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## G = (C2×C10).D8order 320 = 26·5

### 8th non-split extension by C2×C10 of D8 acting via D8/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C2×C10).D8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4⋊Dic5 — C2×C4⋊Dic5 — (C2×C10).D8
 Lower central C5 — C10 — C2×C20 — (C2×C10).D8
 Upper central C1 — C22 — C22×C4 — C4⋊D4

Generators and relations for (C2×C10).D8
G = < a,b,c,d | a2=b10=c8=1, d2=b5, ab=ba, cac-1=ab5, ad=da, cbc-1=dbd-1=b-1, dcd-1=b5c-1 >

Subgroups: 382 in 114 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C2.D8, C2×C4⋊C4, C4⋊D4, C52C8, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C22.D8, C2×C52C8, C4⋊Dic5, C4⋊Dic5, C5×C22⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, D4×C10, D4×C10, C10.D8, C20.55D4, D4⋊Dic5, C2×C4⋊Dic5, C5×C4⋊D4, (C2×C10).D8
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C4○D4, D10, C22.D4, C2×D8, C8.C22, C5⋊D4, C22×D5, C22.D8, D4⋊D5, D42D5, C2×C5⋊D4, C2×D4⋊D5, C23.18D10, D4.9D10, (C2×C10).D8

Smallest permutation representation of (C2×C10).D8
On 160 points
Generators in S160
(1 70)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 113)(12 114)(13 115)(14 116)(15 117)(16 118)(17 119)(18 120)(19 111)(20 112)(21 59)(22 60)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 81)(39 82)(40 83)(41 79)(42 80)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(91 144)(92 145)(93 146)(94 147)(95 148)(96 149)(97 150)(98 141)(99 142)(100 143)(101 138)(102 139)(103 140)(104 131)(105 132)(106 133)(107 134)(108 135)(109 136)(110 137)(121 158)(122 159)(123 160)(124 151)(125 152)(126 153)(127 154)(128 155)(129 156)(130 157)
(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)
(1 130 33 105 30 117 45 92)(2 129 34 104 21 116 46 91)(3 128 35 103 22 115 47 100)(4 127 36 102 23 114 48 99)(5 126 37 101 24 113 49 98)(6 125 38 110 25 112 50 97)(7 124 39 109 26 111 41 96)(8 123 40 108 27 120 42 95)(9 122 31 107 28 119 43 94)(10 121 32 106 29 118 44 93)(11 72 141 69 153 85 138 57)(12 71 142 68 154 84 139 56)(13 80 143 67 155 83 140 55)(14 79 144 66 156 82 131 54)(15 78 145 65 157 81 132 53)(16 77 146 64 158 90 133 52)(17 76 147 63 159 89 134 51)(18 75 148 62 160 88 135 60)(19 74 149 61 151 87 136 59)(20 73 150 70 152 86 137 58)
(1 97 6 92)(2 96 7 91)(3 95 8 100)(4 94 9 99)(5 93 10 98)(11 90 16 85)(12 89 17 84)(13 88 18 83)(14 87 19 82)(15 86 20 81)(21 109 26 104)(22 108 27 103)(23 107 28 102)(24 106 29 101)(25 105 30 110)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 134 56 139)(52 133 57 138)(53 132 58 137)(54 131 59 136)(55 140 60 135)(61 149 66 144)(62 148 67 143)(63 147 68 142)(64 146 69 141)(65 145 70 150)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)

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

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

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

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - - + - image C1 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D8 D10 D10 D10 C5⋊D4 C5⋊D4 C8.C22 D4⋊2D5 D4⋊D5 D4.9D10 kernel (C2×C10).D8 C10.D8 C20.55D4 D4⋊Dic5 C2×C4⋊Dic5 C5×C4⋊D4 C2×C20 C22×C10 C4⋊D4 C20 C2×C10 C4⋊C4 C22×C4 C2×D4 C2×C4 C23 C10 C4 C22 C2 # reps 1 2 1 2 1 1 1 1 2 4 4 2 2 2 4 4 1 4 4 4

Matrix representation of (C2×C10).D8 in GL6(𝔽41)

 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 0 0 0 0 0 14 16
,
 9 0 0 0 0 0 0 32 0 0 0 0 0 0 0 24 0 0 0 0 29 24 0 0 0 0 0 0 29 31 0 0 0 0 35 12
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 0 24 0 0 0 0 12 0 0 0 0 0 0 0 29 31 0 0 0 0 2 12

G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,14,0,0,0,0,0,16],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,0,29,0,0,0,0,24,24,0,0,0,0,0,0,29,35,0,0,0,0,31,12],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,12,0,0,0,0,24,0,0,0,0,0,0,0,29,2,0,0,0,0,31,12] >;

(C2×C10).D8 in GAP, Magma, Sage, TeX

(C_2\times C_{10}).D_8
% in TeX

G:=Group("(C2xC10).D8");
// GroupNames label

G:=SmallGroup(320,660);
// by ID

G=gap.SmallGroup(320,660);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,254,219,1123,297,136,12550]);
// Polycyclic

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

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