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## G = D10⋊1C8.C2order 320 = 26·5

### 6th non-split extension by D10⋊1C8 of C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D10⋊1C8.C2
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — C4⋊C4⋊7D5 — D10⋊1C8.C2
 Lower central C5 — C10 — C2×C20 — D10⋊1C8.C2
 Upper central C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for D101C8.C2
G = < a,b,c,d,e | a2=b4=1, c2=b2, d10=ab2, e2=a, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=ab-1c, ebe-1=b-1c, dcd-1=ece-1=b2c, ede-1=b2d9 >

Subgroups: 366 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C22⋊C8, Q8⋊C4, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C52C8, C40, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C23.20D4, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, Q8×C10, C10.D8, C406C4, D101C8, Q8⋊Dic5, C5×Q8⋊C4, C4⋊C47D5, D103Q8, D101C8.C2
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C4○D8, C8.C22, C22×D5, C23.20D4, C4○D20, D4×D5, D42D5, D10.12D4, SD163D5, Q16⋊D5, D101C8.C2

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

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

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

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

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + - - + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 D10 C4○D8 C4○D20 C8.C22 D4⋊2D5 D4×D5 SD16⋊3D5 Q16⋊D5 kernel D10⋊1C8.C2 C10.D8 C40⋊6C4 D10⋊1C8 Q8⋊Dic5 C5×Q8⋊C4 C4⋊C4⋊7D5 D10⋊3Q8 C2×Dic5 C22×D5 Q8⋊C4 C20 C4⋊C4 C2×C8 C2×Q8 C10 C4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 4 2 2 2 4 8 1 2 2 4 4

Matrix representation of D101C8.C2 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 24 1 0 0 40 17 0 0 0 0 33 24 0 0 40 8
,
 40 0 0 0 0 40 0 0 0 0 32 21 0 0 0 9
,
 19 19 0 0 22 9 0 0 0 0 30 9 0 0 14 11
,
 9 22 0 0 0 32 0 0 0 0 30 3 0 0 14 11
`G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[24,40,0,0,1,17,0,0,0,0,33,40,0,0,24,8],[40,0,0,0,0,40,0,0,0,0,32,0,0,0,21,9],[19,22,0,0,19,9,0,0,0,0,30,14,0,0,9,11],[9,0,0,0,22,32,0,0,0,0,30,14,0,0,3,11] >;`

D101C8.C2 in GAP, Magma, Sage, TeX

`D_{10}\rtimes_1C_8.C_2`
`% in TeX`

`G:=Group("D10:1C8.C2");`
`// GroupNames label`

`G:=SmallGroup(320,442);`
`// by ID`

`G=gap.SmallGroup(320,442);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,64,926,219,184,851,438,102,12550]);`
`// Polycyclic`

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

׿
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