metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40:15D4, Dic5:1SD16, C5:2C8:15D4, C8:8(C5:D4), C5:4(C8:5D4), C4.25(D4xD5), C20.50(C2xD4), (C10xSD16):9C2, (C8xDic5):11C2, (C2xSD16):15D5, (C2xD4).76D10, (C2xC8).264D10, C20:D4.8C2, (C2xQ8).57D10, C2.31(D5xSD16), Dic5:Q8:5C2, C10.48(C2xSD16), C22.272(D4xD5), C2.22(C20:D4), C10.31(C4:1D4), (C2xC20).452C23, (C2xC40).165C22, (C2xDic5).160D4, (Q8xC10).81C22, (D4xC10).101C22, (C2xD20).126C22, (C4xDic5).273C22, (C2xDic10).133C22, C4.9(C2xC5:D4), (C2xQ8:D5):18C2, (C2xC40:C2):30C2, (C2xD4.D5):21C2, (C2xC10).364(C2xD4), (C2xC4).541(C22xD5), (C2xC5:2C8).283C22, SmallGroup(320,802)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40:15D4
G = < a,b,c | a40=b4=c2=1, bab-1=a9, cac=a19, cbc=b-1 >
Subgroups: 654 in 142 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2xC4, C2xC4, D4, Q8, C23, D5, C10, C10, C10, C42, C4:C4, C2xC8, C2xC8, SD16, C2xD4, C2xD4, C2xQ8, C2xQ8, Dic5, Dic5, C20, C20, D10, C2xC10, C2xC10, C4xC8, C4:1D4, C4:Q8, C2xSD16, C2xSD16, C5:2C8, C40, Dic10, D20, C2xDic5, C2xDic5, C5:D4, C2xC20, C2xC20, C5xD4, C5xQ8, C22xD5, C22xC10, C8:5D4, C40:C2, C2xC5:2C8, C4xDic5, C10.D4, D4.D5, Q8:D5, C2xC40, C5xSD16, C2xDic10, C2xD20, C2xC5:D4, D4xC10, Q8xC10, C8xDic5, C2xC40:C2, C2xD4.D5, C20:D4, C2xQ8:D5, Dic5:Q8, C10xSD16, C40:15D4
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2xD4, D10, C4:1D4, C2xSD16, C5:D4, C22xD5, C8:5D4, D4xD5, C2xC5:D4, D5xSD16, C20:D4, C40:15D4
►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 44 150 118)(2 53 151 87)(3 62 152 96)(4 71 153 105)(5 80 154 114)(6 49 155 83)(7 58 156 92)(8 67 157 101)(9 76 158 110)(10 45 159 119)(11 54 160 88)(12 63 121 97)(13 72 122 106)(14 41 123 115)(15 50 124 84)(16 59 125 93)(17 68 126 102)(18 77 127 111)(19 46 128 120)(20 55 129 89)(21 64 130 98)(22 73 131 107)(23 42 132 116)(24 51 133 85)(25 60 134 94)(26 69 135 103)(27 78 136 112)(28 47 137 81)(29 56 138 90)(30 65 139 99)(31 74 140 108)(32 43 141 117)(33 52 142 86)(34 61 143 95)(35 70 144 104)(36 79 145 113)(37 48 146 82)(38 57 147 91)(39 66 148 100)(40 75 149 109)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 101)(42 120)(43 99)(44 118)(45 97)(46 116)(47 95)(48 114)(49 93)(50 112)(51 91)(52 110)(53 89)(54 108)(55 87)(56 106)(57 85)(58 104)(59 83)(60 102)(61 81)(62 100)(63 119)(64 98)(65 117)(66 96)(67 115)(68 94)(69 113)(70 92)(71 111)(72 90)(73 109)(74 88)(75 107)(76 86)(77 105)(78 84)(79 103)(80 82)(121 159)(122 138)(123 157)(124 136)(125 155)(126 134)(127 153)(128 132)(129 151)(131 149)(133 147)(135 145)(137 143)(139 141)(140 160)(142 158)(144 156)(146 154)(148 152)
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), (1,44,150,118)(2,53,151,87)(3,62,152,96)(4,71,153,105)(5,80,154,114)(6,49,155,83)(7,58,156,92)(8,67,157,101)(9,76,158,110)(10,45,159,119)(11,54,160,88)(12,63,121,97)(13,72,122,106)(14,41,123,115)(15,50,124,84)(16,59,125,93)(17,68,126,102)(18,77,127,111)(19,46,128,120)(20,55,129,89)(21,64,130,98)(22,73,131,107)(23,42,132,116)(24,51,133,85)(25,60,134,94)(26,69,135,103)(27,78,136,112)(28,47,137,81)(29,56,138,90)(30,65,139,99)(31,74,140,108)(32,43,141,117)(33,52,142,86)(34,61,143,95)(35,70,144,104)(36,79,145,113)(37,48,146,82)(38,57,147,91)(39,66,148,100)(40,75,149,109), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,101)(42,120)(43,99)(44,118)(45,97)(46,116)(47,95)(48,114)(49,93)(50,112)(51,91)(52,110)(53,89)(54,108)(55,87)(56,106)(57,85)(58,104)(59,83)(60,102)(61,81)(62,100)(63,119)(64,98)(65,117)(66,96)(67,115)(68,94)(69,113)(70,92)(71,111)(72,90)(73,109)(74,88)(75,107)(76,86)(77,105)(78,84)(79,103)(80,82)(121,159)(122,138)(123,157)(124,136)(125,155)(126,134)(127,153)(128,132)(129,151)(131,149)(133,147)(135,145)(137,143)(139,141)(140,160)(142,158)(144,156)(146,154)(148,152)>;
