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## G = C2×D10⋊1C8order 320 = 26·5

### Direct product of C2 and D10⋊1C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D10⋊1C8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — D5×C22×C4 — C2×D10⋊1C8
 Lower central C5 — C10 — C2×D10⋊1C8
 Upper central C1 — C22×C4 — C22×C8

Generators and relations for C2×D101C8
G = < a,b,c,d | a2=b10=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b5c >

Subgroups: 718 in 202 conjugacy classes, 87 normal (29 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C23, D5, C10, C10, C2×C8, C2×C8, C22×C4, C22×C4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C22⋊C8, C22×C8, C22×C8, C23×C4, C52C8, C40, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C2×C22⋊C8, C2×C52C8, C2×C52C8, C2×C40, C2×C40, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, D101C8, C22×C52C8, C22×C40, D5×C22×C4, C2×D101C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, D5, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, D10, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), C4×D5, D20, C5⋊D4, C22×D5, C2×C22⋊C8, C8×D5, C8⋊D5, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, D101C8, D5×C2×C8, C2×C8⋊D5, C2×D10⋊C4, C2×D101C8

Smallest permutation representation of C2×D101C8
On 160 points
Generators in S160
(1 95)(2 96)(3 97)(4 98)(5 99)(6 100)(7 91)(8 92)(9 93)(10 94)(11 90)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 89)(21 106)(22 107)(23 108)(24 109)(25 110)(26 101)(27 102)(28 103)(29 104)(30 105)(31 116)(32 117)(33 118)(34 119)(35 120)(36 111)(37 112)(38 113)(39 114)(40 115)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 136)(52 137)(53 138)(54 139)(55 140)(56 131)(57 132)(58 133)(59 134)(60 135)(61 146)(62 147)(63 148)(64 149)(65 150)(66 141)(67 142)(68 143)(69 144)(70 145)(71 156)(72 157)(73 158)(74 159)(75 160)(76 151)(77 152)(78 153)(79 154)(80 155)
(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)
(1 104)(2 103)(3 102)(4 101)(5 110)(6 109)(7 108)(8 107)(9 106)(10 105)(11 74)(12 73)(13 72)(14 71)(15 80)(16 79)(17 78)(18 77)(19 76)(20 75)(21 93)(22 92)(23 91)(24 100)(25 99)(26 98)(27 97)(28 96)(29 95)(30 94)(31 123)(32 122)(33 121)(34 130)(35 129)(36 128)(37 127)(38 126)(39 125)(40 124)(41 113)(42 112)(43 111)(44 120)(45 119)(46 118)(47 117)(48 116)(49 115)(50 114)(51 148)(52 147)(53 146)(54 145)(55 144)(56 143)(57 142)(58 141)(59 150)(60 149)(61 138)(62 137)(63 136)(64 135)(65 134)(66 133)(67 132)(68 131)(69 140)(70 139)(81 158)(82 157)(83 156)(84 155)(85 154)(86 153)(87 152)(88 151)(89 160)(90 159)
(1 11 45 145 25 160 40 140)(2 12 46 146 26 151 31 131)(3 13 47 147 27 152 32 132)(4 14 48 148 28 153 33 133)(5 15 49 149 29 154 34 134)(6 16 50 150 30 155 35 135)(7 17 41 141 21 156 36 136)(8 18 42 142 22 157 37 137)(9 19 43 143 23 158 38 138)(10 20 44 144 24 159 39 139)(51 91 86 126 66 106 71 111)(52 92 87 127 67 107 72 112)(53 93 88 128 68 108 73 113)(54 94 89 129 69 109 74 114)(55 95 90 130 70 110 75 115)(56 96 81 121 61 101 76 116)(57 97 82 122 62 102 77 117)(58 98 83 123 63 103 78 118)(59 99 84 124 64 104 79 119)(60 100 85 125 65 105 80 120)

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

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

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

104 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I 4J 4K 4L 5A 5B 8A ··· 8H 8I ··· 8P 10A ··· 10N 20A ··· 20P 40A ··· 40AF order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 5 5 8 ··· 8 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 ··· 1 10 10 10 10 1 ··· 1 10 10 10 10 2 2 2 ··· 2 10 ··· 10 2 ··· 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 D4 D5 M4(2) D10 D10 C4×D5 D20 C5⋊D4 C4×D5 C8×D5 C8⋊D5 kernel C2×D10⋊1C8 D10⋊1C8 C22×C5⋊2C8 C22×C40 D5×C22×C4 C2×C4×D5 C22×Dic5 C23×D5 C22×D5 C2×C20 C22×C8 C2×C10 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 4 2 2 16 4 2 4 4 2 4 8 8 4 16 16

Matrix representation of C2×D101C8 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 7 35 0 0 0 7 0
,
 1 0 0 0 0 0 1 0 0 0 0 40 40 0 0 0 0 0 40 1 0 0 0 0 1
,
 1 0 0 0 0 0 40 39 0 0 0 37 1 0 0 0 0 0 24 35 0 0 0 7 17

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,7,7,0,0,0,35,0],[1,0,0,0,0,0,1,40,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,1,1],[1,0,0,0,0,0,40,37,0,0,0,39,1,0,0,0,0,0,24,7,0,0,0,35,17] >;

C2×D101C8 in GAP, Magma, Sage, TeX

C_2\times D_{10}\rtimes_1C_8
% in TeX

G:=Group("C2xD10:1C8");
// GroupNames label

G:=SmallGroup(320,735);
// by ID

G=gap.SmallGroup(320,735);
# by ID

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

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

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