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## G = D20.44D4order 320 = 26·5

### 14th non-split extension by D20 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D20.44D4
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C4○D20 — Q8.10D10 — D20.44D4
 Lower central C5 — C10 — C20 — D20.44D4
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for D20.44D4
G = < a,b,c,d | a20=b2=1, c4=d2=a10, bab=a-1, cac-1=dad-1=a11, cbc-1=dbd-1=a10b, dcd-1=a10c3 >

Subgroups: 870 in 248 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×14], D4, D4 [×10], Q8, Q8 [×2], Q8 [×10], D5 [×3], C10, C10 [×2], C2×C8 [×3], M4(2), M4(2) [×2], D8, SD16 [×2], SD16 [×4], Q16 [×2], Q16 [×7], C2×Q8, C2×Q8 [×7], C4○D4, C4○D4 [×12], Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×3], D10 [×2], D10, C2×C10, C2×C10, C8○D4, C2×Q16 [×3], C4○D8 [×3], C8.C22, C8.C22 [×5], 2- 1+4 [×2], C52C8 [×2], C40 [×2], Dic10 [×2], Dic10 [×2], Dic10 [×5], C4×D5 [×2], C4×D5 [×7], D20 [×2], D20 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C5×Q8 [×2], C5×Q8, Q8○D8, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], Dic20 [×2], C2×C52C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C5⋊Q16 [×4], C5×M4(2), C5×SD16 [×2], C5×Q16 [×2], C2×Dic10, C2×Dic10, C4○D20 [×2], C4○D20 [×3], D42D5 [×2], D42D5 [×2], Q8×D5 [×4], Q8×D5, Q82D5 [×2], Q82D5, Q8×C10, C5×C4○D4, D20.2C4, C8.D10, SD16⋊D5 [×2], SD163D5 [×2], D5×Q16 [×2], Q16⋊D5 [×2], C2×C5⋊Q16, D4.8D10, C5×C8.C22, Q8.10D10, D4.10D10, D20.44D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], Q8○D8, D4×D5 [×2], C23×D5, C2×D4×D5, D20.44D4

Smallest permutation representation of D20.44D4
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 76)(2 75)(3 74)(4 73)(5 72)(6 71)(7 70)(8 69)(9 68)(10 67)(11 66)(12 65)(13 64)(14 63)(15 62)(16 61)(17 80)(18 79)(19 78)(20 77)(21 136)(22 135)(23 134)(24 133)(25 132)(26 131)(27 130)(28 129)(29 128)(30 127)(31 126)(32 125)(33 124)(34 123)(35 122)(36 121)(37 140)(38 139)(39 138)(40 137)(41 142)(42 141)(43 160)(44 159)(45 158)(46 157)(47 156)(48 155)(49 154)(50 153)(51 152)(52 151)(53 150)(54 149)(55 148)(56 147)(57 146)(58 145)(59 144)(60 143)(81 105)(82 104)(83 103)(84 102)(85 101)(86 120)(87 119)(88 118)(89 117)(90 116)(91 115)(92 114)(93 113)(94 112)(95 111)(96 110)(97 109)(98 108)(99 107)(100 106)
(1 115 33 147 11 105 23 157)(2 106 34 158 12 116 24 148)(3 117 35 149 13 107 25 159)(4 108 36 160 14 118 26 150)(5 119 37 151 15 109 27 141)(6 110 38 142 16 120 28 152)(7 101 39 153 17 111 29 143)(8 112 40 144 18 102 30 154)(9 103 21 155 19 113 31 145)(10 114 22 146 20 104 32 156)(41 71 86 139 51 61 96 129)(42 62 87 130 52 72 97 140)(43 73 88 121 53 63 98 131)(44 64 89 132 54 74 99 122)(45 75 90 123 55 65 100 133)(46 66 91 134 56 76 81 124)(47 77 92 125 57 67 82 135)(48 68 93 136 58 78 83 126)(49 79 94 127 59 69 84 137)(50 70 95 138 60 80 85 128)
(1 140 11 130)(2 131 12 121)(3 122 13 132)(4 133 14 123)(5 124 15 134)(6 135 16 125)(7 126 17 136)(8 137 18 127)(9 128 19 138)(10 139 20 129)(21 80 31 70)(22 71 32 61)(23 62 33 72)(24 73 34 63)(25 64 35 74)(26 75 36 65)(27 66 37 76)(28 77 38 67)(29 68 39 78)(30 79 40 69)(41 156 51 146)(42 147 52 157)(43 158 53 148)(44 149 54 159)(45 160 55 150)(46 151 56 141)(47 142 57 152)(48 153 58 143)(49 144 59 154)(50 155 60 145)(81 109 91 119)(82 120 92 110)(83 111 93 101)(84 102 94 112)(85 113 95 103)(86 104 96 114)(87 115 97 105)(88 106 98 116)(89 117 99 107)(90 108 100 118)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 D10 D10 Q8○D8 D4×D5 D4×D5 D20.44D4 kernel D20.44D4 D20.2C4 C8.D10 SD16⋊D5 SD16⋊3D5 D5×Q16 Q16⋊D5 C2×C5⋊Q16 D4.8D10 C5×C8.C22 Q8.10D10 D4.10D10 Dic10 D20 C5⋊D4 C8.C22 M4(2) SD16 Q16 C2×Q8 C4○D4 C5 C4 C22 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 1 1 2 2 2 4 4 2 2 2 2 2 2

