metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8:5(C4xD5), C40:20(C2xC4), C8:D5:5C4, (C4xD5).2Q8, C4.31(Q8xD5), C2.D8:10D5, C40:6C4:19C2, (C2xC8).65D10, C20.22(C2xQ8), C4:C4.171D10, C22.91(D4xD5), D10.17(C4:C4), C10.D8:20C2, C2.5(D8:D5), C20.Q8:20C2, C10.40(C8:C22), C5:3(M4(2):C4), Dic5.18(C4:C4), (C2xC40).143C22, C20.109(C22xC4), (C2xC20).297C23, C2.5(Q16:D5), (C2xDic5).223D4, (C22xD5).121D4, C10.69(C8.C22), C4:Dic5.123C22, C4.81(C2xC4xD5), C5:2C8:5(C2xC4), (D5xC4:C4).8C2, C2.16(D5xC4:C4), (C5xC2.D8):7C2, C10.38(C2xC4:C4), (C4xD5).7(C2xC4), C4:C4:7D5.8C2, (C2xC8:D5).4C2, (C2xC4xD5).41C22, (C2xC10).302(C2xD4), (C5xC4:C4).90C22, (C2xC5:2C8).68C22, (C2xC4).400(C22xD5), SmallGroup(320,508)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40:20(C2xC4)
G = < a,b,c | a40=b2=c4=1, bab=a29, cac-1=a31, bc=cb >
Subgroups: 430 in 118 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2xC4, C2xC4, C23, D5, C10, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, M4(2), C22xC4, Dic5, Dic5, C20, C20, D10, D10, C2xC10, C4.Q8, C2.D8, C2.D8, C2xC4:C4, C42:C2, C2xM4(2), C5:2C8, C40, C4xD5, C4xD5, C2xDic5, C2xDic5, C2xC20, C2xC20, C22xD5, M4(2):C4, C8:D5, C2xC5:2C8, C4xDic5, C10.D4, C4:Dic5, D10:C4, C5xC4:C4, C2xC40, C2xC4xD5, C2xC4xD5, C10.D8, C20.Q8, C40:6C4, C5xC2.D8, D5xC4:C4, C4:C4:7D5, C2xC8:D5, C40:20(C2xC4)
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C23, D5, C4:C4, C22xC4, C2xD4, C2xQ8, D10, C2xC4:C4, C8:C22, C8.C22, C4xD5, C22xD5, M4(2):C4, C2xC4xD5, D4xD5, Q8xD5, D5xC4:C4, D8:D5, Q16:D5, C40:20(C2xC4)
►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 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 61)(42 50)(43 79)(44 68)(45 57)(47 75)(48 64)(49 53)(51 71)(52 60)(54 78)(55 67)(58 74)(59 63)(62 70)(65 77)(69 73)(72 80)(81 89)(82 118)(83 107)(84 96)(86 114)(87 103)(88 92)(90 110)(91 99)(93 117)(94 106)(97 113)(98 102)(100 120)(101 109)(104 116)(108 112)(111 119)(122 150)(123 139)(124 128)(125 157)(126 146)(127 135)(129 153)(130 142)(132 160)(133 149)(134 138)(136 156)(137 145)(140 152)(143 159)(144 148)(147 155)(154 158)
(1 115 151 46)(2 106 152 77)(3 97 153 68)(4 88 154 59)(5 119 155 50)(6 110 156 41)(7 101 157 72)(8 92 158 63)(9 83 159 54)(10 114 160 45)(11 105 121 76)(12 96 122 67)(13 87 123 58)(14 118 124 49)(15 109 125 80)(16 100 126 71)(17 91 127 62)(18 82 128 53)(19 113 129 44)(20 104 130 75)(21 95 131 66)(22 86 132 57)(23 117 133 48)(24 108 134 79)(25 99 135 70)(26 90 136 61)(27 81 137 52)(28 112 138 43)(29 103 139 74)(30 94 140 65)(31 85 141 56)(32 116 142 47)(33 107 143 78)(34 98 144 69)(35 89 145 60)(36 120 146 51)(37 111 147 42)(38 102 148 73)(39 93 149 64)(40 84 150 55)
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), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,61)(42,50)(43,79)(44,68)(45,57)(47,75)(48,64)(49,53)(51,71)(52,60)(54,78)(55,67)(58,74)(59,63)(62,70)(65,77)(69,73)(72,80)(81,89)(82,118)(83,107)(84,96)(86,114)(87,103)(88,92)(90,110)(91,99)(93,117)(94,106)(97,113)(98,102)(100,120)(101,109)(104,116)(108,112)(111,119)(122,150)(123,139)(124,128)(125,157)(126,146)(127,135)(129,153)(130,142)(132,160)(133,149)(134,138)(136,156)(137,145)(140,152)(143,159)(144,148)(147,155)(154,158), (1,115,151,46)(2,106,152,77)(3,97,153,68)(4,88,154,59)(5,119,155,50)(6,110,156,41)(7,101,157,72)(8,92,158,63)(9,83,159,54)(10,114,160,45)(11,105,121,76)(12,96,122,67)(13,87,123,58)(14,118,124,49)(15,109,125,80)(16,100,126,71)(17,91,127,62)(18,82,128,53)(19,113,129,44)(20,104,130,75)(21,95,131,66)(22,86,132,57)(23,117,133,48)(24,108,134,79)(25,99,135,70)(26,90,136,61)(27,81,137,52)(28,112,138,43)(29,103,139,74)(30,94,140,65)(31,85,141,56)(32,116,142,47)(33,107,143,78)(34,98,144,69)(35,89,145,60)(36,120,146,51)(37,111,147,42)(38,102,148,73)(39,93,149,64)(40,84,150,55)>;
