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G = D10.13D8order 320 = 26·5

2nd non-split extension by D10 of D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D10.13D8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — D5×C4⋊C4 — D10.13D8
 Lower central C5 — C10 — C2×C20 — D10.13D8
 Upper central C1 — C22 — C2×C4 — C2.D8

Generators and relations for D10.13D8
G = < a,b,c,d | a10=b2=c8=1, d2=a5, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a5b, dcd-1=c-1 >

Subgroups: 574 in 114 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C22⋊C8, D4⋊C4, C2.D8, C2.D8, C2×C4⋊C4, C4⋊D4, C52C8, C40, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22.D8, C2×C52C8, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C10.D8, D206C4, D101C8, D205C4, C5×C2.D8, D5×C4⋊C4, C4⋊D20, D10.13D8
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C4○D4, D10, C22.D4, C2×D8, C8.C22, C22×D5, C22.D8, C4○D20, D4×D5, Q82D5, D10.13D4, D5×D8, Q16⋊D5, D10.13D8

Smallest permutation representation of D10.13D8
On 160 points
Generators in S160
```(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)
(1 152)(2 151)(3 160)(4 159)(5 158)(6 157)(7 156)(8 155)(9 154)(10 153)(11 134)(12 133)(13 132)(14 131)(15 140)(16 139)(17 138)(18 137)(19 136)(20 135)(21 44)(22 43)(23 42)(24 41)(25 50)(26 49)(27 48)(28 47)(29 46)(30 45)(31 145)(32 144)(33 143)(34 142)(35 141)(36 150)(37 149)(38 148)(39 147)(40 146)(51 99)(52 98)(53 97)(54 96)(55 95)(56 94)(57 93)(58 92)(59 91)(60 100)(61 120)(62 119)(63 118)(64 117)(65 116)(66 115)(67 114)(68 113)(69 112)(70 111)(71 110)(72 109)(73 108)(74 107)(75 106)(76 105)(77 104)(78 103)(79 102)(80 101)(81 123)(82 122)(83 121)(84 130)(85 129)(86 128)(87 127)(88 126)(89 125)(90 124)
(1 53 13 73 33 90 50 64)(2 54 14 74 34 81 41 65)(3 55 15 75 35 82 42 66)(4 56 16 76 36 83 43 67)(5 57 17 77 37 84 44 68)(6 58 18 78 38 85 45 69)(7 59 19 79 39 86 46 70)(8 60 20 80 40 87 47 61)(9 51 11 71 31 88 48 62)(10 52 12 72 32 89 49 63)(21 118 158 98 138 109 149 125)(22 119 159 99 139 110 150 126)(23 120 160 100 140 101 141 127)(24 111 151 91 131 102 142 128)(25 112 152 92 132 103 143 129)(26 113 153 93 133 104 144 130)(27 114 154 94 134 105 145 121)(28 115 155 95 135 106 146 122)(29 116 156 96 136 107 147 123)(30 117 157 97 137 108 148 124)
(1 138 6 133)(2 139 7 134)(3 140 8 135)(4 131 9 136)(5 132 10 137)(11 156 16 151)(12 157 17 152)(13 158 18 153)(14 159 19 154)(15 160 20 155)(21 38 26 33)(22 39 27 34)(23 40 28 35)(24 31 29 36)(25 32 30 37)(41 150 46 145)(42 141 47 146)(43 142 48 147)(44 143 49 148)(45 144 50 149)(51 96 56 91)(52 97 57 92)(53 98 58 93)(54 99 59 94)(55 100 60 95)(61 106 66 101)(62 107 67 102)(63 108 68 103)(64 109 69 104)(65 110 70 105)(71 116 76 111)(72 117 77 112)(73 118 78 113)(74 119 79 114)(75 120 80 115)(81 126 86 121)(82 127 87 122)(83 128 88 123)(84 129 89 124)(85 130 90 125)```

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

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

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

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 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H order 1 2 2 2 2 2 2 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 10 10 40 2 2 4 4 8 20 20 20 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 4 4 4 4 4 type + + + + + + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D8 D10 D10 C4○D20 C8.C22 Q8⋊2D5 D4×D5 D5×D8 Q16⋊D5 kernel D10.13D8 C10.D8 D20⋊6C4 D10⋊1C8 D20⋊5C4 C5×C2.D8 D5×C4⋊C4 C4⋊D20 C2×Dic5 C22×D5 C2.D8 C20 D10 C4⋊C4 C2×C8 C4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 4 4 4 2 8 1 2 2 4 4

Matrix representation of D10.13D8 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 35 35 0 0 0 0 6 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 9 0 0 0 0 32 0 0 0 0 0 0 0 35 35 0 0 0 0 40 6 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 7 0 0 0 0 35 24
,
 0 32 0 0 0 0 32 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 4 0 0 0 0 0 1

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,35,40,0,0,0,0,35,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,35,0,0,0,0,7,24],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,4,1] >;`

D10.13D8 in GAP, Magma, Sage, TeX

`D_{10}._{13}D_8`
`% in TeX`

`G:=Group("D10.13D8");`
`// GroupNames label`

`G:=SmallGroup(320,509);`
`// by ID`

`G=gap.SmallGroup(320,509);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,254,219,268,851,102,12550]);`
`// Polycyclic`

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

׿
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