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## G = C22.D40order 320 = 26·5

### 3rd non-split extension by C22 of D40 acting via D40/D20=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C22.D40
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C20⋊7D4 — C22.D40
 Lower central C5 — C10 — C2×C20 — C22.D40
 Upper central C1 — C22 — C22×C4 — C22⋊C8

Generators and relations for C22.D40
G = < a,b,c,d | a2=b2=c40=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 542 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C2.D8, C2×C4⋊C4, C4⋊D4, C40, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C22.D8, C4⋊Dic5, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C2×C40, C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, C405C4, D205C4, C5×C22⋊C8, C2×C4⋊Dic5, C207D4, C22.D40
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C4○D4, D10, C22.D4, C2×D8, C8.C22, D20, C22×D5, C22.D8, D40, C2×D20, D42D5, C22.D20, C2×D40, C8.D10, C22.D40

Smallest permutation representation of C22.D40
On 160 points
Generators in S160
```(1 100)(2 42)(3 102)(4 44)(5 104)(6 46)(7 106)(8 48)(9 108)(10 50)(11 110)(12 52)(13 112)(14 54)(15 114)(16 56)(17 116)(18 58)(19 118)(20 60)(21 120)(22 62)(23 82)(24 64)(25 84)(26 66)(27 86)(28 68)(29 88)(30 70)(31 90)(32 72)(33 92)(34 74)(35 94)(36 76)(37 96)(38 78)(39 98)(40 80)(41 156)(43 158)(45 160)(47 122)(49 124)(51 126)(53 128)(55 130)(57 132)(59 134)(61 136)(63 138)(65 140)(67 142)(69 144)(71 146)(73 148)(75 150)(77 152)(79 154)(81 137)(83 139)(85 141)(87 143)(89 145)(91 147)(93 149)(95 151)(97 153)(99 155)(101 157)(103 159)(105 121)(107 123)(109 125)(111 127)(113 129)(115 131)(117 133)(119 135)
(1 156)(2 157)(3 158)(4 159)(5 160)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 129)(15 130)(16 131)(17 132)(18 133)(19 134)(20 135)(21 136)(22 137)(23 138)(24 139)(25 140)(26 141)(27 142)(28 143)(29 144)(30 145)(31 146)(32 147)(33 148)(34 149)(35 150)(36 151)(37 152)(38 153)(39 154)(40 155)(41 100)(42 101)(43 102)(44 103)(45 104)(46 105)(47 106)(48 107)(49 108)(50 109)(51 110)(52 111)(53 112)(54 113)(55 114)(56 115)(57 116)(58 117)(59 118)(60 119)(61 120)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 99)
(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)
(1 155 156 40)(2 39 157 154)(3 153 158 38)(4 37 159 152)(5 151 160 36)(6 35 121 150)(7 149 122 34)(8 33 123 148)(9 147 124 32)(10 31 125 146)(11 145 126 30)(12 29 127 144)(13 143 128 28)(14 27 129 142)(15 141 130 26)(16 25 131 140)(17 139 132 24)(18 23 133 138)(19 137 134 22)(20 21 135 136)(41 80 100 99)(42 98 101 79)(43 78 102 97)(44 96 103 77)(45 76 104 95)(46 94 105 75)(47 74 106 93)(48 92 107 73)(49 72 108 91)(50 90 109 71)(51 70 110 89)(52 88 111 69)(53 68 112 87)(54 86 113 67)(55 66 114 85)(56 84 115 65)(57 64 116 83)(58 82 117 63)(59 62 118 81)(60 120 119 61)```

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

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

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

59 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 ··· 20H 20I 20J 20K 20L 40A ··· 40P 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 40 ··· 40 size 1 1 1 1 2 2 40 2 2 4 20 20 20 20 40 2 2 4 4 4 4 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

59 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D8 D10 D10 D20 D20 D40 C8.C22 D4⋊2D5 C8.D10 kernel C22.D40 C40⋊5C4 D20⋊5C4 C5×C22⋊C8 C2×C4⋊Dic5 C20⋊7D4 C2×C20 C22×C10 C22⋊C8 C20 C2×C10 C2×C8 C22×C4 C2×C4 C23 C22 C10 C4 C2 # reps 1 2 2 1 1 1 1 1 2 4 4 4 2 4 4 16 1 4 4

Matrix representation of C22.D40 in GL4(𝔽41) generated by

 13 31 0 0 25 28 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 32 0 0 0 34 9 0 0 0 0 2 35 0 0 6 3
,
 32 0 0 0 0 32 0 0 0 0 39 6 0 0 20 2
`G:=sub<GL(4,GF(41))| [13,25,0,0,31,28,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[32,34,0,0,0,9,0,0,0,0,2,6,0,0,35,3],[32,0,0,0,0,32,0,0,0,0,39,20,0,0,6,2] >;`

C22.D40 in GAP, Magma, Sage, TeX

`C_2^2.D_{40}`
`% in TeX`

`G:=Group("C2^2.D40");`
`// GroupNames label`

`G:=SmallGroup(320,363);`
`// by ID`

`G=gap.SmallGroup(320,363);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,219,310,1123,136,12550]);`
`// Polycyclic`

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

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