metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.13D20, Q8.13D20, C20.63C24, C40.12C23, D40.14C22, D20.26C23, M4(2).28D10, Dic20.10C22, Dic10.26C23, C8○D4⋊5D5, C5⋊1(Q8○D8), (C5×D4).25D4, C20.75(C2×D4), C4.29(C2×D20), (C5×Q8).25D4, C4○D4.40D10, (C2×C8).102D10, D40⋊7C2⋊13C2, C22.5(C2×D20), C8.54(C22×D5), C4.60(C23×D5), (C2×Dic20)⋊15C2, C8.D10⋊12C2, (C2×C40).70C22, C40⋊C2.2C22, C2.32(C22×D20), C10.30(C22×D4), D4.10D10⋊4C2, (C2×C20).517C23, C4○D20.27C22, (C5×M4(2)).30C22, (C2×Dic10).200C22, (C5×C8○D4)⋊5C2, (C2×C10).10(C2×D4), (C5×C4○D4).47C22, (C2×C4).228(C22×D5), SmallGroup(320,1425)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 902 in 248 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2 [×5], C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], C5, C8, C8 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×8], Q8, Q8 [×12], D5 [×2], C10, C10 [×3], C2×C8 [×3], M4(2) [×3], D8, SD16 [×6], Q16 [×9], C2×Q8 [×8], C4○D4, C4○D4 [×12], Dic5 [×6], C20, C20 [×3], D10 [×2], C2×C10 [×3], C8○D4, C2×Q16 [×3], C4○D8 [×3], C8.C22 [×6], 2- (1+4) [×2], C40, C40 [×3], Dic10 [×6], Dic10 [×6], C4×D5 [×6], D20 [×2], C2×Dic5 [×6], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, Q8○D8, C40⋊C2 [×6], D40, Dic20 [×9], C2×C40 [×3], C5×M4(2) [×3], C2×Dic10 [×6], C4○D20 [×6], D4⋊2D5 [×6], Q8×D5 [×2], C5×C4○D4, D40⋊7C2 [×3], C2×Dic20 [×3], C8.D10 [×6], C5×C8○D4, D4.10D10 [×2], D4.13D20
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, D20 [×4], C22×D5 [×7], Q8○D8, C2×D20 [×6], C23×D5, C22×D20, D4.13D20
Generators and relations
G = < a,b,c,d | a4=b2=d2=1, c20=a2, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=a2c19 >
►On 160 points
Generators in S160
(1 46 21 66)(2 47 22 67)(3 48 23 68)(4 49 24 69)(5 50 25 70)(6 51 26 71)(7 52 27 72)(8 53 28 73)(9 54 29 74)(10 55 30 75)(11 56 31 76)(12 57 32 77)(13 58 33 78)(14 59 34 79)(15 60 35 80)(16 61 36 41)(17 62 37 42)(18 63 38 43)(19 64 39 44)(20 65 40 45)(81 160 101 140)(82 121 102 141)(83 122 103 142)(84 123 104 143)(85 124 105 144)(86 125 106 145)(87 126 107 146)(88 127 108 147)(89 128 109 148)(90 129 110 149)(91 130 111 150)(92 131 112 151)(93 132 113 152)(94 133 114 153)(95 134 115 154)(96 135 116 155)(97 136 117 156)(98 137 118 157)(99 138 119 158)(100 139 120 159)
(1 128)(2 129)(3 130)(4 131)(5 132)(6 133)(7 134)(8 135)(9 136)(10 137)(11 138)(12 139)(13 140)(14 141)(15 142)(16 143)(17 144)(18 145)(19 146)(20 147)(21 148)(22 149)(23 150)(24 151)(25 152)(26 153)(27 154)(28 155)(29 156)(30 157)(31 158)(32 159)(33 160)(34 121)(35 122)(36 123)(37 124)(38 125)(39 126)(40 127)(41 84)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 92)(50 93)(51 94)(52 95)(53 96)(54 97)(55 98)(56 99)(57 100)(58 101)(59 102)(60 103)(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 81)(79 82)(80 83)
(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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 50)(42 49)(43 48)(44 47)(45 46)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)(61 70)(62 69)(63 68)(64 67)(65 66)(81 116)(82 115)(83 114)(84 113)(85 112)(86 111)(87 110)(88 109)(89 108)(90 107)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)(97 100)(98 99)(117 120)(118 119)(121 154)(122 153)(123 152)(124 151)(125 150)(126 149)(127 148)(128 147)(129 146)(130 145)(131 144)(132 143)(133 142)(134 141)(135 140)(136 139)(137 138)(155 160)(156 159)(157 158)
G:=sub<Sym(160)| (1,46,21,66)(2,47,22,67)(3,48,23,68)(4,49,24,69)(5,50,25,70)(6,51,26,71)(7,52,27,72)(8,53,28,73)(9,54,29,74)(10,55,30,75)(11,56,31,76)(12,57,32,77)(13,58,33,78)(14,59,34,79)(15,60,35,80)(16,61,36,41)(17,62,37,42)(18,63,38,43)(19,64,39,44)(20,65,40,45)(81,160,101,140)(82,121,102,141)(83,122,103,142)(84,123,104,143)(85,124,105,144)(86,125,106,145)(87,126,107,146)(88,127,108,147)(89,128,109,148)(90,129,110,149)(91,130,111,150)(92,131,112,151)(93,132,113,152)(94,133,114,153)(95,134,115,154)(96,135,116,155)(97,136,117,156)(98,137,118,157)(99,138,119,158)(100,139,120,159), (1,128)(2,129)(3,130)(4,131)(5,132)(6,133)(7,134)(8,135)(9,136)(10,137)(11,138)(12,139)(13,140)(14,141)(15,142)(16,143)