direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D20⋊8C4, C10⋊4(C4×D4), C4⋊C4⋊52D10, (C2×D20)⋊29C4, D20⋊32(C2×C4), C20⋊6(C22×C4), (C2×Dic5)⋊24D4, Dic5⋊10(C2×D4), D10⋊4(C22×C4), C10.34(C23×C4), (C2×C10).47C24, C22.131(D4×D5), C10.41(C22×D4), (C2×C20).579C23, (C4×Dic5)⋊64C22, (C22×D20).17C2, (C22×C4).317D10, D10⋊C4⋊59C22, C22.23(C23×D5), (C2×D20).260C22, C23.326(C22×D5), (C22×C20).215C22, (C22×C10).396C23, C22.34(Q8⋊2D5), (C2×Dic5).372C23, (C23×D5).110C22, (C22×D5).164C23, (C22×Dic5).234C22, C5⋊4(C2×C4×D4), C4⋊2(C2×C4×D5), C2.4(C2×D4×D5), (C2×C4)⋊9(C4×D5), (C10×C4⋊C4)⋊9C2, (C2×C4⋊C4)⋊26D5, (C2×C4×Dic5)⋊5C2, (C2×C20)⋊21(C2×C4), (C2×C4×D5)⋊67C22, (D5×C22×C4)⋊19C2, (C5×C4⋊C4)⋊44C22, C2.15(D5×C22×C4), C22.73(C2×C4×D5), C2.2(C2×Q8⋊2D5), C10.108(C2×C4○D4), (C2×C10).387(C2×D4), (C22×D5)⋊16(C2×C4), (C2×D10⋊C4)⋊32C2, (C2×C4).266(C22×D5), (C2×C10).196(C4○D4), (C2×C10).253(C22×C4), SmallGroup(320,1175)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D20⋊8C4
G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b11, dcd-1=b10c >
Subgroups: 1518 in 426 conjugacy classes, 175 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C2×C4×D4, C4×Dic5, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C22×Dic5, C22×C20, C22×C20, C23×D5, D20⋊8C4, C2×C4×Dic5, C2×D10⋊C4, C10×C4⋊C4, D5×C22×C4, C22×D20, C2×D20⋊8C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, C24, D10, C4×D4, C23×C4, C22×D4, C2×C4○D4, C4×D5, C22×D5, C2×C4×D4, C2×C4×D5, D4×D5, Q8⋊2D5, C23×D5, D20⋊8C4, D5×C22×C4, C2×D4×D5, C2×Q8⋊2D5, C2×D20⋊8C4
►On 160 points
Generators in S160
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 21)(20 22)(41 131)(42 132)(43 133)(44 134)(45 135)(46 136)(47 137)(48 138)(49 139)(50 140)(51 121)(52 122)(53 123)(54 124)(55 125)(56 126)(57 127)(58 128)(59 129)(60 130)(61 104)(62 105)(63 106)(64 107)(65 108)(66 109)(67 110)(68 111)(69 112)(70 113)(71 114)(72 115)(73 116)(74 117)(75 118)(76 119)(77 120)(78 101)(79 102)(80 103)(81 153)(82 154)(83 155)(84 156)(85 157)(86 158)(87 159)(88 160)(89 141)(90 142)(91 143)(92 144)(93 145)(94 146)(95 147)(96 148)(97 149)(98 150)(99 151)(100 152)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(41 45)(42 44)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(74 80)(75 79)(76 78)(81 87)(82 86)(83 85)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)(121 125)(122 124)(126 140)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)(141 151)(142 150)(143 149)(144 148)(145 147)(152 160)(153 159)(154 158)(155 157)
(1 82 126 118)(2 93 127 109)(3 84 128 120)(4 95 129 111)(5 86 130 102)(6 97 131 113)(7 88 132 104)(8 99 133 115)(9 90 134 106)(10 81 135 117)(11 92 136 108)(12 83 137 119)(13 94 138 110)(14 85 139 101)(15 96 140 112)(16 87 121 103)(17 98 122 114)(18 89 123 105)(19 100 124 116)(20 91 125 107)(21 152 54 73)(22 143 55 64)(23 154 56 75)(24 145 57 66)(25 156 58 77)(26 147 59 68)(27 158 60 79)(28 149 41 70)(29 160 42 61)(30 151 43 72)(31 142 44 63)(32 153 45 74)(33 144 46 65)(34 155 47 76)(35 146 48 67)(36 157 49 78)(37 148 50 69)(38 159 51 80)(39 150 52 71)(40 141 53 62)
