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## G = C2×C4.12D20order 320 = 26·5

### Direct product of C2 and C4.12D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×C4.12D20
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×Dic10 — C22×Dic10 — C2×C4.12D20
 Lower central C5 — C10 — C2×C10 — C2×C4.12D20
 Upper central C1 — C22 — C22×C4 — C2×M4(2)

Generators and relations for C2×C4.12D20
G = < a,b,c,d | a2=b20=1, c4=d2=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b5c3 >

Subgroups: 478 in 146 conjugacy classes, 63 normal (39 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C10, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C4.10D4, C2×M4(2), C2×M4(2), C22×Q8, C52C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C2×C4.10D4, C2×C52C8, C4.Dic5, C4.Dic5, C2×C40, C5×M4(2), C5×M4(2), C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, C4.12D20, C2×C4.Dic5, C10×M4(2), C22×Dic10, C2×C4.12D20
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C4.10D4, C2×C22⋊C4, C4×D5, D20, C5⋊D4, C22×D5, C2×C4.10D4, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C4.12D20, C2×D10⋊C4, C2×C4.12D20

Smallest permutation representation of C2×C4.12D20
On 160 points
Generators in S160
(1 122)(2 123)(3 124)(4 125)(5 126)(6 127)(7 128)(8 129)(9 130)(10 131)(11 132)(12 133)(13 134)(14 135)(15 136)(16 137)(17 138)(18 139)(19 140)(20 121)(21 99)(22 100)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)(41 150)(42 151)(43 152)(44 153)(45 154)(46 155)(47 156)(48 157)(49 158)(50 159)(51 160)(52 141)(53 142)(54 143)(55 144)(56 145)(57 146)(58 147)(59 148)(60 149)(61 112)(62 113)(63 114)(64 115)(65 116)(66 117)(67 118)(68 119)(69 120)(70 101)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)
(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)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 159 127 45 11 149 137 55)(2 158 128 44 12 148 138 54)(3 157 129 43 13 147 139 53)(4 156 130 42 14 146 140 52)(5 155 131 41 15 145 121 51)(6 154 132 60 16 144 122 50)(7 153 133 59 17 143 123 49)(8 152 134 58 18 142 124 48)(9 151 135 57 19 141 125 47)(10 150 136 56 20 160 126 46)(21 105 94 79 31 115 84 69)(22 104 95 78 32 114 85 68)(23 103 96 77 33 113 86 67)(24 102 97 76 34 112 87 66)(25 101 98 75 35 111 88 65)(26 120 99 74 36 110 89 64)(27 119 100 73 37 109 90 63)(28 118 81 72 38 108 91 62)(29 117 82 71 39 107 92 61)(30 116 83 70 40 106 93 80)
(1 102 11 112)(2 101 12 111)(3 120 13 110)(4 119 14 109)(5 118 15 108)(6 117 16 107)(7 116 17 106)(8 115 18 105)(9 114 19 104)(10 113 20 103)(21 157 31 147)(22 156 32 146)(23 155 33 145)(24 154 34 144)(25 153 35 143)(26 152 36 142)(27 151 37 141)(28 150 38 160)(29 149 39 159)(30 148 40 158)(41 96 51 86)(42 95 52 85)(43 94 53 84)(44 93 54 83)(45 92 55 82)(46 91 56 81)(47 90 57 100)(48 89 58 99)(49 88 59 98)(50 87 60 97)(61 122 71 132)(62 121 72 131)(63 140 73 130)(64 139 74 129)(65 138 75 128)(66 137 76 127)(67 136 77 126)(68 135 78 125)(69 134 79 124)(70 133 80 123)

