metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊5(C4×D5), C40⋊20(C2×C4), C8⋊D5⋊5C4, (C4×D5).2Q8, C4.31(Q8×D5), C2.D8⋊10D5, C40⋊6C4⋊19C2, (C2×C8).65D10, C20.22(C2×Q8), C4⋊C4.171D10, C22.91(D4×D5), 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), (C2×C40).143C22, C20.109(C22×C4), (C2×C20).297C23, C2.5(Q16⋊D5), (C2×Dic5).223D4, (C22×D5).121D4, C10.69(C8.C22), C4⋊Dic5.123C22, C4.81(C2×C4×D5), C5⋊2C8⋊5(C2×C4), (D5×C4⋊C4).8C2, C2.16(D5×C4⋊C4), (C5×C2.D8)⋊7C2, C10.38(C2×C4⋊C4), (C4×D5).7(C2×C4), C4⋊C4⋊7D5.8C2, (C2×C8⋊D5).4C2, (C2×C4×D5).41C22, (C2×C10).302(C2×D4), (C5×C4⋊C4).90C22, (C2×C5⋊2C8).68C22, (C2×C4).400(C22×D5), SmallGroup(320,508)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊20(C2×C4)
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 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8, C2×C8, M4(2) [×4], C22×C4 [×2], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4.Q8 [×2], C2.D8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C5⋊2C8 [×2], C40 [×2], C4×D5 [×4], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, M4(2)⋊C4, C8⋊D5 [×4], C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5 [×2], D10⋊C4, C5×C4⋊C4 [×2], C2×C40, C2×C4×D5, C2×C4×D5, C10.D8, C20.Q8, C40⋊6C4, C5×C2.D8, D5×C4⋊C4, C4⋊C4⋊7D5, C2×C8⋊D5, C40⋊20(C2×C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C8⋊C22, C8.C22, C4×D5 [×2], C22×D5, M4(2)⋊C4, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, D8⋊D5, Q16⋊D5, C40⋊20(C2×C4)
►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 77)(42 66)(43 55)(45 73)(46 62)(47 51)(48 80)(49 69)(50 58)(52 76)(53 65)(56 72)(57 61)(59 79)(60 68)(63 75)(67 71)(70 78)(81 93)(83 111)(84 100)(85 89)(86 118)(87 107)(88 96)(90 114)(91 103)(94 110)(95 99)(97 117)(98 106)(101 113)(104 120)(105 109)(108 116)(115 119)(121 149)(122 138)(123 127)(124 156)(125 145)(126 134)(128 152)(129 141)(131 159)(132 148)(133 137)(135 155)(136 144)(139 151)(142 158)(143 147)(146 154)(153 157)
(1 44 82 160)(2 75 83 151)(3 66 84 142)(4 57 85 133)(5 48 86 124)(6 79 87 155)(7 70 88 146)(8 61 89 137)(9 52 90 128)(10 43 91 159)(11 74 92 150)(12 65 93 141)(13 56 94 132)(14 47 95 123)(15 78 96 154)(16 69 97 145)(17 60 98 136)(18 51 99 127)(19 42 100 158)(20 73 101 149)(21 64 102 140)(22 55 103 131)(23 46 104 122)(24 77 105 153)(25 68 106 144)(26 59 107 135)(27 50 108 126)(28 41 109 157)(29 72 110 148)(30 63 111 139)(31 54 112 130)(32 45 113 121)(33 76 114 152)(34 67 115 143)(35 58 116 134)(36 49 117 125)(37 80 118 156)(38 71 119 147)(39 62 120 138)(40 53 81 129)
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)(41,77)(42,66)(43,55)(45,73)(46,62)(47,51)(48,80)(49,69)(50,58)(52,76)(53,65)(56,72)(57,61)(59,79)(60,68)(63,75)(67,71)(70,78)(81,93)(83,111)(84,100)(85,89)(86,118)(87,107)(88,96)(90,114)(91,103)(94,110)(95,99)(97,117)(98,106)(101,113)(104,120)(105,109)(108,116)(115,119)(121,149)(122,138)(123,127)(124,156)(125,145)(126,134)(128,152)(129,141)(131,159)(132,148)(133,137)(135,155)(136,144)(139,151)(142,158)(143,147)(146,154)(153,157), (1,44,82,160)(2,75,83,151)(3,66,84,142)(4,57,85,133)(5,48,86,124)(6,79,87,155)(7,70,88,146)(8,61,89,137)(9,52,90,128)(10,43,91,159)(11,74,92,150)(12,65,93,141)(13,56,94,132)(14,47,95,123)(15,78,96,154)(16,69,97,145)(17,60,98,136)(18,51,99,127)(19,42,100,158)(20,73,101,149)(21,64,102,140)(22,55,103,131)(23,46,104,122)(24,77,105,153)(25,68,106,144)(26,59,107,135)(27,50,108,126)(28,41,109,157)(29,72,110,148)(30,63,111,139)(31,54,112,130)(32,45,113,121)(33,76,114,152)(34,67,115,143)(35,58,116,134)(36,49,117,125)(37,80,118,156)(38,71,119,147)(39,62,120,138)(40,53,81,129)>;