G:=sub<Sym(160)| (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,21)(20,22)(41,131)(42,132)(43,133)(44,134)(45,135)(46,136)(47,137)(48,138)(49,139)(50,140)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,127)(58,128)(59,129)(60,130)(61,104)(62,105)(63,106)(64,107)(65,108)(66,109)(67,110)(68,111)(69,112)(70,113)(71,114)(72,115)(73,116)(74,117)(75,118)(76,119)(77,120)(78,101)(79,102)(80,103)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,141)(90,142)(91,143)(92,144)(93,145)(94,146)(95,147)(96,148)(97,149)(98,150)(99,151)(100,152), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(41,45)(42,44)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78)(81,87)(82,86)(83,85)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)(121,125)(122,124)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(141,151)(142,150)(143,149)(144,148)(145,147)(152,160)(153,159)(154,158)(155,157), (1,82,126,118)(2,93,127,109)(3,84,128,120)(4,95,129,111)(5,86,130,102)(6,97,131,113)(7,88,132,104)(8,99,133,115)(9,90,134,106)(10,81,135,117)(11,92,136,108)(12,83,137,119)(13,94,138,110)(14,85,139,101)(15,96,140,112)(16,87,121,103)(17,98,122,114)(18,89,123,105)(19,100,124,116)(20,91,125,107)(21,152,54,73)(22,143,55,64)(23,154,56,75)(24,145,57,66)(25,156,58,77)(26,147,59,68)(27,158,60,79)(28,149,41,70)(29,160,42,61)(30,151,43,72)(31,142,44,63)(32,153,45,74)(33,144,46,65)(34,155,47,76)(35,146,48,67)(36,157,49,78)(37,148,50,69)(38,159,51,80)(39,150,52,71)(40,141,53,62)>;
G:=Group( (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,21)(20,22)(41,131)(42,132)(43,133)(44,134)(45,135)(46,136)(47,137)(48,138)(49,139)(50,140)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,127)(58,128)(59,129)(60,130)(61,104)(62,105)(63,106)(64,107)(65,108)(66,109)(67,110)(68,111)(69,112)(70,113)(71,114)(72,115)(73,116)(74,117)(75,118)(76,119)(77,120)(78,101)(79,102)(80,103)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,141)(90,142)(91,143)(92,144)(93,145)(94,146)(95,147)(96,148)(97,149)(98,150)(99,151)(100,152), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(41,45)(42,44)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78)(81,87)(82,86)(83,85)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)(121,125)(122,124)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(141,151)(142,150)(143,149)(144,148)(145,147)(152,160)(153,159)(154,158)(155,157), (1,82,126,118)(2,93,127,109)(3,84,128,120)(4,95,129,111)(5,86,130,102)(6,97,131,113)(7,88,132,104)(8,99,133,115)(9,90,134,106)(10,81,135,117)(11,92,136,108)(12,83,137,119)(13,94,138,110)(14,85,139,101)(15,96,140,112)(16,87,121,103)(17,98,122,114)(18,89,123,105)(19,100,124,116)(20,91,125,107)(21,152,54,73)(22,143,55,64)(23,154,56,75)(24,145,57,66)(25,156,58,77)(26,147,59,68)(27,158,60,79)(28,149,41,70)(29,160,42,61)(30,151,43,72)(31,142,44,63)(32,153,45,74)(33,144,46,65)(34,155,47,76)(35,146,48,67)(36,157,49,78)(37,148,50,69)(38,159,51,80)(39,150,52,71)(40,141,53,62) );
