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G = D20⋊8Q8order 320 = 26·5

6th semidirect product of D20 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D20⋊8Q8
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×Q8×D5 — D20⋊8Q8
 Lower central C5 — C2×C10 — D20⋊8Q8
 Upper central C1 — C22 — C4⋊Q8

Generators and relations for D208Q8
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=c-1 >

Subgroups: 966 in 280 conjugacy classes, 115 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×13], C22, C22 [×8], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×18], D4 [×4], Q8 [×16], C23 [×2], D5 [×4], C10 [×3], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×8], C22×C4 [×6], C2×D4, C2×Q8 [×2], C2×Q8 [×13], Dic5 [×4], Dic5 [×4], C20 [×4], C20 [×5], D10 [×4], D10 [×4], C2×C10, C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C4⋊Q8, C4⋊Q8 [×2], C22×Q8 [×2], Dic10 [×4], Dic10 [×8], C4×D5 [×12], D20 [×4], C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×4], C22×D5 [×2], D4×Q8, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5 [×2], D10⋊C4 [×6], C4×C20, C5×C4⋊C4 [×4], C2×Dic10, C2×Dic10 [×4], C2×C4×D5 [×6], C2×D20, Q8×D5 [×8], Q8×C10 [×2], C4×Dic10, C4×D20, C20⋊Q8 [×2], D208C4 [×2], D10⋊Q8 [×4], D103Q8 [×2], C5×C4⋊Q8, C2×Q8×D5 [×2], D208Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×6], C24, D10 [×7], C22×D4, C22×Q8, 2- 1+4, C22×D5 [×7], D4×Q8, D4×D5 [×2], Q8×D5 [×2], C23×D5, C2×D4×D5, C2×Q8×D5, Q8.10D10, D208Q8

Smallest permutation representation of D208Q8
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(81 89)(82 88)(83 87)(84 86)(90 100)(91 99)(92 98)(93 97)(94 96)(101 105)(102 104)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(121 133)(122 132)(123 131)(124 130)(125 129)(126 128)(134 140)(135 139)(136 138)(141 157)(142 156)(143 155)(144 154)(145 153)(146 152)(147 151)(148 150)(158 160)
(1 88 106 140)(2 89 107 121)(3 90 108 122)(4 91 109 123)(5 92 110 124)(6 93 111 125)(7 94 112 126)(8 95 113 127)(9 96 114 128)(10 97 115 129)(11 98 116 130)(12 99 117 131)(13 100 118 132)(14 81 119 133)(15 82 120 134)(16 83 101 135)(17 84 102 136)(18 85 103 137)(19 86 104 138)(20 87 105 139)(21 55 143 78)(22 56 144 79)(23 57 145 80)(24 58 146 61)(25 59 147 62)(26 60 148 63)(27 41 149 64)(28 42 150 65)(29 43 151 66)(30 44 152 67)(31 45 153 68)(32 46 154 69)(33 47 155 70)(34 48 156 71)(35 49 157 72)(36 50 158 73)(37 51 159 74)(38 52 160 75)(39 53 141 76)(40 54 142 77)
(1 59 106 62)(2 50 107 73)(3 41 108 64)(4 52 109 75)(5 43 110 66)(6 54 111 77)(7 45 112 68)(8 56 113 79)(9 47 114 70)(10 58 115 61)(11 49 116 72)(12 60 117 63)(13 51 118 74)(14 42 119 65)(15 53 120 76)(16 44 101 67)(17 55 102 78)(18 46 103 69)(19 57 104 80)(20 48 105 71)(21 136 143 84)(22 127 144 95)(23 138 145 86)(24 129 146 97)(25 140 147 88)(26 131 148 99)(27 122 149 90)(28 133 150 81)(29 124 151 92)(30 135 152 83)(31 126 153 94)(32 137 154 85)(33 128 155 96)(34 139 156 87)(35 130 157 98)(36 121 158 89)(37 132 159 100)(38 123 160 91)(39 134 141 82)(40 125 142 93)```

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4I 4J 4K 4L 4M 4N 4O 4P 4Q 5A 5B 10A ··· 10F 20A ··· 20L 20M ··· 20T order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 10 10 10 10 2 2 2 2 4 ··· 4 10 10 10 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4 8 ··· 8

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + - + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 Q8 D5 D10 D10 D10 2- 1+4 D4×D5 Q8×D5 Q8.10D10 kernel D20⋊8Q8 C4×Dic10 C4×D20 C20⋊Q8 D20⋊8C4 D10⋊Q8 D10⋊3Q8 C5×C4⋊Q8 C2×Q8×D5 Dic10 D20 C4⋊Q8 C42 C4⋊C4 C2×Q8 C10 C4 C4 C2 # reps 1 1 1 2 2 4 2 1 2 4 4 2 2 8 4 1 4 4 4

Matrix representation of D208Q8 in GL6(𝔽41)

 8 4 0 0 0 0 35 33 0 0 0 0 0 0 35 1 0 0 0 0 5 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 37 40 0 0 0 0 0 0 40 40 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 1 0
,
 33 37 0 0 0 0 26 8 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 30 1 0 0 0 0 1 11

`G:=sub<GL(6,GF(41))| [8,35,0,0,0,0,4,33,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,37,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[33,26,0,0,0,0,37,8,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,1,0,0,0,0,1,11] >;`

D208Q8 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_8Q_8`
`% in TeX`

`G:=Group("D20:8Q8");`
`// GroupNames label`

`G:=SmallGroup(320,1399);`
`// by ID`

`G=gap.SmallGroup(320,1399);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,1571,297,136,12550]);`
`// Polycyclic`

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

׿
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