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## G = D20.30D4order 320 = 26·5

### 13rd non-split extension by D20 of D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D20.30D4
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C4○D20 — Q8.10D10 — D20.30D4
 Lower central C5 — C10 — C20 — D20.30D4
 Upper central C1 — C2 — C2×C4 — C2×Q16

Generators and relations for D20.30D4
G = < a,b,c,d | a20=b2=1, c4=d2=a10, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=a10c3 >

Subgroups: 886 in 248 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, Q8, D5, C10, C10, C2×C8, C2×C8, M4(2), D8, SD16, Q16, Q16, C2×Q8, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C8○D4, C2×Q16, C2×Q16, C4○D8, C8.C22, 2- 1+4, C52C8, C40, Dic10, Dic10, Dic10, C4×D5, C4×D5, D20, D20, D20, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×Q8, C5×Q8, Q8○D8, C8×D5, C8⋊D5, C40⋊C2, D40, Dic20, C4.Dic5, Q8⋊D5, C5⋊Q16, C2×C40, C5×Q16, C4○D20, C4○D20, C4○D20, Q8×D5, Q8×D5, Q82D5, Q82D5, Q8×C10, D20.3C4, D407C2, D5×Q16, Q16⋊D5, Q8.D10, C20.C23, C10×Q16, Q8.10D10, D20.30D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C22×D5, Q8○D8, D4×D5, C23×D5, C2×D4×D5, D20.30D4

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

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

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

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

50 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 ··· 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 4 4 5 5 8 8 8 8 8 10 ··· 10 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 2 10 10 20 20 2 2 4 4 4 4 10 10 20 20 2 2 2 2 4 20 20 2 ··· 2 4 4 4 4 8 ··· 8 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 Q8○D8 D4×D5 D4×D5 D20.30D4 kernel D20.30D4 D20.3C4 D40⋊7C2 D5×Q16 Q16⋊D5 Q8.D10 C20.C23 C10×Q16 Q8.10D10 Dic10 D20 C5⋊D4 C2×Q16 C2×C8 Q16 C2×Q8 C5 C4 C22 C1 # reps 1 1 1 2 4 2 2 1 2 1 1 2 2 2 8 4 2 2 2 8

Matrix representation of D20.30D4 in GL4(𝔽41) generated by

 0 0 6 23 0 0 18 21 35 18 0 0 23 20 0 0
,
 8 17 0 15 24 33 15 0 0 26 8 17 26 0 24 33
,
 12 0 29 0 0 12 0 29 12 0 12 0 0 12 0 12
,
 15 0 17 8 0 15 33 24 17 8 26 0 33 24 0 26
`G:=sub<GL(4,GF(41))| [0,0,35,23,0,0,18,20,6,18,0,0,23,21,0,0],[8,24,0,26,17,33,26,0,0,15,8,24,15,0,17,33],[12,0,12,0,0,12,0,12,29,0,12,0,0,29,0,12],[15,0,17,33,0,15,8,24,17,33,26,0,8,24,0,26] >;`

D20.30D4 in GAP, Magma, Sage, TeX

`D_{20}._{30}D_4`
`% in TeX`

`G:=Group("D20.30D4");`
`// GroupNames label`

`G:=SmallGroup(320,1438);`
`// by ID`

`G=gap.SmallGroup(320,1438);`
`# by ID`

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

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

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