,90),(3,99),(4,88),(5,97),(6,86),(7,95),(8,84),(9,93),(10,82),(11,91),(12,100),(13,89),(14,98),(15,87),(16,96),(17,85),(18,94),(19,83),(20,92),(21,77),(22,66),(23,75),(24,64),(25,73),(26,62),(27,71),(28,80),(29,69),(30,78),(31,67),(32,76),(33,65),(34,74),(35,63),(36,72),(37,61),(38,70),(39,79),(40,68),(41,111),(42,120),(43,109),(44,118),(45,107),(46,116),(47,105),(48,114),(49,103),(50,112),(51,101),(52,110),(53,119),(54,108),(55,117),(56,106),(57,115),(58,104),(59,113),(60,102)]])

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 10G ··· 10N 12A 12B 15A 15B 20A 20B 20C 20D 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 10 ··· 10 12 12 15 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 1 1 6 6 6 6 10 10 10 10 15 15 15 15 2 2 2 30 30 2 2 2 2 2 20 20 20 20 2 ··· 2 12 ··· 12 4 4 4 4 4 4 4 4 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D5 D6 D6 D6 D10 D10 D10 S3×D4 S3×D5 D4×D5 C2×S3×D5 C2×S3×D5 C20⋊D6 kernel C2×C20⋊D6 C20⋊D6 C2×C15⋊D4 C6×D20 C10×D12 C2×C4×D15 C22×S3×D5 C2×D20 D30 C2×D12 D20 C2×C20 C22×D5 D12 C2×C12 C22×S3 C10 C2×C4 C6 C4 C22 C2 # reps 1 8 2 1 1 1 2 1 4 2 4 1 2 8 2 4 2 2 4 4 2 8

Matrix representation of C2×C20⋊D6 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 1 18
,
 0 1 0 0 0 0 60 60 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 43 60
,
 0 60 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 18 1

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,18],[0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,1,43,0,0,0,0,0,60],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,18,0,0,0,0,0,1] >;

C2×C20⋊D6 in GAP, Magma, Sage, TeX

C_2\times C_{20}\rtimes D_6
% in TeX

G:=Group("C2xC20:D6");
// GroupNames label

G:=SmallGroup(480,1089);
// by ID

G=gap.SmallGroup(480,1089);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,675,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^6=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^9,d*c*d=c^-1>;
// generators/relations

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