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## G = D20.23D4order 320 = 26·5

### 6th non-split extension by D20 of D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.23D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C4×D20 — D20.23D4
 Lower central C5 — C10 — C2×C20 — D20.23D4
 Upper central C1 — C22 — C42 — C4.4D4

Generators and relations for D20.23D4
G = < a,b,c,d | a20=b2=c4=1, d2=a10, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a15b, dcd-1=a10c-1 >

Subgroups: 518 in 124 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C52C8, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, D4.2D4, C2×C52C8, C4⋊Dic5, D10⋊C4, D4⋊D5, Q8⋊D5, C4×C20, C5×C22⋊C4, C2×C4×D5, C2×D20, D4×C10, Q8×C10, C203C8, D4⋊Dic5, Q8⋊Dic5, C4×D20, C2×D4⋊D5, C2×Q8⋊D5, C5×C4.4D4, D20.23D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8⋊C22, C5⋊D4, C22×D5, D4.2D4, D4×D5, D42D5, C2×C5⋊D4, C202D4, D4⋊D10, D4.8D10, D20.23D4

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

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

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

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

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 10G 10H 10I 10J 20A ··· 20L 20M 20N 20O 20P order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 size 1 1 1 1 8 20 20 2 2 2 2 4 8 20 20 2 2 20 20 20 20 2 ··· 2 8 8 8 8 4 ··· 4 8 8 8 8

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 C5⋊D4 C8⋊C22 D4×D5 D4⋊2D5 D4⋊D10 D4.8D10 kernel D20.23D4 C20⋊3C8 D4⋊Dic5 Q8⋊Dic5 C4×D20 C2×D4⋊D5 C2×Q8⋊D5 C5×C4.4D4 D20 C2×C20 C4.4D4 C20 C42 C2×D4 C2×Q8 C10 C2×C4 C10 C4 C4 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 8 1 2 2 4 4

Matrix representation of D20.23D4 in GL6(𝔽41)

 35 40 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 9 38 0 0 0 0 0 32
,
 18 6 0 0 0 0 35 23 0 0 0 0 0 0 15 15 0 0 0 0 15 26 0 0 0 0 0 0 30 13 0 0 0 0 16 11

`G:=sub<GL(6,GF(41))| [35,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,38,32],[18,35,0,0,0,0,6,23,0,0,0,0,0,0,15,15,0,0,0,0,15,26,0,0,0,0,0,0,30,16,0,0,0,0,13,11] >;`

D20.23D4 in GAP, Magma, Sage, TeX

`D_{20}._{23}D_4`
`% in TeX`

`G:=Group("D20.23D4");`
`// GroupNames label`

`G:=SmallGroup(320,684);`
`// by ID`

`G=gap.SmallGroup(320,684);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,344,254,219,1123,297,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=a^10,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=a^10*c^-1>;`
`// generators/relations`

׿
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