G:=sub<Sym(160)| (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,131)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,121)(21,99)(22,100)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,150)(42,151)(43,152)(44,153)(45,154)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111), (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)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,159,127,45,11,149,137,55)(2,158,128,44,12,148,138,54)(3,157,129,43,13,147,139,53)(4,156,130,42,14,146,140,52)(5,155,131,41,15,145,121,51)(6,154,132,60,16,144,122,50)(7,153,133,59,17,143,123,49)(8,152,134,58,18,142,124,48)(9,151,135,57,19,141,125,47)(10,150,136,56,20,160,126,46)(21,105,94,79,31,115,84,69)(22,104,95,78,32,114,85,68)(23,103,96,77,33,113,86,67)(24,102,97,76,34,112,87,66)(25,101,98,75,35,111,88,65)(26,120,99,74,36,110,89,64)(27,119,100,73,37,109,90,63)(28,118,81,72,38,108,91,62)(29,117,82,71,39,107,92,61)(30,116,83,70,40,106,93,80), (1,102,11,112)(2,101,12,111)(3,120,13,110)(4,119,14,109)(5,118,15,108)(6,117,16,107)(7,116,17,106)(8,115,18,105)(9,114,19,104)(10,113,20,103)(21,157,31,147)(22,156,32,146)(23,155,33,145)(24,154,34,144)(25,153,35,143)(26,152,36,142)(27,151,37,141)(28,150,38,160)(29,149,39,159)(30,148,40,158)(41,96,51,86)(42,95,52,85)(43,94,53,84)(44,93,54,83)(45,92,55,82)(46,91,56,81)(47,90,57,100)(48,89,58,99)(49,88,59,98)(50,87,60,97)(61,122,71,132)(62,121,72,131)(63,140,73,130)(64,139,74,129)(65,138,75,128)(66,137,76,127)(67,136,77,126)(68,135,78,125)(69,134,79,124)(70,133,80,123)>;

G:=Group( (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,131)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,121)(21,99)(22,100)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,150)(42,151)(43,152)(44,153)(45,154)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111), (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)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,159,127,45,11,149,137,55)(2,158,128,44,12,148,138,54)(3,157,129,43,13,147,139,53)(4,156,130,42,14,146,140,52)(5,155,131,41,15,145,121,51)(6,154,132,60,16,144,122,50)(7,153,133,59,17,143,123,49)(8,152,134,58,18,142,124,48)(9,151,135,57,19,141,125,47)(10,150,136,56,20,160,126,46)(21,105,94,79,31,115,84,69)(22,104,95,78,32,114,85,68)(23,103,96,77,33,113,86,67)(24,102,97,76,34,112,87,66)(25,101,98,75,35,111,88,65)(26,120,99,74,36,110,89,64)(27,119,100,73,37,109,90,63)(28,118,81,72,38,108,91,62)(29,117,82,71,39,107,92,61)(30,116,83,70,40,106,93,80), (1,102,11,112)(2,101,12,111)(3,120,13,110)(4,119,14,109)(5,118,15,108)(6,117,16,107)(7,116,17,106)(8,115,18,105)(9,114,19,104)(10,113,20,103)(21,157,31,147)(22,156,32,146)(23,155,33,145)(24,154,34,144)(25,153,35,143)(26,152,36,142)(27,151,37,141)(28,150,38,160)(29,149,39,159)(30,148,40,158)(41,96,51,86)(42,95,52,85)(43,94,53,84)(44,93,54,83)(45,92,55,82)(46,91,56,81)(47,90,57,100)(48,89,58,99)(49,88,59,98)(50,87,60,97)(61,122,71,132)(62,121,72,131)(63,140,73,130)(64,139,74,129)(65,138,75,128)(66,137,76,127)(67,136,77,126)(68,135,78,125)(69,134,79,124)(70,133,80,123) );