Matrix representation of D20.44D4 in GL8(𝔽41)

 6 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 0 29 12 0 0 0 0 0 0 12 12 0 0 0 0 12 29 0 0 0 0 0 0 29 29 0 0
,
 0 40 35 7 0 0 0 0 40 0 2 6 0 0 0 0 0 0 35 6 0 0 0 0 0 0 1 6 0 0 0 0 0 0 0 0 0 1 32 32 0 0 0 0 40 0 32 9 0 0 0 0 9 9 0 1 0 0 0 0 9 32 40 0
,
 35 40 6 0 0 0 0 0 1 0 0 6 0 0 0 0 35 40 6 1 0 0 0 0 1 0 40 0 0 0 0 0 0 0 0 0 9 9 0 1 0 0 0 0 32 9 1 0 0 0 0 0 40 0 32 9 0 0 0 0 0 1 32 32
,
 6 1 35 39 0 0 0 0 40 0 2 6 0 0 0 0 6 1 35 40 0 0 0 0 40 0 1 0 0 0 0 0 0 0 0 0 40 0 32 9 0 0 0 0 0 1 32 32 0 0 0 0 32 32 0 40 0 0 0 0 9 32 40 0

`G:=sub<GL(8,GF(41))| [6,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,29,0,0,0,0,0,0,29,29,0,0,0,0,29,12,0,0,0,0,0,0,12,12,0,0],[0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,35,2,35,1,0,0,0,0,7,6,6,6,0,0,0,0,0,0,0,0,0,40,9,9,0,0,0,0,1,0,9,32,0,0,0,0,32,32,0,40,0,0,0,0,32,9,1,0],[35,1,35,1,0,0,0,0,40,0,40,0,0,0,0,0,6,0,6,40,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,0,9,32,40,0,0,0,0,0,9,9,0,1,0,0,0,0,0,1,32,32,0,0,0,0,1,0,9,32],[6,40,6,40,0,0,0,0,1,0,1,0,0,0,0,0,35,2,35,1,0,0,0,0,39,6,40,0,0,0,0,0,0,0,0,0,40,0,32,9,0,0,0,0,0,1,32,32,0,0,0,0,32,32,0,40,0,0,0,0,9,32,40,0] >;`

D20.44D4 in GAP, Magma, Sage, TeX

`D_{20}._{44}D_4`
`% in TeX`

`G:=Group("D20.44D4");`
`// GroupNames label`

`G:=SmallGroup(320,1451);`
`// by ID`

`G=gap.SmallGroup(320,1451);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,184,570,185,136,438,235,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=1,c^4=d^2=a^10,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^11,c*b*c^-1=d*b*d^-1=a^10*b,d*c*d^-1=a^10*c^3>;`
`// generators/relations`

׿
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