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), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,61)(42,50)(43,79)(44,68)(45,57)(47,75)(48,64)(49,53)(51,71)(52,60)(54,78)(55,67)(58,74)(59,63)(62,70)(65,77)(69,73)(72,80)(81,89)(82,118)(83,107)(84,96)(86,114)(87,103)(88,92)(90,110)(91,99)(93,117)(94,106)(97,113)(98,102)(100,120)(101,109)(104,116)(108,112)(111,119)(122,150)(123,139)(124,128)(125,157)(126,146)(127,135)(129,153)(130,142)(132,160)(133,149)(134,138)(136,156)(137,145)(140,152)(143,159)(144,148)(147,155)(154,158), (1,115,151,46)(2,106,152,77)(3,97,153,68)(4,88,154,59)(5,119,155,50)(6,110,156,41)(7,101,157,72)(8,92,158,63)(9,83,159,54)(10,114,160,45)(11,105,121,76)(12,96,122,67)(13,87,123,58)(14,118,124,49)(15,109,125,80)(16,100,126,71)(17,91,127,62)(18,82,128,53)(19,113,129,44)(20,104,130,75)(21,95,131,66)(22,86,132,57)(23,117,133,48)(24,108,134,79)(25,99,135,70)(26,90,136,61)(27,81,137,52)(28,112,138,43)(29,103,139,74)(30,94,140,65)(31,85,141,56)(32,116,142,47)(33,107,143,78)(34,98,144,69)(35,89,145,60)(36,120,146,51)(37,111,147,42)(38,102,148,73)(39,93,149,64)(40,84,150,55) );
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)], [(2,30),(3,19),(4,8),(5,37),(6,26),(7,15),(9,33),(10,22),(12,40),(13,29),(14,18),(16,36),(17,25),(20,32),(23,39),(24,28),(27,35),(34,38),(41,61),(42,50),(43,79),(44,68),(45,57),(47,75),(48,64),(49,53),(51,71),(52,60),(54,78),(55,67),(58,74),(59,63),(62,70),(65,77),(69,73),(72,80),(81,89),(82,118),(83,107),(84,96),(86,114),(87,103),(88,92),(90,110),(91,99),(93,117),(94,106),(97,113),(98,102),(100,120),(101,109),(104,116),(108,112),(111,119),(122,150),(123,139),(124,128),(125,157),(126,146),(127,135),(129,153),(130,142),(132,160),(133,149),(134,138),(136,156),(137,145),(140,152),(143,159),(144,148),(147,155),(154,158)], [(1,115,151,46),(2,106,152,77),(3,97,153,68),(4,88,154,59),(5,119,155,50),(6,110,156,41),(7,101,157,72),(8,92,158,63),(9,83,159,54),(10,114,160,45),(11,105,121,76),(12,96,122,67),(13,87,123,58),(14,118,124,49),(15,109,125,80),(16,100,126,71),(17,91,127,62),(18,82,128,53),(19,113,129,44),(20,104,130,75),(21,95,131,66),(22,86,132,57),(23,117,133,48),(24,108,134,79),(25,99,135,70),(26,90,136,61),(27,81,137,52),(28,112,138,43),(29,103,139,74),(30,94,140,65),(31,85,141,56),(32,116,142,47),(33,107,143,78),(34,98,144,69),(35,89,145,60),(36,120,146,51),(37,111,147,42),(38,102,148,73),(39,93,149,64),(40,84,150,55)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | D4 | D5 | D10 | D10 | C4xD5 | C8:C22 | C8.C22 | Q8xD5 | D4xD5 | D8:D5 | Q16:D5 |
| kernel | C40:20(C2xC4) | C10.D8 | C20.Q8 | C40:6C4 | C5xC2.D8 | D5xC4:C4 | C4:C4:7D5 | C2xC8:D5 | C8:D5 | C4xD5 | C2xDic5 | C22xD5 | C2.D8 | C4:C4 | C2xC8 | C8 | C10 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 1 | 1 | 2 | 4 | 2 | 8 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C40:20(C2xC4) ►in GL6(F41)
| 8 | 10 | 0 | 0 | 0 | 0 |
| 14 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 36 | 1 | 36 |
| 0 | 0 | 5 | 36 | 5 | 36 |
| 0 | 0 | 40 | 5 | 1 | 36 |
| 0 | 0 | 36 | 5 | 5 | 36 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 34 | 40 |
| 10 | 20 | 0 | 0 | 0 | 0 |
| 38 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 0 | 27 | 0 |
| 0 | 0 | 0 | 25 | 0 | 27 |
| 0 | 0 | 27 | 0 | 16 | 0 |
| 0 | 0 | 0 | 27 | 0 | 16 |
G:=sub<GL(6,GF(41))| [8,14,0,0,0,0,10,33,0,0,0,0,0,0,1,5,40,36,0,0,36,36,5,5,0,0,1,5,1,5,0,0,36,36,36,36],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40],[10,38,0,0,0,0,20,31,0,0,0,0,0,0,25,0,27,0,0,0,0,25,0,27,0,0,27,0,16,0,0,0,0,27,0,16] >;
C40:20(C2xC4) in GAP, Magma, Sage, TeX
C_{40}\rtimes_{20}(C_2\times C_4) % in TeX
G:=Group("C40:20(C2xC4)"); // GroupNames label
G:=SmallGroup(320,508);
// by ID
G=gap.SmallGroup(320,508);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,219,58,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^29,c*a*c^-1=a^31,b*c=c*b>;
// generators/relations