metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8.8D10, C20.59D8, C40.45D4, Q16.9D10, C40.30C23, D40.12C22, Dic20.13C22, C4○D8⋊1D5, C5⋊5(C4○D16), C5⋊D16⋊7C2, D8.D5⋊7C2, C5⋊Q32⋊7C2, C10.69(C2×D8), (C2×C10).10D8, C5⋊SD32⋊7C2, D40⋊7C2⋊5C2, C4.32(D4⋊D5), (C2×C8).253D10, C20.192(C2×D4), (C2×C20).186D4, C8.21(C5⋊D4), (C5×D8).8C22, C8.36(C22×D5), (C2×C40).43C22, C22.1(D4⋊D5), (C5×Q16).9C22, C5⋊2C16.10C22, (C5×C4○D8)⋊1C2, (C2×C5⋊2C16)⋊3C2, C2.24(C2×D4⋊D5), C4.18(C2×C5⋊D4), (C2×C4).146(C5⋊D4), SmallGroup(320,821)
Series: Derived ►Chief ►Lower central ►Upper central
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, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C16, C2×C8, D8, D8, SD16, Q16, Q16, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C16, D16, SD32, Q32, C4○D8, C4○D8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C4○D16, C5⋊2C16, C40⋊C2, D40, Dic20, C2×C40, C5×D8, C5×SD16, C5×Q16, C4○D20, C5×C4○D4, C2×C5⋊2C16, C5⋊D16, D8.D5, C5⋊SD32, C5⋊Q32, D40⋊7C2, C5×C4○D8, C40.30C23
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, C5⋊D4, C22×D5, C4○D16, D4⋊D5, C2×C5⋊D4, C2×D4⋊D5, 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 74)(42 73)(43 72)(44 71)(45 70)(46 69)(47 68)(48 67)(49 66)(50 65)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)(75 80)(76 79)(77 78)(81 87)(82 86)(83 85)(88 120)(89 119)(90 118)(91 117)(92 116)(93 115)(94 114)(95 113)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 105)(121 136)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)(137 160)(138 159)(139 158)(140 157)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 150)(148 149)
(1 84 21 104)(2 85 22 105)(3 86 23 106)(4 87 24 107)(5 88 25 108)(6 89 26 109)(7 90 27 110)(8 91 28 111)(9 92 29 112)(10 93 30 113)(11 94 31 114)(12 95 32 115)(13 96 33 116)(14 97 34 117)(15 98 35 118)(16 99 36 119)(17 100 37 120)(18 101 38 81)(19 102 39 82)(20 103 40 83)(41 132 61 152)(42 133 62 153)(43 134 63 154)(44 135 64 155)(45 136 65 156)(46 137 66 157)(47 138 67 158)(48 139 68 159)(49 140 69 160)(50 141 70 121)(51 142 71 122)(52 143 72 123)(53 144 73 124)(54 145 74 125)(55 146 75 126)(56 147 76 127)(57 148 77 128)(58 149 78 129)(59 150 79 130)(60 151 80 131)
(1 45)(2 76)(3 67)(4 58)(5 49)(6 80)(7 71)(8 62)(9 53)(10 44)(11 75)(12 66)(13 57)(14 48)(15 79)(16 70)(17 61)(18 52)(19 43)(20 74)(21 65)(22 56)(23 47)(24 78)(25 69)(26 60)(27 51)(28 42)(29 73)(30 64)(31 55)(32 46)(33 77)(34 68)(35 59)(36 50)(37 41)(38 72)(39 63)(40 54)(81 123)(82 154)(83 145)(84 136)(85 127)(86 158)(87 149)(88 140)(89 131)(90 122)(91 153)(92 144)(93 135)(94 126)(95 157)(96 148)(97 139)(98 130)(99 121)(100 152)(101 143)(102 134)(103 125)(104 156)(105 147)(106 138)(107 129)(108 160)(109 151)(110 142)(111 133)(112 124)(113 155)(114 146)(115 137)(116 128)(117 159)(118 150)(119 141)(120 132)
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)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(75,80)(76,79)(77,78)(81,87)(82,86)(83,85)(88,120)(89,119)(90,118)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)(137,160)(138,159)(139,158)(140,157)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149), (1,84,21,104)(2,85,22,105)(3,86,23,106)(4,87,24,107)(5,88,25,108)(6,89,26,109)(7,90,27,110)(8,91,28,111)(9,92,29,112)(10,93,30,113)(11,94,31,114)(12,95,32,115)(13,96,33,116)(14,97,34,117)(15,98,35,118)(16,99,36,119)(17,100,37,120)(18,101,38,81)(19,102,39,82)(20,103,40,83)(41,132,61,152)(42,133,62,153)(43,134,63,154)(44,135,64,155)(45,136,65,156)(46,137,66,157)(47,138,67,158)(48,139,68,159)(49,140,69,160)(50,141,70,121)(51,142,71,122)(52,143,72,123)(53,144,73,124)(54,145,74,125)(55,146,75,126)(56,147,76,127)(57,148,77,128)(58,149,78,129)(59,150,79,130)(60,151,80,131), (1,45)(2,76)(3,67)(4,58)(5,49)(6,80)(7,71)(8,62)(9,53)(10,44)(11,75)(12,66)(13,57)(14,48)(15,79)(16,70)(17,61)(18,52)(19,43)(20,74)(21,65)(22,56)(23,47)(24,78)(25,69)(26,60)(27,51)(28,42)(29,73)(30,64)(31,55)(32,46)(33,77)(34,68)(35,59)(36,50)(37,41)(38,72)(39,63)(40,54)(81,123)(82,154)(83,145)(84,136)(85,127)(86,158)(87,149)(88,140)(89,131)(90,122)(91,153)(92,144)(93,135)(94,126)(95,157)(96,148)(97,139)(98,130)(99,121)(100,152)(101,143)(102,134)(103,125)(104,156)(105,147)(106,138)(107,129)(108,160)(109,151)(110,142)(111,133)(112,124)(113,155)(114,146)(115,137)(116,128)(117,159)(118,150)(119,141)(120,132)>;