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)(41,77)(42,66)(43,55)(45,73)(46,62)(47,51)(48,80)(49,69)(50,58)(52,76)(53,65)(56,72)(57,61)(59,79)(60,68)(63,75)(67,71)(70,78)(81,93)(83,111)(84,100)(85,89)(86,118)(87,107)(88,96)(90,114)(91,103)(94,110)(95,99)(97,117)(98,106)(101,113)(104,120)(105,109)(108,116)(115,119)(121,149)(122,138)(123,127)(124,156)(125,145)(126,134)(128,152)(129,141)(131,159)(132,148)(133,137)(135,155)(136,144)(139,151)(142,158)(143,147)(146,154)(153,157), (1,44,82,160)(2,75,83,151)(3,66,84,142)(4,57,85,133)(5,48,86,124)(6,79,87,155)(7,70,88,146)(8,61,89,137)(9,52,90,128)(10,43,91,159)(11,74,92,150)(12,65,93,141)(13,56,94,132)(14,47,95,123)(15,78,96,154)(16,69,97,145)(17,60,98,136)(18,51,99,127)(19,42,100,158)(20,73,101,149)(21,64,102,140)(22,55,103,131)(23,46,104,122)(24,77,105,153)(25,68,106,144)(26,59,107,135)(27,50,108,126)(28,41,109,157)(29,72,110,148)(30,63,111,139)(31,54,112,130)(32,45,113,121)(33,76,114,152)(34,67,115,143)(35,58,116,134)(36,49,117,125)(37,80,118,156)(38,71,119,147)(39,62,120,138)(40,53,81,129) );
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),(41,77),(42,66),(43,55),(45,73),(46,62),(47,51),(48,80),(49,69),(50,58),(52,76),(53,65),(56,72),(57,61),(59,79),(60,68),(63,75),(67,71),(70,78),(81,93),(83,111),(84,100),(85,89),(86,118),(87,107),(88,96),(90,114),(91,103),(94,110),(95,99),(97,117),(98,106),(101,113),(104,120),(105,109),(108,116),(115,119),(121,149),(122,138),(123,127),(124,156),(125,145),(126,134),(128,152),(129,141),(131,159),(132,148),(133,137),(135,155),(136,144),(139,151),(142,158),(143,147),(146,154),(153,157)], [(1,44,82,160),(2,75,83,151),(3,66,84,142),(4,57,85,133),(5,48,86,124),(6,79,87,155),(7,70,88,146),(8,61,89,137),(9,52,90,128),(10,43,91,159),(11,74,92,150),(12,65,93,141),(13,56,94,132),(14,47,95,123),(15,78,96,154),(16,69,97,145),(17,60,98,136),(18,51,99,127),(19,42,100,158),(20,73,101,149),(21,64,102,140),(22,55,103,131),(23,46,104,122),(24,77,105,153),(25,68,106,144),(26,59,107,135),(27,50,108,126),(28,41,109,157),(29,72,110,148),(30,63,111,139),(31,54,112,130),(32,45,113,121),(33,76,114,152),(34,67,115,143),(35,58,116,134),(36,49,117,125),(37,80,118,156),(38,71,119,147),(39,62,120,138),(40,53,81,129)])
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 | C4×D5 | C8⋊C22 | C8.C22 | Q8×D5 | D4×D5 | D8⋊D5 | Q16⋊D5 |
| kernel | C40⋊20(C2×C4) | C10.D8 | C20.Q8 | C40⋊6C4 | C5×C2.D8 | D5×C4⋊C4 | C4⋊C4⋊7D5 | C2×C8⋊D5 | C8⋊D5 | C4×D5 | C2×Dic5 | C22×D5 | C2.D8 | C4⋊C4 | C2×C8 | 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(C2×C4) ►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(C2×C4) 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