G=PermutationGroup([[(1,122),(2,123),(3,124),(4,125),(5,126),(6,127),(7,128),(8,129),(9,130),(10,131),(11,132),(12,133),(13,134),(14,135),(15,136),(16,137),(17,138),(18,139),(19,140),(20,121),(21,99),(22,100),(23,81),(24,82),(25,83),(26,84),(27,85),(28,86),(29,87),(30,88),(31,89),(32,90),(33,91),(34,92),(35,93),(36,94),(37,95),(38,96),(39,97),(40,98),(41,150),(42,151),(43,152),(44,153),(45,154),(46,155),(47,156),(48,157),(49,158),(50,159),(51,160),(52,141),(53,142),(54,143),(55,144),(56,145),(57,146),(58,147),(59,148),(60,149),(61,112),(62,113),(63,114),(64,115),(65,116),(66,117),(67,118),(68,119),(69,120),(70,101),(71,102),(72,103),(73,104),(74,105),(75,106),(76,107),(77,108),(78,109),(79,110),(80,111)], [(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),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,159,127,45,11,149,137,55),(2,158,128,44,12,148,138,54),(3,157,129,43,13,147,139,53),(4,156,130,42,14,146,140,52),(5,155,131,41,15,145,121,51),(6,154,132,60,16,144,122,50),(7,153,133,59,17,143,123,49),(8,152,134,58,18,142,124,48),(9,151,135,57,19,141,125,47),(10,150,136,56,20,160,126,46),(21,105,94,79,31,115,84,69),(22,104,95,78,32,114,85,68),(23,103,96,77,33,113,86,67),(24,102,97,76,34,112,87,66),(25,101,98,75,35,111,88,65),(26,120,99,74,36,110,89,64),(27,119,100,73,37,109,90,63),(28,118,81,72,38,108,91,62),(29,117,82,71,39,107,92,61),(30,116,83,70,40,106,93,80)], [(1,102,11,112),(2,101,12,111),(3,120,13,110),(4,119,14,109),(5,118,15,108),(6,117,16,107),(7,116,17,106),(8,115,18,105),(9,114,19,104),(10,113,20,103),(21,157,31,147),(22,156,32,146),(23,155,33,145),(24,154,34,144),(25,153,35,143),(26,152,36,142),(27,151,37,141),(28,150,38,160),(29,149,39,159),(30,148,40,158),(41,96,51,86),(42,95,52,85),(43,94,53,84),(44,93,54,83),(45,92,55,82),(46,91,56,81),(47,90,57,100),(48,89,58,99),(49,88,59,98),(50,87,60,97),(61,122,71,132),(62,121,72,131),(63,140,73,130),(64,139,74,129),(65,138,75,128),(66,137,76,127),(67,136,77,126),(68,135,78,125),(69,134,79,124),(70,133,80,123)]])

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 1 1 2 2 2 2 2 2 20 20 20 20 2 2 4 4 4 4 20 20 20 20 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - - image C1 C2 C2 C2 C2 C4 C4 D4 D5 D10 D10 C4×D5 D20 C5⋊D4 C4×D5 C4.10D4 C4.12D20 kernel C2×C4.12D20 C4.12D20 C2×C4.Dic5 C10×M4(2) C22×Dic10 C2×Dic10 C22×Dic5 C2×C20 C2×M4(2) M4(2) C22×C4 C2×C4 C2×C4 C2×C4 C23 C10 C2 # reps 1 4 1 1 1 4 4 4 2 4 2 4 8 8 4 2 8

Matrix representation of C2×C4.12D20 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 40 0 0 0 0 36 6 0 0 0 0 0 0 9 11 0 0 0 0 30 14 0 0 0 0 0 0 39 11 0 0 0 0 14 25
,
 35 3 0 0 0 0 15 6 0 0 0 0 0 0 1 24 4 39 0 0 17 40 27 39 0 0 8 17 23 1 0 0 10 9 3 18
,
 28 27 0 0 0 0 12 13 0 0 0 0 0 0 0 32 0 0 0 0 32 0 0 0 0 0 0 0 32 0 0 0 0 0 4 9

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,36,0,0,0,0,40,6,0,0,0,0,0,0,9,30,0,0,0,0,11,14,0,0,0,0,0,0,39,14,0,0,0,0,11,25],[35,15,0,0,0,0,3,6,0,0,0,0,0,0,1,17,8,10,0,0,24,40,17,9,0,0,4,27,23,3,0,0,39,39,1,18],[28,12,0,0,0,0,27,13,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,32,4,0,0,0,0,0,9] >;

C2×C4.12D20 in GAP, Magma, Sage, TeX

C_2\times C_4._{12}D_{20}
% in TeX

G:=Group("C2xC4.12D20");
// GroupNames label

G:=SmallGroup(320,763);
// by ID

G=gap.SmallGroup(320,763);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,422,58,1123,136,438,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=1,c^4=d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^5*c^3>;
// generators/relations

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