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 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C8, C2×C4, C2×C4 [×6], D4 [×4], Q8 [×2], C23 [×2], D5, C10 [×3], C10, C22⋊C4 [×2], C4⋊C4 [×3], C2×C8, C2×C8, M4(2) [×2], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C5⋊2C8, C40 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×C10, C8⋊D4, C8⋊D5 [×2], C2×C5⋊2C8, C10.D4, C4⋊Dic5 [×2], D10⋊C4, C23.D5, C2×C40, C5×SD16 [×2], 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 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8⋊C22, C8.C22, C5⋊D4 [×2], 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)
(1 146 102 66)(2 145 103 65)(3 144 104 64)(4 143 105 63)(5 142 106 62)(6 141 107 61)(7 140 108 60)(8 139 109 59)(9 138 110 58)(10 137 111 57)(11 136 112 56)(12 135 113 55)(13 134 114 54)(14 133 115 53)(15 132 116 52)(16 131 117 51)(17 130 118 50)(18 129 119 49)(19 128 120 48)(20 127 81 47)(21 126 82 46)(22 125 83 45)(23 124 84 44)(24 123 85 43)(25 122 86 42)(26 121 87 41)(27 160 88 80)(28 159 89 79)(29 158 90 78)(30 157 91 77)(31 156 92 76)(32 155 93 75)(33 154 94 74)(34 153 95 73)(35 152 96 72)(36 151 97 71)(37 150 98 70)(38 149 99 69)(39 148 100 68)(40 147 101 67)
(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)(41 141)(42 130)(43 159)(44 148)(45 137)(46 126)(47 155)(48 144)(49 133)(50 122)(51 151)(52 140)(53 129)(54 158)(55 147)(56 136)(57 125)(58 154)(59 143)(60 132)(61 121)(62 150)(63 139)(64 128)(65 157)(66 146)(67 135)(68 124)(69 153)(70 142)(71 131)(72 160)(73 149)(74 138)(75 127)(76 156)(77 145)(78 134)(79 123)(80 152)(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)
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), (1,146,102,66)(2,145,103,65)(3,144,104,64)(4,143,105,63)(5,142,106,62)(6,141,107,61)(7,140,108,60)(8,139,109,59)(9,138,110,58)(10,137,111,57)(11,136,112,56)(12,135,113,55)(13,134,114,54)(14,133,115,53)(15,132,116,52)(16,131,117,51)(17,130,118,50)(18,129,119,49)(19,128,120,48)(20,127,81,47)(21,126,82,46)(22,125,83,45)(23,124,84,44)(24,123,85,43)(25,122,86,42)(26,121,87,41)(27,160,88,80)(28,159,89,79)(29,158,90,78)(30,157,91,77)(31,156,92,76)(32,155,93,75)(33,154,94,74)(34,153,95,73)(35,152,96,72)(36,151,97,71)(37,150,98,70)(38,149,99,69)(39,148,100,68)(40,147,101,67), (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)(41,141)(42,130)(43,159)(44,148)(45,137)(46,126)(47,155)(48,144)(49,133)(50,122)(51,151)(52,140)(53,129)(54,158)(55,147)(56,136)(57,125)(58,154)(59,143)(60,132)(61,121)(62,150)(63,139)(64,128)(65,157)(66,146)(67,135)(68,124)(69,153)(70,142)(71,131)(72,160)(73,149)(74,138)(75,127)(76,156)(77,145)(78,134)(79,123)(80,152)(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)>;
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), (1,146,102,66)(2,145,103,65)(3,144,104,64)(4,143,105,63)(5,142,106,62)(6,141,107,61)(7,140,108,60)(8,139,109,59)(9,138,110,58)(10,137,111,57)(11,136,112,56)(12,135,113,55)(13,134,114,54)(14,133,115,53)(15,132,116,52)(16,131,117,51)(17,130,118,50)(18,129,119,49)(19,128,120,48)(20,127,81,47)(21,126,82,46)(22,125,83,45)(23,124,84,44)(24,123,85,43)(25,122,86,42)(26,121,87,41)(27,160,88,80)(28,159,89,79)(29,158,90,78)(30,157,91,77)(31,156,92,76)(32,155,93,75)(33,154,94,74)(34,153,95,73)(35,152,96,72)(36,151,97,71)(37,150,98,70)(38,149,99,69)(39,148,100,68)(40,147,101,67), (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)(41,141)(42,130)(43,159)(44,148)(45,137)(46,126)(47,155)(48,144)(49,133)(50,122)(51,151)(52,140)(53,129)(54,158)(55,147)(56,136)(57,125)(58,154)(59,143)(60,132)(61,121)(62,150)(63,139)(64,128)(65,157)(66,146)(67,135)(68,124)(69,153)(70,142)(71,131)(72,160)(73,149)(74,138)(75,127)(76,156)(77,145)(78,134)(79,123)(80,152)(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) );
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)], [(1,146,102,66),(2,145,103,65),(3,144,104,64),(4,143,105,63),(5,142,106,62),(6,141,107,61),(7,140,108,60),(8,139,109,59),(9,138,110,58),(10,137,111,57),(11,136,112,56),(12,135,113,55),(13,134,114,54),(14,133,115,53),(15,132,116,52),(16,131,117,51),(17,130,118,50),(18,129,119,49),(19,128,120,48),(20,127,81,47),(21,126,82,46),(22,125,83,45),(23,124,84,44),(24,123,85,43),(25,122,86,42),(26,121,87,41),(27,160,88,80),(28,159,89,79),(29,158,90,78),(30,157,91,77),(31,156,92,76),(32,155,93,75),(33,154,94,74),(34,153,95,73),(35,152,96,72),(36,151,97,71),(37,150,98,70),(38,149,99,69),(39,148,100,68),(40,147,101,67)], [(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),(41,141),(42,130),(43,159),(44,148),(45,137),(46,126),(47,155),(48,144),(49,133),(50,122),(51,151),(52,140),(53,129),(54,158),(55,147),(56,136),(57,125),(58,154),(59,143),(60,132),(61,121),(62,150),(63,139),(64,128),(65,157),(66,146),(67,135),(68,124),(69,153),(70,142),(71,131),(72,160),(73,149),(74,138),(75,127),(76,156),(77,145),(78,134),(79,123),(80,152),(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)])
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