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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 [×2], C2 [×12], C3, C4 [×2], C4 [×2], C22, C22 [×38], C5, S3 [×8], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×5], D4 [×16], C23 [×21], D5 [×8], C10, C10 [×2], C10 [×4], Dic3 [×2], C12 [×2], D6 [×4], D6 [×26], C2×C6, C2×C6 [×8], C15, C22×C4, C2×D4 [×12], C24 [×2], Dic5 [×2], C20 [×2], D10 [×4], D10 [×26], C2×C10, C2×C10 [×8], C4×S3 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3 [×2], C22×S3 [×17], C22×C6 [×2], C5×S3 [×4], C3×D5 [×4], D15 [×4], C30, C30 [×2], C22×D4, C4×D5 [×4], D20 [×4], C2×Dic5, C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5 [×2], C22×D5 [×17], C22×C10 [×2], S3×C2×C4, C2×D12, S3×D4 [×8], C2×C3⋊D4 [×2], C6×D4, S3×C23 [×2], Dic15 [×2], C60 [×2], S3×D5 [×16], C6×D5 [×4], C6×D5 [×4], S3×C10 [×4], S3×C10 [×4], D30 [×6], C2×C30, C2×C4×D5, C2×D20, D4×D5 [×8], C2×C5⋊D4 [×2], D4×C10, C23×D5 [×2], C2×S3×D4, C15⋊D4 [×8], C3×D20 [×4], C5×D12 [×4], C4×D15 [×4], C2×Dic15, C2×C60, C2×S3×D5 [×8], C2×S3×D5 [×8], D5×C2×C6 [×2], S3×C2×C10 [×2], C22×D15, C2×D4×D5, C20⋊D6 [×8], C2×C15⋊D4 [×2], C6×D20, C10×D12, C2×C4×D15, C22×S3×D5 [×2], C2×C20⋊D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], C22×S3 [×7], C22×D4, C22×D5 [×7], S3×D4 [×2], S3×C23, S3×D5, D4×D5 [×2], C23×D5, C2×S3×D4, C2×S3×D5 [×3], C2×D4×D5, C20⋊D6 [×2], C22×S3×D5, C2×C20⋊D6

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

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

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

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

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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