metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊8D4, C5⋊7(C8⋊D4), C8⋊5(C5⋊D4), (C2×SD16)⋊1D5, C40⋊5C4⋊27C2, (C2×C8).90D10, D10⋊3Q8⋊6C2, (C10×SD16)⋊3C2, (C2×D4).75D10, C20.178(C2×D4), (C2×Q8).56D10, Q8⋊Dic5⋊31C2, D4⋊Dic5⋊36C2, C20⋊2D4.10C2, (C2×Dic5).81D4, (C22×D5).47D4, C22.271(D4×D5), C20.102(C4○D4), C4.33(D4⋊2D5), C2.30(D40⋊C2), C2.20(C20⋊2D4), C10.80(C8⋊C22), (C2×C20).451C23, (C2×C40).115C22, (Q8×C10).80C22, C10.117(C4⋊D4), C2.31(SD16⋊D5), (D4×C10).100C22, C10.51(C8.C22), C4⋊Dic5.178C22, (C2×C8⋊D5)⋊3C2, C4.83(C2×C5⋊D4), (C2×C4×D5).55C22, (C2×C10).363(C2×D4), (C2×C4).540(C22×D5), (C2×C5⋊2C8).160C22, SmallGroup(320,801)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊8D4
G = < a,b,c | a40=b4=c2=1, bab-1=a-1, cac=a29, cbc=b-1 >
Subgroups: 486 in 120 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, C8⋊D4, C8⋊D5, C2×C5⋊2C8, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C2×C40, C5×SD16, C2×C4×D5, C2×C5⋊D4, D4×C10, Q8×C10, C40⋊5C4, D4⋊Dic5, Q8⋊Dic5, C2×C8⋊D5, C20⋊2D4, D10⋊3Q8, C10×SD16, C40⋊8D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8⋊C22, C8.C22, C5⋊D4, C22×D5, C8⋊D4, D4×D5, D4⋊2D5, C2×C5⋊D4, D40⋊C2, SD16⋊D5, C20⋊2D4, C40⋊8D4
►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 95 64 142)(2 94 65 141)(3 93 66 140)(4 92 67 139)(5 91 68 138)(6 90 69 137)(7 89 70 136)(8 88 71 135)(9 87 72 134)(10 86 73 133)(11 85 74 132)(12 84 75 131)(13 83 76 130)(14 82 77 129)(15 81 78 128)(16 120 79 127)(17 119 80 126)(18 118 41 125)(19 117 42 124)(20 116 43 123)(21 115 44 122)(22 114 45 121)(23 113 46 160)(24 112 47 159)(25 111 48 158)(26 110 49 157)(27 109 50 156)(28 108 51 155)(29 107 52 154)(30 106 53 153)(31 105 54 152)(32 104 55 151)(33 103 56 150)(34 102 57 149)(35 101 58 148)(36 100 59 147)(37 99 60 146)(38 98 61 145)(39 97 62 144)(40 96 63 143)
(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)(81 136)(82 125)(83 154)(84 143)(85 132)(86 121)(87 150)(88 139)(89 128)(90 157)(91 146)(92 135)(93 124)(94 153)(95 142)(96 131)(97 160)(98 149)(99 138)(100 127)(101 156)(102 145)(103 134)(104 123)(105 152)(106 141)(107 130)(108 159)(109 148)(110 137)(111 126)(112 155)(113 144)(114 133)(115 122)(116 151)(117 140)(118 129)(119 158)(120 147)
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), (1,95,64,142)(2,94,65,141)(3,93,66,140)(4,92,67,139)(5,91,68,138)(6,90,69,137)(7,89,70,136)(8,88,71,135)(9,87,72,134)(10,86,73,133)(11,85,74,132)(12,84,75,131)(13,83,76,130)(14,82,77,129)(15,81,78,128)(16,120,79,127)(17,119,80,126)(18,118,41,125)(19,117,42,124)(20,116,43,123)(21,115,44,122)(22,114,45,121)(23,113,46,160)(24,112,47,159)(25,111,48,158)(26,110,49,157)(27,109,50,156)(28,108,51,155)(29,107,52,154)(30,106,53,153)(31,105,54,152)(32,104,55,151)(33,103,56,150)(34,102,57,149)(35,101,58,148)(36,100,59,147)(37,99,60,146)(38,98,61,145)(39,97,62,144)(40,96,63,143), (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)(81,136)(82,125)(83,154)(84,143)(85,132)(86,121)(87,150)(88,139)(89,128)(90,157)(91,146)(92,135)(93,124)(94,153)(95,142)(96,131)(97,160)(98,149)(99,138)(100,127)(101,156)(102,145)(103,134)(104,123)(105,152)(106,141)(107,130)(108,159)(109,148)(110,137)(111,126)(112,155)(113,144)(114,133)(115,122)(116,151)(117,140)(118,129)(119,158)(120,147)>;
