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## G = C40.30C23order 320 = 26·5

### 23rd non-split extension by C40 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C40.30C23
 Chief series C1 — C5 — C10 — C20 — C40 — D40 — D40⋊7C2 — C40.30C23
 Lower central C5 — C10 — C20 — C40 — C40.30C23
 Upper central C1 — C4 — C2×C4 — C2×C8 — C4○D8

Generators and relations for C40.30C23
G = < a,b,c,d | a40=b2=d2=1, c2=a20, bab=a-1, ac=ca, dad=a31, bc=cb, dbd=a25b, cd=dc >

Subgroups: 350 in 84 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C5, C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], D5, C10, C10 [×2], C16 [×2], C2×C8, D8, D8, SD16 [×2], Q16, Q16, C4○D4 [×2], Dic5, C20 [×2], C20, D10, C2×C10, C2×C10, C2×C16, D16, SD32 [×2], Q32, C4○D8, C4○D8, C40 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4 [×2], C5×Q8, C4○D16, C52C16 [×2], C40⋊C2, D40, Dic20, C2×C40, C5×D8, C5×SD16, C5×Q16, C4○D20, C5×C4○D4, C2×C52C16, C5⋊D16, D8.D5, C5⋊SD32, C5⋊Q32, D407C2, C5×C4○D8, C40.30C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], C2×D8, C5⋊D4 [×2], C22×D5, C4○D16, D4⋊D5 [×2], C2×C5⋊D4, C2×D4⋊D5, C40.30C23

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

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 D8 D8 D10 D10 D10 C5⋊D4 C5⋊D4 C4○D16 D4⋊D5 D4⋊D5 C40.30C23 kernel C40.30C23 C2×C5⋊2C16 C5⋊D16 D8.D5 C5⋊SD32 C5⋊Q32 D40⋊7C2 C5×C4○D8 C40 C2×C20 C4○D8 C20 C2×C10 C2×C8 D8 Q16 C8 C2×C4 C5 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 8 2 2 8

Matrix representation of C40.30C23 in GL4(𝔽241) generated by

 190 52 0 0 190 0 0 0 0 0 219 22 0 0 230 0
,
 51 1 0 0 51 190 0 0 0 0 0 219 0 0 230 0
,
 240 0 0 0 0 240 0 0 0 0 64 0 0 0 0 64
,
 76 138 0 0 152 165 0 0 0 0 129 54 0 0 156 112
`G:=sub<GL(4,GF(241))| [190,190,0,0,52,0,0,0,0,0,219,230,0,0,22,0],[51,51,0,0,1,190,0,0,0,0,0,230,0,0,219,0],[240,0,0,0,0,240,0,0,0,0,64,0,0,0,0,64],[76,152,0,0,138,165,0,0,0,0,129,156,0,0,54,112] >;`

C40.30C23 in GAP, Magma, Sage, TeX

`C_{40}._{30}C_2^3`
`% in TeX`

`G:=Group("C40.30C2^3");`
`// GroupNames label`

`G:=SmallGroup(320,821);`
`// by ID`

`G=gap.SmallGroup(320,821);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,675,185,192,1684,438,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^40=b^2=d^2=1,c^2=a^20,b*a*b=a^-1,a*c=c*a,d*a*d=a^31,b*c=c*b,d*b*d=a^25*b,c*d=d*c>;`
`// generators/relations`

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