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## G = C2×C8.D10order 320 = 26·5

### Direct product of C2 and C8.D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×C8.D10
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — C2×C4○D20 — C2×C8.D10
 Lower central C5 — C10 — C20 — C2×C8.D10
 Upper central C1 — C22 — C22×C4 — C2×M4(2)

Generators and relations for C2×C8.D10
G = < a,b,c,d | a2=b8=1, c10=d2=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b-1, dcd-1=c9 >

Subgroups: 958 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×7], Q8 [×13], C23, C23, D5 [×2], C10, C10 [×2], C10 [×2], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8 [×10], C4○D4 [×6], Dic5 [×6], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C40 [×4], Dic10 [×6], Dic10 [×7], C4×D5 [×4], D20 [×2], D20, C2×Dic5 [×7], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C22×D5, C22×C10, C2×C8.C22, C40⋊C2 [×8], Dic20 [×8], C2×C40 [×2], C5×M4(2) [×4], C2×Dic10, C2×Dic10 [×6], C2×Dic10 [×3], C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C22×Dic5, C2×C5⋊D4, C22×C20, C2×C40⋊C2 [×2], C2×Dic20 [×2], C8.D10 [×8], C10×M4(2), C22×Dic10, C2×C4○D20, C2×C8.D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8.C22 [×2], C22×D4, D20 [×4], C22×D5 [×7], C2×C8.C22, C2×D20 [×6], C23×D5, C8.D10 [×2], C22×D20, C2×C8.D10

Smallest permutation representation of C2×C8.D10
On 160 points
Generators in S160
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 129)(22 130)(23 131)(24 132)(25 133)(26 134)(27 135)(28 136)(29 137)(30 138)(31 139)(32 140)(33 121)(34 122)(35 123)(36 124)(37 125)(38 126)(39 127)(40 128)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 101)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(55 108)(56 109)(57 110)(58 111)(59 112)(60 113)(81 149)(82 150)(83 151)(84 152)(85 153)(86 154)(87 155)(88 156)(89 157)(90 158)(91 159)(92 160)(93 141)(94 142)(95 143)(96 144)(97 145)(98 146)(99 147)(100 148)
(1 157 38 103 11 147 28 113)(2 148 39 114 12 158 29 104)(3 159 40 105 13 149 30 115)(4 150 21 116 14 160 31 106)(5 141 22 107 15 151 32 117)(6 152 23 118 16 142 33 108)(7 143 24 109 17 153 34 119)(8 154 25 120 18 144 35 110)(9 145 26 111 19 155 36 101)(10 156 27 102 20 146 37 112)(41 66 90 137 51 76 100 127)(42 77 91 128 52 67 81 138)(43 68 92 139 53 78 82 129)(44 79 93 130 54 69 83 140)(45 70 94 121 55 80 84 131)(46 61 95 132 56 71 85 122)(47 72 96 123 57 62 86 133)(48 63 97 134 58 73 87 124)(49 74 98 125 59 64 88 135)(50 65 99 136 60 75 89 126)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 24 31 34)(22 33 32 23)(25 40 35 30)(26 29 36 39)(27 38 37 28)(41 97 51 87)(42 86 52 96)(43 95 53 85)(44 84 54 94)(45 93 55 83)(46 82 56 92)(47 91 57 81)(48 100 58 90)(49 89 59 99)(50 98 60 88)(61 78 71 68)(62 67 72 77)(63 76 73 66)(64 65 74 75)(69 70 79 80)(101 148 111 158)(102 157 112 147)(103 146 113 156)(104 155 114 145)(105 144 115 154)(106 153 116 143)(107 142 117 152)(108 151 118 141)(109 160 119 150)(110 149 120 159)(121 140 131 130)(122 129 132 139)(123 138 133 128)(124 127 134 137)(125 136 135 126)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4J 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 5 5 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 20 20 2 2 2 2 20 ··· 20 2 2 4 4 4 4 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 C2 C2 D4 D4 D5 D10 D10 D10 D20 D20 C8.C22 C8.D10 kernel C2×C8.D10 C2×C40⋊C2 C2×Dic20 C8.D10 C10×M4(2) C22×Dic10 C2×C4○D20 C2×C20 C22×C10 C2×M4(2) C2×C8 M4(2) C22×C4 C2×C4 C23 C10 C2 # reps 1 2 2 8 1 1 1 3 1 2 4 8 2 12 4 2 8

Matrix representation of C2×C8.D10 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
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 2 28 0 0 0 0 13 39 0 0
,
 0 34 0 0 0 0 6 6 0 0 0 0 0 0 6 35 34 8 0 0 6 1 33 27 0 0 33 7 35 6 0 0 34 32 35 40
,
 1 0 0 0 0 0 5 40 0 0 0 0 0 0 0 1 35 32 0 0 1 0 27 6 0 0 35 32 28 2 0 0 27 6 39 13

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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,2,13,0,0,0,0,28,39,0,0,1,0,0,0,0,0,0,1,0,0],[0,6,0,0,0,0,34,6,0,0,0,0,0,0,6,6,33,34,0,0,35,1,7,32,0,0,34,33,35,35,0,0,8,27,6,40],[1,5,0,0,0,0,0,40,0,0,0,0,0,0,0,1,35,27,0,0,1,0,32,6,0,0,35,27,28,39,0,0,32,6,2,13] >;

C2×C8.D10 in GAP, Magma, Sage, TeX

C_2\times C_8.D_{10}
% in TeX

G:=Group("C2xC8.D10");
// GroupNames label

G:=SmallGroup(320,1419);
// by ID

G=gap.SmallGroup(320,1419);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,80,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^10=d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations

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