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), (1,95,64,142)(2,94,65,141)(3,93,66,140)(4,92,67,139)(5,91,68,138)(6,90,69,137)(7,89,70,136)(8,88,71,135)(9,87,72,134)(10,86,73,133)(11,85,74,132)(12,84,75,131)(13,83,76,130)(14,82,77,129)(15,81,78,128)(16,120,79,127)(17,119,80,126)(18,118,41,125)(19,117,42,124)(20,116,43,123)(21,115,44,122)(22,114,45,121)(23,113,46,160)(24,112,47,159)(25,111,48,158)(26,110,49,157)(27,109,50,156)(28,108,51,155)(29,107,52,154)(30,106,53,153)(31,105,54,152)(32,104,55,151)(33,103,56,150)(34,102,57,149)(35,101,58,148)(36,100,59,147)(37,99,60,146)(38,98,61,145)(39,97,62,144)(40,96,63,143), (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)(81,136)(82,125)(83,154)(84,143)(85,132)(86,121)(87,150)(88,139)(89,128)(90,157)(91,146)(92,135)(93,124)(94,153)(95,142)(96,131)(97,160)(98,149)(99,138)(100,127)(101,156)(102,145)(103,134)(104,123)(105,152)(106,141)(107,130)(108,159)(109,148)(110,137)(111,126)(112,155)(113,144)(114,133)(115,122)(116,151)(117,140)(118,129)(119,158)(120,147) );
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)], [(1,95,64,142),(2,94,65,141),(3,93,66,140),(4,92,67,139),(5,91,68,138),(6,90,69,137),(7,89,70,136),(8,88,71,135),(9,87,72,134),(10,86,73,133),(11,85,74,132),(12,84,75,131),(13,83,76,130),(14,82,77,129),(15,81,78,128),(16,120,79,127),(17,119,80,126),(18,118,41,125),(19,117,42,124),(20,116,43,123),(21,115,44,122),(22,114,45,121),(23,113,46,160),(24,112,47,159),(25,111,48,158),(26,110,49,157),(27,109,50,156),(28,108,51,155),(29,107,52,154),(30,106,53,153),(31,105,54,152),(32,104,55,151),(33,103,56,150),(34,102,57,149),(35,101,58,148),(36,100,59,147),(37,99,60,146),(38,98,61,145),(39,97,62,144),(40,96,63,143)], [(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),(81,136),(82,125),(83,154),(84,143),(85,132),(86,121),(87,150),(88,139),(89,128),(90,157),(91,146),(92,135),(93,124),(94,153),(95,142),(96,131),(97,160),(98,149),(99,138),(100,127),(101,156),(102,145),(103,134),(104,123),(105,152),(106,141),(107,130),(108,159),(109,148),(110,137),(111,126),(112,155),(113,144),(114,133),(115,122),(116,151),(117,140),(118,129),(119,158),(120,147)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 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 | 5 | 5 | 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 | 20 | 2 | 2 | 8 | 20 | 40 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | C8⋊C22 | C8.C22 | D4⋊2D5 | D4×D5 | D40⋊C2 | SD16⋊D5 |
| kernel | C40⋊8D4 | C40⋊5C4 | D4⋊Dic5 | Q8⋊Dic5 | C2×C8⋊D5 | C20⋊2D4 | D10⋊3Q8 | C10×SD16 | C40 | C2×Dic5 | C22×D5 | C2×SD16 | C20 | C2×C8 | C2×D4 | C2×Q8 | C8 | C10 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C40⋊8D4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 28 | 17 | 13 |
| 0 | 0 | 13 | 28 | 28 | 13 |
| 0 | 0 | 24 | 28 | 24 | 28 |
| 0 | 0 | 13 | 28 | 13 | 28 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 9 | 13 | 23 |
| 0 | 0 | 11 | 27 | 19 | 28 |
| 0 | 0 | 13 | 23 | 27 | 32 |
| 0 | 0 | 19 | 28 | 30 | 14 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 6 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,13,24,13,0,0,28,28,28,28,0,0,17,28,24,13,0,0,13,13,28,28],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,14,11,13,19,0,0,9,27,23,28,0,0,13,19,27,30,0,0,23,28,32,14],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40] >;
C40⋊8D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_8D_4 % in TeX
G:=Group("C40:8D4"); // GroupNames label
G:=SmallGroup(320,801);
// by ID
G=gap.SmallGroup(320,801);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,254,555,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^29,c*b*c=b^-1>;
// generators/relations