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## G = C40.31C23order 320 = 26·5

### 24th non-split extension by C40 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C40.31C23
 Chief series C1 — C5 — C10 — C20 — C40 — Dic20 — C2×Dic20 — C40.31C23
 Lower central C5 — C10 — C20 — C40 — C40.31C23
 Upper central C1 — C2 — C2×C4 — C2×C8 — C4○D8

Generators and relations for C40.31C23
G = < a,b,c,d | a40=c2=d2=1, b2=a20, bab-1=a-1, ac=ca, dad=a31, bc=cb, dbd=a25b, dcd=a20c >

Subgroups: 302 in 82 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×3], C22, C22, C5, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], C10, C10 [×2], C16 [×2], C2×C8, D8, SD16, Q16, Q16 [×3], C2×Q8, C4○D4, Dic5 [×2], C20 [×2], C20, C2×C10, C2×C10, M5(2), SD32 [×2], Q32 [×2], C2×Q16, C4○D8, C40 [×2], Dic10 [×3], C2×Dic5, C2×C20, C2×C20, C5×D4 [×2], C5×Q8, Q32⋊C2, C52C16 [×2], Dic20 [×2], Dic20, C2×C40, C5×D8, C5×SD16, C5×Q16, C2×Dic10, C5×C4○D4, C20.4C8, D8.D5 [×2], C5⋊Q32 [×2], C2×Dic20, C5×C4○D8, C40.31C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], C2×D8, C5⋊D4 [×2], C22×D5, Q32⋊C2, D4⋊D5 [×2], C2×C5⋊D4, C2×D4⋊D5, C40.31C23

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 D4 D4 D5 D8 D8 D10 D10 D10 C5⋊D4 C5⋊D4 Q32⋊C2 D4⋊D5 D4⋊D5 C40.31C23 kernel C40.31C23 C20.4C8 D8.D5 C5⋊Q32 C2×Dic20 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 2 2 1 1 1 1 2 2 2 2 2 2 4 4 2 2 2 8

Matrix representation of C40.31C23 in GL6(𝔽241)

 190 51 0 0 0 0 190 240 0 0 0 0 0 0 11 230 0 0 0 0 11 11 0 0 0 0 0 0 11 230 0 0 0 0 11 11
,
 73 92 0 0 0 0 225 168 0 0 0 0 0 0 218 240 239 46 0 0 240 23 46 2 0 0 239 46 218 240 0 0 46 2 240 23
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(241))| [190,190,0,0,0,0,51,240,0,0,0,0,0,0,11,11,0,0,0,0,230,11,0,0,0,0,0,0,11,11,0,0,0,0,230,11],[73,225,0,0,0,0,92,168,0,0,0,0,0,0,218,240,239,46,0,0,240,23,46,2,0,0,239,46,218,240,0,0,46,2,240,23],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1] >;`

C40.31C23 in GAP, Magma, Sage, TeX

`C_{40}._{31}C_2^3`
`% in TeX`

`G:=Group("C40.31C2^3");`
`// GroupNames label`

`G:=SmallGroup(320,822);`
`// by ID`

`G=gap.SmallGroup(320,822);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,254,387,675,185,192,1684,438,102,12550]);`
`// Polycyclic`

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

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