metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8.9D10, C40.33D4, C20.52D8, C40.31C23, Q16.10D10, Dic20.14C22, D8.D5⋊6C2, C4○D8.3D5, C5⋊Q32⋊6C2, (C2×C10).11D8, C10.70(C2×D8), (C2×C8).99D10, C8.8(C5⋊D4), C5⋊5(Q32⋊C2), C20.4C8⋊7C2, C4.25(D4⋊D5), C20.193(C2×D4), (C2×C20).187D4, (C5×D8).9C22, C8.37(C22×D5), (C2×Dic20)⋊22C2, C5⋊2C16.4C22, C22.6(D4⋊D5), (C2×C40).105C22, (C5×Q16).10C22, C2.25(C2×D4⋊D5), (C5×C4○D8).4C2, C4.19(C2×C5⋊D4), (C2×C4).82(C5⋊D4), SmallGroup(320,822)
Series: Derived ►Chief ►Lower central ►Upper central
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, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C10, C10, C16, C2×C8, D8, SD16, Q16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, M5(2), SD32, Q32, C2×Q16, C4○D8, C40, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, Q32⋊C2, C5⋊2C16, Dic20, Dic20, C2×C40, C5×D8, C5×SD16, C5×Q16, C2×Dic10, C5×C4○D4, C20.4C8, D8.D5, C5⋊Q32, C2×Dic20, C5×C4○D8, C40.31C23
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, C5⋊D4, C22×D5, Q32⋊C2, D4⋊D5, C2×C5⋊D4, C2×D4⋊D5, 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)(41 132 61 152)(42 131 62 151)(43 130 63 150)(44 129 64 149)(45 128 65 148)(46 127 66 147)(47 126 67 146)(48 125 68 145)(49 124 69 144)(50 123 70 143)(51 122 71 142)(52 121 72 141)(53 160 73 140)(54 159 74 139)(55 158 75 138)(56 157 76 137)(57 156 77 136)(58 155 78 135)(59 154 79 134)(60 153 80 133)
(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)(81 139)(82 140)(83 141)(84 142)(85 143)(86 144)(87 145)(88 146)(89 147)(90 148)(91 149)(92 150)(93 151)(94 152)(95 153)(96 154)(97 155)(98 156)(99 157)(100 158)(101 159)(102 160)(103 121)(104 122)(105 123)(106 124)(107 125)(108 126)(109 127)(110 128)(111 129)(112 130)(113 131)(114 132)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)
(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)(121 136)(122 127)(123 158)(124 149)(125 140)(126 131)(128 153)(129 144)(130 135)(132 157)(133 148)(134 139)(137 152)(138 143)(141 156)(142 147)(145 160)(146 151)(150 155)(154 159)
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)(41,132,61,152)(42,131,62,151)(43,130,63,150)(44,129,64,149)(45,128,65,148)(46,127,66,147)(47,126,67,146)(48,125,68,145)(49,124,69,144)(50,123,70,143)(51,122,71,142)(52,121,72,141)(53,160,73,140)(54,159,74,139)(55,158,75,138)(56,157,76,137)(57,156,77,136)(58,155,78,135)(59,154,79,134)(60,153,80,133), (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)(81,139)(82,140)(83,141)(84,142)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,154)(97,155)(98,156)(99,157)(100,158)(101,159)(102,160)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138), (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)(121,136)(122,127)(123,158)(124,149)(125,140)(126,131)(128,153)(129,144)(130,135)(132,157)(133,148)(134,139)(137,152)(138,143)(141,156)(142,147)(145,160)(146,151)(150,155)(154,159)>;
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)(41,132,61,152)(42,131,62,151)(43,130,63,150)(44,129,64,149)(45,128,65,148)(46,127,66,147)(47,126,67,146)(48,125,68,145)(49,124,69,144)(50,123,70,143)(51,122,71,142)(52,121,72,141)(53,160,73,140)(54,159,74,139)(55,158,75,138)(56,157,76,137)(57,156,77,136)(58,155,78,135)(59,154,79,134)(60,153,80,133), (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)(81,139)(82,140)(83,141)(84,142)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,154)(97,155)(98,156)(99,157)(100,158)(101,159)(102,160)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138), (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)(121,136)(122,127)(123,158)(124,149)(125,140)(126,131)(128,153)(129,144)(130,135)(132,157)(133,148)(134,139)(137,152)(138,143)(141,156)(142,147)(145,160)(146,151)(150,155)(154,159) );
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),(41,132,61,152),(42,131,62,151),(43,130,63,150),(44,129,64,149),(45,128,65,148),(46,127,66,147),(47,126,67,146),(48,125,68,145),(49,124,69,144),(50,123,70,143),(51,122,71,142),(52,121,72,141),(53,160,73,140),(54,159,74,139),(55,158,75,138),(56,157,76,137),(57,156,77,136),(58,155,78,135),(59,154,79,134),(60,153,80,133)], [(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),(81,139),(82,140),(83,141),(84,142),(85,143),(86,144),(87,145),(88,146),(89,147),(90,148),(91,149),(92,150),(93,151),(94,152),(95,153),(96,154),(97,155),(98,156),(99,157),(100,158),(101,159),(102,160),(103,121),(104,122),(105,123),(106,124),(107,125),(108,126),(109,127),(110,128),(111,129),(112,130),(113,131),(114,132),(115,133),(116,134),(117,135),(118,136),(119,137),(120,138)], [(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),(121,136),(122,127),(123,158),(124,149),(125,140),(126,131),(128,153),(129,144),(130,135),(132,157),(133,148),(134,139),(137,152),(138,143),(141,156),(142,147),(145,160),(146,151),(150,155),(154,159)]])
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