G=PermutationGroup([[(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,39),(18,40),(19,21),(20,22),(41,131),(42,132),(43,133),(44,134),(45,135),(46,136),(47,137),(48,138),(49,139),(50,140),(51,121),(52,122),(53,123),(54,124),(55,125),(56,126),(57,127),(58,128),(59,129),(60,130),(61,104),(62,105),(63,106),(64,107),(65,108),(66,109),(67,110),(68,111),(69,112),(70,113),(71,114),(72,115),(73,116),(74,117),(75,118),(76,119),(77,120),(78,101),(79,102),(80,103),(81,153),(82,154),(83,155),(84,156),(85,157),(86,158),(87,159),(88,160),(89,141),(90,142),(91,143),(92,144),(93,145),(94,146),(95,147),(96,148),(97,149),(98,150),(99,151),(100,152)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,39),(22,38),(23,37),(24,36),(25,35),(26,34),(27,33),(28,32),(29,31),(41,45),(42,44),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68),(74,80),(75,79),(76,78),(81,87),(82,86),(83,85),(88,100),(89,99),(90,98),(91,97),(92,96),(93,95),(101,119),(102,118),(103,117),(104,116),(105,115),(106,114),(107,113),(108,112),(109,111),(121,125),(122,124),(126,140),(127,139),(128,138),(129,137),(130,136),(131,135),(132,134),(141,151),(142,150),(143,149),(144,148),(145,147),(152,160),(153,159),(154,158),(155,157)], [(1,82,126,118),(2,93,127,109),(3,84,128,120),(4,95,129,111),(5,86,130,102),(6,97,131,113),(7,88,132,104),(8,99,133,115),(9,90,134,106),(10,81,135,117),(11,92,136,108),(12,83,137,119),(13,94,138,110),(14,85,139,101),(15,96,140,112),(16,87,121,103),(17,98,122,114),(18,89,123,105),(19,100,124,116),(20,91,125,107),(21,152,54,73),(22,143,55,64),(23,154,56,75),(24,145,57,66),(25,156,58,77),(26,147,59,68),(27,158,60,79),(28,149,41,70),(29,160,42,61),(30,151,43,72),(31,142,44,63),(32,153,45,74),(33,144,46,65),(34,155,47,76),(35,146,48,67),(36,157,49,78),(37,148,50,69),(38,159,51,80),(39,150,52,71),(40,141,53,62)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4L | 4M | ··· | 4T | 4U | 4V | 4W | 4X | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 10 | ··· | 10 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | D10 | C4×D5 | D4×D5 | Q8⋊2D5 |
| kernel | C2×D20⋊8C4 | D20⋊8C4 | C2×C4×Dic5 | C2×D10⋊C4 | C10×C4⋊C4 | D5×C22×C4 | C22×D20 | C2×D20 | C2×Dic5 | C2×C4⋊C4 | C2×C10 | C4⋊C4 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 8 | 1 | 2 | 1 | 2 | 1 | 16 | 4 | 2 | 4 | 8 | 6 | 16 | 4 | 4 |
Matrix representation of C2×D20⋊8C4 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 34 | 40 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 9 | 0 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 34 | 40 | 0 | 0 |
| 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,34,1,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,9,0],[40,0,0,0,0,0,34,7,0,0,0,40,7,0,0,0,0,0,40,0,0,0,0,0,1],[1,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,40,0] >;
C2×D20⋊8C4 in GAP, Magma, Sage, TeX
C_2\times D_{20}\rtimes_8C_4 % in TeX
G:=Group("C2xD20:8C4"); // GroupNames label
G:=SmallGroup(320,1175);
// by ID
G=gap.SmallGroup(320,1175);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^11,d*c*d^-1=b^10*c>;
// generators/relations