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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 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×7], Q8 [×2], C23, D5, C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, C22×C4, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×3], C2×C10, C22⋊C8, Q8⋊C4, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C52C8, C40, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C23.20D4, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5 [×2], D10⋊C4 [×2], 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 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], 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
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 101)(10 102)(11 103)(12 104)(13 105)(14 106)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(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 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 61)(60 62)(81 143)(82 144)(83 145)(84 146)(85 147)(86 148)(87 149)(88 150)(89 151)(90 152)(91 153)(92 154)(93 155)(94 156)(95 157)(96 158)(97 159)(98 160)(99 141)(100 142)
(1 57 103 69)(2 156 104 84)(3 59 105 71)(4 158 106 86)(5 41 107 73)(6 160 108 88)(7 43 109 75)(8 142 110 90)(9 45 111 77)(10 144 112 92)(11 47 113 79)(12 146 114 94)(13 49 115 61)(14 148 116 96)(15 51 117 63)(16 150 118 98)(17 53 119 65)(18 152 120 100)(19 55 101 67)(20 154 102 82)(21 81 139 153)(22 56 140 68)(23 83 121 155)(24 58 122 70)(25 85 123 157)(26 60 124 72)(27 87 125 159)(28 42 126 74)(29 89 127 141)(30 44 128 76)(31 91 129 143)(32 46 130 78)(33 93 131 145)(34 48 132 80)(35 95 133 147)(36 50 134 62)(37 97 135 149)(38 52 136 64)(39 99 137 151)(40 54 138 66)
(1 33 103 131)(2 132 104 34)(3 35 105 133)(4 134 106 36)(5 37 107 135)(6 136 108 38)(7 39 109 137)(8 138 110 40)(9 21 111 139)(10 140 112 22)(11 23 113 121)(12 122 114 24)(13 25 115 123)(14 124 116 26)(15 27 117 125)(16 126 118 28)(17 29 119 127)(18 128 120 30)(19 31 101 129)(20 130 102 32)(41 149 73 97)(42 98 74 150)(43 151 75 99)(44 100 76 152)(45 153 77 81)(46 82 78 154)(47 155 79 83)(48 84 80 156)(49 157 61 85)(50 86 62 158)(51 159 63 87)(52 88 64 160)(53 141 65 89)(54 90 66 142)(55 143 67 91)(56 92 68 144)(57 145 69 93)(58 94 70 146)(59 147 71 95)(60 96 72 148)
(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 10 113 102)(2 101 114 9)(3 8 115 120)(4 119 116 7)(5 6 117 118)(11 20 103 112)(12 111 104 19)(13 18 105 110)(14 109 106 17)(15 16 107 108)(21 34 129 122)(22 121 130 33)(23 32 131 140)(24 139 132 31)(25 30 133 138)(26 137 134 29)(27 28 135 136)(35 40 123 128)(36 127 124 39)(37 38 125 126)(41 64 63 42)(43 62 65 60)(44 59 66 61)(45 80 67 58)(46 57 68 79)(47 78 69 56)(48 55 70 77)(49 76 71 54)(50 53 72 75)(51 74 73 52)(81 156 143 94)(82 93 144 155)(83 154 145 92)(84 91 146 153)(85 152 147 90)(86 89 148 151)(87 150 149 88)(95 142 157 100)(96 99 158 141)(97 160 159 98)

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

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

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

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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