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## G = C40.28D4order 320 = 26·5

### 28th non-split extension by C40 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C40.28D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×D20 — C2×D40 — C40.28D4
 Lower central C5 — C10 — C2×C20 — C40.28D4
 Upper central C1 — C22 — C2×C4 — C2×Q16

Generators and relations for C40.28D4
G = < a,b,c | a40=b4=c2=1, bab-1=a9, cac=a-1, cbc=a20b-1 >

Subgroups: 622 in 130 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×4], D4 [×4], Q8 [×4], C23 [×2], D5 [×2], C10, C10 [×2], C42, C22⋊C4 [×4], C2×C8, C2×C8, D8 [×2], SD16 [×4], Q16 [×2], C2×D4 [×2], C2×Q8 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C4×C8, C4.4D4 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C52C8 [×2], C40 [×2], D20 [×4], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C22×D5 [×2], C8.12D4, D40 [×2], C2×C52C8, C4×Dic5, D10⋊C4 [×4], Q8⋊D5 [×4], C2×C40, C5×Q16 [×2], C2×D20 [×2], Q8×C10 [×2], C8×Dic5, C2×D40, C2×Q8⋊D5 [×2], C20.23D4 [×2], C10×Q16, C40.28D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C41D4, C4○D8 [×2], C5⋊D4 [×2], C22×D5, C8.12D4, D4×D5 [×2], C2×C5⋊D4, Q8.D10 [×2], C20⋊D4, C40.28D4

Smallest permutation representation of C40.28D4
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)
(1 135 100 57)(2 144 101 66)(3 153 102 75)(4 122 103 44)(5 131 104 53)(6 140 105 62)(7 149 106 71)(8 158 107 80)(9 127 108 49)(10 136 109 58)(11 145 110 67)(12 154 111 76)(13 123 112 45)(14 132 113 54)(15 141 114 63)(16 150 115 72)(17 159 116 41)(18 128 117 50)(19 137 118 59)(20 146 119 68)(21 155 120 77)(22 124 81 46)(23 133 82 55)(24 142 83 64)(25 151 84 73)(26 160 85 42)(27 129 86 51)(28 138 87 60)(29 147 88 69)(30 156 89 78)(31 125 90 47)(32 134 91 56)(33 143 92 65)(34 152 93 74)(35 121 94 43)(36 130 95 52)(37 139 96 61)(38 148 97 70)(39 157 98 79)(40 126 99 48)
(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(41 131)(42 130)(43 129)(44 128)(45 127)(46 126)(47 125)(48 124)(49 123)(50 122)(51 121)(52 160)(53 159)(54 158)(55 157)(56 156)(57 155)(58 154)(59 153)(60 152)(61 151)(62 150)(63 149)(64 148)(65 147)(66 146)(67 145)(68 144)(69 143)(70 142)(71 141)(72 140)(73 139)(74 138)(75 137)(76 136)(77 135)(78 134)(79 133)(80 132)(81 119)(82 118)(83 117)(84 116)(85 115)(86 114)(87 113)(88 112)(89 111)(90 110)(91 109)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)```

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 C4○D8 C5⋊D4 D4×D5 D4×D5 Q8.D10 kernel C40.28D4 C8×Dic5 C2×D40 C2×Q8⋊D5 C20.23D4 C10×Q16 C5⋊2C8 C40 C2×Dic5 C2×Q16 C2×C8 C2×Q8 C10 C8 C4 C22 C2 # reps 1 1 1 2 2 1 2 2 2 2 2 4 8 8 2 2 8

Matrix representation of C40.28D4 in GL4(𝔽41) generated by

 34 7 0 0 34 1 0 0 0 0 0 30 0 0 15 17
,
 3 3 0 0 24 38 0 0 0 0 32 39 0 0 40 9
,
 34 7 0 0 40 7 0 0 0 0 40 0 0 0 9 1
`G:=sub<GL(4,GF(41))| [34,34,0,0,7,1,0,0,0,0,0,15,0,0,30,17],[3,24,0,0,3,38,0,0,0,0,32,40,0,0,39,9],[34,40,0,0,7,7,0,0,0,0,40,9,0,0,0,1] >;`

C40.28D4 in GAP, Magma, Sage, TeX

`C_{40}._{28}D_4`
`% in TeX`

`G:=Group("C40.28D4");`
`// GroupNames label`

`G:=SmallGroup(320,818);`
`// by ID`

`G=gap.SmallGroup(320,818);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,555,184,1684,438,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=a^20*b^-1>;`
`// generators/relations`

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