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## G = C40.70C23order 320 = 26·5

### 16th non-split extension by C40 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C40.70C23
 Chief series C1 — C5 — C10 — C20 — C40 — C5⋊2C16 — C2×C5⋊2C16 — C40.70C23
 Lower central C5 — C10 — C40.70C23
 Upper central C1 — C8 — C8○D4

Generators and relations for C40.70C23
G = < a,b,c,d | a40=c2=d2=1, b2=a25, bab-1=a9, ac=ca, ad=da, bc=cb, bd=db, dcd=a20c >

Subgroups: 134 in 84 conjugacy classes, 67 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, C10, C10, C16, C2×C8, M4(2), C4○D4, C20, C20, C2×C10, C2×C16, M5(2), C8○D4, C40, C40, C2×C20, C5×D4, C5×Q8, D4○C16, C52C16, C52C16, C2×C40, C5×M4(2), C5×C4○D4, C2×C52C16, C20.4C8, C5×C8○D4, C40.70C23
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C2×C8, C22×C4, Dic5, D10, C22×C8, C52C8, C2×Dic5, C22×D5, D4○C16, C2×C52C8, C22×Dic5, C22×C52C8, C40.70C23

Smallest permutation representation of C40.70C23
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)
(1 108 26 93 11 118 36 103 21 88 6 113 31 98 16 83)(2 117 27 102 12 87 37 112 22 97 7 82 32 107 17 92)(3 86 28 111 13 96 38 81 23 106 8 91 33 116 18 101)(4 95 29 120 14 105 39 90 24 115 9 100 34 85 19 110)(5 104 30 89 15 114 40 99 25 84 10 109 35 94 20 119)(41 122 66 147 51 132 76 157 61 142 46 127 71 152 56 137)(42 131 67 156 52 141 77 126 62 151 47 136 72 121 57 146)(43 140 68 125 53 150 78 135 63 160 48 145 73 130 58 155)(44 149 69 134 54 159 79 144 64 129 49 154 74 139 59 124)(45 158 70 143 55 128 80 153 65 138 50 123 75 148 60 133)
(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(1 79)(2 80)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)(81 157)(82 158)(83 159)(84 160)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)(91 127)(92 128)(93 129)(94 130)(95 131)(96 132)(97 133)(98 134)(99 135)(100 136)(101 137)(102 138)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)(109 145)(110 146)(111 147)(112 148)(113 149)(114 150)(115 151)(116 152)(117 153)(118 154)(119 155)(120 156)```

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 8E ··· 8J 10A 10B 10C ··· 10H 16A ··· 16H 16I ··· 16T 20A 20B 20C 20D 20E ··· 20J 40A ··· 40H 40I ··· 40T order 1 2 2 2 2 4 4 4 4 4 5 5 8 8 8 8 8 ··· 8 10 10 10 ··· 10 16 ··· 16 16 ··· 16 20 20 20 20 20 ··· 20 40 ··· 40 40 ··· 40 size 1 1 2 2 2 1 1 2 2 2 2 2 1 1 1 1 2 ··· 2 2 2 4 ··· 4 5 ··· 5 10 ··· 10 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + - - image C1 C2 C2 C2 C4 C4 C8 C8 D5 D10 Dic5 Dic5 C5⋊2C8 C5⋊2C8 D4○C16 C40.70C23 kernel C40.70C23 C2×C5⋊2C16 C20.4C8 C5×C8○D4 C5×M4(2) C5×C4○D4 C5×D4 C5×Q8 C8○D4 C2×C8 M4(2) C4○D4 D4 Q8 C5 C1 # reps 1 3 3 1 6 2 12 4 2 6 6 2 12 4 8 8

Matrix representation of C40.70C23 in GL4(𝔽241) generated by

 177 131 0 0 110 110 0 0 0 0 8 0 0 0 0 8
,
 180 129 0 0 151 61 0 0 0 0 197 0 0 0 0 197
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 240
,
 240 0 0 0 0 240 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(241))| [177,110,0,0,131,110,0,0,0,0,8,0,0,0,0,8],[180,151,0,0,129,61,0,0,0,0,197,0,0,0,0,197],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,240],[240,0,0,0,0,240,0,0,0,0,0,1,0,0,1,0] >;`

C40.70C23 in GAP, Magma, Sage, TeX

`C_{40}._{70}C_2^3`
`% in TeX`

`G:=Group("C40.70C2^3");`
`// GroupNames label`

`G:=SmallGroup(320,767);`
`// by ID`

`G=gap.SmallGroup(320,767);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,80,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^40=c^2=d^2=1,b^2=a^25,b*a*b^-1=a^9,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^20*c>;`
`// generators/relations`

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