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)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(75,80)(76,79)(77,78)(81,87)(82,86)(83,85)(88,120)(89,119)(90,118)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)(137,160)(138,159)(139,158)(140,157)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149), (1,84,21,104)(2,85,22,105)(3,86,23,106)(4,87,24,107)(5,88,25,108)(6,89,26,109)(7,90,27,110)(8,91,28,111)(9,92,29,112)(10,93,30,113)(11,94,31,114)(12,95,32,115)(13,96,33,116)(14,97,34,117)(15,98,35,118)(16,99,36,119)(17,100,37,120)(18,101,38,81)(19,102,39,82)(20,103,40,83)(41,132,61,152)(42,133,62,153)(43,134,63,154)(44,135,64,155)(45,136,65,156)(46,137,66,157)(47,138,67,158)(48,139,68,159)(49,140,69,160)(50,141,70,121)(51,142,71,122)(52,143,72,123)(53,144,73,124)(54,145,74,125)(55,146,75,126)(56,147,76,127)(57,148,77,128)(58,149,78,129)(59,150,79,130)(60,151,80,131), (1,45)(2,76)(3,67)(4,58)(5,49)(6,80)(7,71)(8,62)(9,53)(10,44)(11,75)(12,66)(13,57)(14,48)(15,79)(16,70)(17,61)(18,52)(19,43)(20,74)(21,65)(22,56)(23,47)(24,78)(25,69)(26,60)(27,51)(28,42)(29,73)(30,64)(31,55)(32,46)(33,77)(34,68)(35,59)(36,50)(37,41)(38,72)(39,63)(40,54)(81,123)(82,154)(83,145)(84,136)(85,127)(86,158)(87,149)(88,140)(89,131)(90,122)(91,153)(92,144)(93,135)(94,126)(95,157)(96,148)(97,139)(98,130)(99,121)(100,152)(101,143)(102,134)(103,125)(104,156)(105,147)(106,138)(107,129)(108,160)(109,151)(110,142)(111,133)(112,124)(113,155)(114,146)(115,137)(116,128)(117,159)(118,150)(119,141)(120,132) );
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),(41,74),(42,73),(43,72),(44,71),(45,70),(46,69),(47,68),(48,67),(49,66),(50,65),(51,64),(52,63),(53,62),(54,61),(55,60),(56,59),(57,58),(75,80),(76,79),(77,78),(81,87),(82,86),(83,85),(88,120),(89,119),(90,118),(91,117),(92,116),(93,115),(94,114),(95,113),(96,112),(97,111),(98,110),(99,109),(100,108),(101,107),(102,106),(103,105),(121,136),(122,135),(123,134),(124,133),(125,132),(126,131),(127,130),(128,129),(137,160),(138,159),(139,158),(140,157),(141,156),(142,155),(143,154),(144,153),(145,152),(146,151),(147,150),(148,149)], [(1,84,21,104),(2,85,22,105),(3,86,23,106),(4,87,24,107),(5,88,25,108),(6,89,26,109),(7,90,27,110),(8,91,28,111),(9,92,29,112),(10,93,30,113),(11,94,31,114),(12,95,32,115),(13,96,33,116),(14,97,34,117),(15,98,35,118),(16,99,36,119),(17,100,37,120),(18,101,38,81),(19,102,39,82),(20,103,40,83),(41,132,61,152),(42,133,62,153),(43,134,63,154),(44,135,64,155),(45,136,65,156),(46,137,66,157),(47,138,67,158),(48,139,68,159),(49,140,69,160),(50,141,70,121),(51,142,71,122),(52,143,72,123),(53,144,73,124),(54,145,74,125),(55,146,75,126),(56,147,76,127),(57,148,77,128),(58,149,78,129),(59,150,79,130),(60,151,80,131)], [(1,45),(2,76),(3,67),(4,58),(5,49),(6,80),(7,71),(8,62),(9,53),(10,44),(11,75),(12,66),(13,57),(14,48),(15,79),(16,70),(17,61),(18,52),(19,43),(20,74),(21,65),(22,56),(23,47),(24,78),(25,69),(26,60),(27,51),(28,42),(29,73),(30,64),(31,55),(32,46),(33,77),(34,68),(35,59),(36,50),(37,41),(38,72),(39,63),(40,54),(81,123),(82,154),(83,145),(84,136),(85,127),(86,158),(87,149),(88,140),(89,131),(90,122),(91,153),(92,144),(93,135),(94,126),(95,157),(96,148),(97,139),(98,130),(99,121),(100,152),(101,143),(102,134),(103,125),(104,156),(105,147),(106,138),(107,129),(108,160),(109,151),(110,142),(111,133),(112,124),(113,155),(114,146),(115,137),(116,128),(117,159),(118,150),(119,141),(120,132)]])
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