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## G = D20.35C23order 320 = 26·5

### 16th non-split extension by D20 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D20.35C23
 Chief series C1 — C5 — C10 — C20 — D20 — C4○D20 — D4.10D10 — D20.35C23
 Lower central C5 — C10 — C20 — D20.35C23
 Upper central C1 — C2 — C4○D4 — 2- 1+4

Generators and relations for D20.35C23
G = < a,b,c,d,e | a20=b2=e2=1, c2=d2=a10, bab=a-1, ac=ca, ad=da, eae=a11, bc=cb, bd=db, ebe=a15b, dcd-1=a10c, ce=ec, de=ed >

Subgroups: 726 in 248 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, Q8, D5, C10, C10, C2×C8, M4(2), D8, SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, Dic5, C20, C20, C20, D10, C2×C10, C2×C10, C8○D4, C2×Q16, C4○D8, C8.C22, 2- 1+4, 2- 1+4, C52C8, C52C8, Dic10, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×D4, C5×Q8, C5×Q8, C5×Q8, Q8○D8, C2×C52C8, C4.Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C2×Dic10, C4○D20, D42D5, Q8×D5, Q8×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C5×C4○D4, C20.C23, C2×C5⋊Q16, D4.Dic5, D4.8D10, D4.9D10, D4.10D10, C5×2- 1+4, D20.35C23
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C5⋊D4, C22×D5, Q8○D8, C2×C5⋊D4, C23×D5, C22×C5⋊D4, D20.35C23

Smallest permutation representation of D20.35C23
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 23)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 60)(58 59)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(93 100)(94 99)(95 98)(96 97)(101 109)(102 108)(103 107)(104 106)(110 120)(111 119)(112 118)(113 117)(114 116)(121 136)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)(137 140)(138 139)(142 160)(143 159)(144 158)(145 157)(146 156)(147 155)(148 154)(149 153)(150 152)
(1 97 11 87)(2 98 12 88)(3 99 13 89)(4 100 14 90)(5 81 15 91)(6 82 16 92)(7 83 17 93)(8 84 18 94)(9 85 19 95)(10 86 20 96)(21 104 31 114)(22 105 32 115)(23 106 33 116)(24 107 34 117)(25 108 35 118)(26 109 36 119)(27 110 37 120)(28 111 38 101)(29 112 39 102)(30 113 40 103)(41 131 51 121)(42 132 52 122)(43 133 53 123)(44 134 54 124)(45 135 55 125)(46 136 56 126)(47 137 57 127)(48 138 58 128)(49 139 59 129)(50 140 60 130)(61 148 71 158)(62 149 72 159)(63 150 73 160)(64 151 74 141)(65 152 75 142)(66 153 76 143)(67 154 77 144)(68 155 78 145)(69 156 79 146)(70 157 80 147)
(1 59 11 49)(2 60 12 50)(3 41 13 51)(4 42 14 52)(5 43 15 53)(6 44 16 54)(7 45 17 55)(8 46 18 56)(9 47 19 57)(10 48 20 58)(21 63 31 73)(22 64 32 74)(23 65 33 75)(24 66 34 76)(25 67 35 77)(26 68 36 78)(27 69 37 79)(28 70 38 80)(29 71 39 61)(30 72 40 62)(81 123 91 133)(82 124 92 134)(83 125 93 135)(84 126 94 136)(85 127 95 137)(86 128 96 138)(87 129 97 139)(88 130 98 140)(89 131 99 121)(90 132 100 122)(101 157 111 147)(102 158 112 148)(103 159 113 149)(104 160 114 150)(105 141 115 151)(106 142 116 152)(107 143 117 153)(108 144 118 154)(109 145 119 155)(110 146 120 156)
(1 40)(2 31)(3 22)(4 33)(5 24)(6 35)(7 26)(8 37)(9 28)(10 39)(11 30)(12 21)(13 32)(14 23)(15 34)(16 25)(17 36)(18 27)(19 38)(20 29)(41 64)(42 75)(43 66)(44 77)(45 68)(46 79)(47 70)(48 61)(49 72)(50 63)(51 74)(52 65)(53 76)(54 67)(55 78)(56 69)(57 80)(58 71)(59 62)(60 73)(81 107)(82 118)(83 109)(84 120)(85 111)(86 102)(87 113)(88 104)(89 115)(90 106)(91 117)(92 108)(93 119)(94 110)(95 101)(96 112)(97 103)(98 114)(99 105)(100 116)(121 141)(122 152)(123 143)(124 154)(125 145)(126 156)(127 147)(128 158)(129 149)(130 160)(131 151)(132 142)(133 153)(134 144)(135 155)(136 146)(137 157)(138 148)(139 159)(140 150)```

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

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

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

56 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 ··· 10L 20A ··· 20T 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 20 ··· 20 size 1 1 2 2 2 4 20 2 2 2 2 4 4 4 20 20 20 2 2 10 10 20 20 20 2 2 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 8 type + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 C5⋊D4 C5⋊D4 Q8○D8 D20.35C23 kernel D20.35C23 C20.C23 C2×C5⋊Q16 D4.Dic5 D4.8D10 D4.9D10 D4.10D10 C5×2- 1+4 C5×D4 C5×Q8 2- 1+4 C2×Q8 C4○D4 D4 Q8 C5 C1 # reps 1 3 3 1 3 3 1 1 3 1 2 6 8 12 4 2 2

Matrix representation of D20.35C23 in GL6(𝔽41)

 34 1 0 0 0 0 40 0 0 0 0 0 0 0 1 21 0 0 0 0 37 40 0 0 0 0 10 33 1 18 0 0 29 2 9 40
,
 1 34 0 0 0 0 0 40 0 0 0 0 0 0 1 21 0 0 0 0 0 40 0 0 0 0 0 33 1 18 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 14 12 13 0 0 0 0 4 0 17 0 0 29 37 27 19 0 0 0 40 0 37
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 30 2 9 0 0 0 0 1 0 37 0 0 32 25 11 10 0 0 0 21 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 24 6 0 0 0 0 34 17 0 0 0 0 38 5 24 11 0 0 20 24 26 17

`G:=sub<GL(6,GF(41))| [34,40,0,0,0,0,1,0,0,0,0,0,0,0,1,37,10,29,0,0,21,40,33,2,0,0,0,0,1,9,0,0,0,0,18,40],[1,0,0,0,0,0,34,40,0,0,0,0,0,0,1,0,0,0,0,0,21,40,33,0,0,0,0,0,1,0,0,0,0,0,18,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,0,29,0,0,0,12,4,37,40,0,0,13,0,27,0,0,0,0,17,19,37],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,0,32,0,0,0,2,1,25,21,0,0,9,0,11,0,0,0,0,37,10,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,34,38,20,0,0,6,17,5,24,0,0,0,0,24,26,0,0,0,0,11,17] >;`

D20.35C23 in GAP, Magma, Sage, TeX

`D_{20}._{35}C_2^3`
`% in TeX`

`G:=Group("D20.35C2^3");`
`// GroupNames label`

`G:=SmallGroup(320,1510);`
`// by ID`

`G=gap.SmallGroup(320,1510);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,675,1684,235,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^20=b^2=e^2=1,c^2=d^2=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^11,b*c=c*b,b*d=d*b,e*b*e=a^15*b,d*c*d^-1=a^10*c,c*e=e*c,d*e=e*d>;`
`// generators/relations`

׿
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