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), (1,44,150,118)(2,53,151,87)(3,62,152,96)(4,71,153,105)(5,80,154,114)(6,49,155,83)(7,58,156,92)(8,67,157,101)(9,76,158,110)(10,45,159,119)(11,54,160,88)(12,63,121,97)(13,72,122,106)(14,41,123,115)(15,50,124,84)(16,59,125,93)(17,68,126,102)(18,77,127,111)(19,46,128,120)(20,55,129,89)(21,64,130,98)(22,73,131,107)(23,42,132,116)(24,51,133,85)(25,60,134,94)(26,69,135,103)(27,78,136,112)(28,47,137,81)(29,56,138,90)(30,65,139,99)(31,74,140,108)(32,43,141,117)(33,52,142,86)(34,61,143,95)(35,70,144,104)(36,79,145,113)(37,48,146,82)(38,57,147,91)(39,66,148,100)(40,75,149,109), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,101)(42,120)(43,99)(44,118)(45,97)(46,116)(47,95)(48,114)(49,93)(50,112)(51,91)(52,110)(53,89)(54,108)(55,87)(56,106)(57,85)(58,104)(59,83)(60,102)(61,81)(62,100)(63,119)(64,98)(65,117)(66,96)(67,115)(68,94)(69,113)(70,92)(71,111)(72,90)(73,109)(74,88)(75,107)(76,86)(77,105)(78,84)(79,103)(80,82)(121,159)(122,138)(123,157)(124,136)(125,155)(126,134)(127,153)(128,132)(129,151)(131,149)(133,147)(135,145)(137,143)(139,141)(140,160)(142,158)(144,156)(146,154)(148,152) );
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)], [(1,44,150,118),(2,53,151,87),(3,62,152,96),(4,71,153,105),(5,80,154,114),(6,49,155,83),(7,58,156,92),(8,67,157,101),(9,76,158,110),(10,45,159,119),(11,54,160,88),(12,63,121,97),(13,72,122,106),(14,41,123,115),(15,50,124,84),(16,59,125,93),(17,68,126,102),(18,77,127,111),(19,46,128,120),(20,55,129,89),(21,64,130,98),(22,73,131,107),(23,42,132,116),(24,51,133,85),(25,60,134,94),(26,69,135,103),(27,78,136,112),(28,47,137,81),(29,56,138,90),(30,65,139,99),(31,74,140,108),(32,43,141,117),(33,52,142,86),(34,61,143,95),(35,70,144,104),(36,79,145,113),(37,48,146,82),(38,57,147,91),(39,66,148,100),(40,75,149,109)], [(2,20),(3,39),(4,18),(5,37),(6,16),(7,35),(8,14),(9,33),(10,12),(11,31),(13,29),(15,27),(17,25),(19,23),(22,40),(24,38),(26,36),(28,34),(30,32),(41,101),(42,120),(43,99),(44,118),(45,97),(46,116),(47,95),(48,114),(49,93),(50,112),(51,91),(52,110),(53,89),(54,108),(55,87),(56,106),(57,85),(58,104),(59,83),(60,102),(61,81),(62,100),(63,119),(64,98),(65,117),(66,96),(67,115),(68,94),(69,113),(70,92),(71,111),(72,90),(73,109),(74,88),(75,107),(76,86),(77,105),(78,84),(79,103),(80,82),(121,159),(122,138),(123,157),(124,136),(125,155),(126,134),(127,153),(128,132),(129,151),(131,149),(133,147),(135,145),(137,143),(139,141),(140,160),(142,158),(144,156),(146,154),(148,152)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 40 | 2 | 2 | 8 | 10 | 10 | 10 | 10 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 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 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | SD16 | D10 | D10 | D10 | C5:D4 | D4xD5 | D4xD5 | D5xSD16 |
| kernel | C40:15D4 | C8xDic5 | C2xC40:C2 | C2xD4.D5 | C20:D4 | C2xQ8:D5 | Dic5:Q8 | C10xSD16 | C5:2C8 | C40 | C2xDic5 | C2xSD16 | Dic5 | C2xC8 | C2xD4 | C2xQ8 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 8 | 2 | 2 | 8 |
Matrix representation of C40:15D4 ►in GL6(F41)
| 0 | 35 | 0 | 0 | 0 | 0 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 26 | 0 | 0 |
| 0 | 0 | 15 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 26 |
| 0 | 0 | 0 | 0 | 15 | 15 |
| 34 | 35 | 0 | 0 | 0 | 0 |
| 8 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 34 | 35 | 0 | 0 | 0 | 0 |
| 8 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [0,7,0,0,0,0,35,7,0,0,0,0,0,0,15,15,0,0,0,0,26,15,0,0,0,0,0,0,15,15,0,0,0,0,26,15],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
C40:15D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_{15}D_4 % in TeX
G:=Group("C40:15D4"); // GroupNames label
G:=SmallGroup(320,802);
// by ID
G=gap.SmallGroup(320,802);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,555,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^9,c*a*c=a^19,c*b*c=b^-1>;
// generators/relations