(17,144)(18,145)(19,146)(20,147)(21,148)(22,149)(23,150)(24,151)(25,152)(26,153)(27,154)(28,155)(29,156)(30,157)(31,158)(32,159)(33,160)(34,121)(35,122)(36,123)(37,124)(38,125)(39,126)(40,127)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,96)(54,97)(55,98)(56,99)(57,100)(58,101)(59,102)(60,103)(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)(78,81)(79,82)(80,83), (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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,50)(42,49)(43,48)(44,47)(45,46)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)(81,116)(82,115)(83,114)(84,113)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)(117,120)(118,119)(121,154)(122,153)(123,152)(124,151)(125,150)(126,149)(127,148)(128,147)(129,146)(130,145)(131,144)(132,143)(133,142)(134,141)(135,140)(136,139)(137,138)(155,160)(156,159)(157,158)>;
G:=Group( (1,46,21,66)(2,47,22,67)(3,48,23,68)(4,49,24,69)(5,50,25,70)(6,51,26,71)(7,52,27,72)(8,53,28,73)(9,54,29,74)(10,55,30,75)(11,56,31,76)(12,57,32,77)(13,58,33,78)(14,59,34,79)(15,60,35,80)(16,61,36,41)(17,62,37,42)(18,63,38,43)(19,64,39,44)(20,65,40,45)(81,160,101,140)(82,121,102,141)(83,122,103,142)(84,123,104,143)(85,124,105,144)(86,125,106,145)(87,126,107,146)(88,127,108,147)(89,128,109,148)(90,129,110,149)(91,130,111,150)(92,131,112,151)(93,132,113,152)(94,133,114,153)(95,134,115,154)(96,135,116,155)(97,136,117,156)(98,137,118,157)(99,138,119,158)(100,139,120,159), (1,128)(2,129)(3,130)(4,131)(5,132)(6,133)(7,134)(8,135)(9,136)(10,137)(11,138)(12,139)(13,140)(14,141)(15,142)(16,143)(17,144)(18,145)(19,146)(20,147)(21,148)(22,149)(23,150)(24,151)(25,152)(26,153)(27,154)(28,155)(29,156)(30,157)(31,158)(32,159)(33,160)(34,121)(35,122)(36,123)(37,124)(38,125)(39,126)(40,127)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,96)(54,97)(55,98)(56,99)(57,100)(58,101)(59,102)(60,103)(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)(78,81)(79,82)(80,83), (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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,50)(42,49)(43,48)(44,47)(45,46)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)(81,116)(82,115)(83,114)(84,113)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)(117,120)(118,119)(121,154)(122,153)(123,152)(124,151)(125,150)(126,149)(127,148)(128,147)(129,146)(130,145)(131,144)(132,143)(133,142)(134,141)(135,140)(136,139)(137,138)(155,160)(156,159)(157,158) );
G=PermutationGroup([(1,46,21,66),(2,47,22,67),(3,48,23,68),(4,49,24,69),(5,50,25,70),(6,51,26,71),(7,52,27,72),(8,53,28,73),(9,54,29,74),(10,55,30,75),(11,56,31,76),(12,57,32,77),(13,58,33,78),(14,59,34,79),(15,60,35,80),(16,61,36,41),(17,62,37,42),(18,63,38,43),(19,64,39,44),(20,65,40,45),(81,160,101,140),(82,121,102,141),(83,122,103,142),(84,123,104,143),(85,124,105,144),(86,125,106,145),(87,126,107,146),(88,127,108,147),(89,128,109,148),(90,129,110,149),(91,130,111,150),(92,131,112,151),(93,132,113,152),(94,133,114,153),(95,134,115,154),(96,135,116,155),(97,136,117,156),(98,137,118,157),(99,138,119,158),(100,139,120,159)], [(1,128),(2,129),(3,130),(4,131),(5,132),(6,133),(7,134),(8,135),(9,136),(10,137),(11,138),(12,139),(13,140),(14,141),(15,142),(16,143),(17,144),(18,145),(19,146),(20,147),(21,148),(22,149),(23,150),(24,151),(25,152),(26,153),(27,154),(28,155),(29,156),(30,157),(31,158),(32,159),(33,160),(34,121),(35,122),(36,123),(37,124),(38,125),(39,126),(40,127),(41,84),(42,85),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(49,92),(50,93),(51,94),(52,95),(53,96),(54,97),(55,98),(56,99),(57,100),(58,101),(59,102),(60,103),(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),(78,81),(79,82),(80,83)], [(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)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(41,50),(42,49),(43,48),(44,47),(45,46),(51,80),(52,79),(53,78),(54,77),(55,76),(56,75),(57,74),(58,73),(59,72),(60,71),(61,70),(62,69),(63,68),(64,67),(65,66),(81,116),(82,115),(83,114),(84,113),(85,112),(86,111),(87,110),(88,109),(89,108),(90,107),(91,106),(92,105),(93,104),(94,103),(95,102),(96,101),(97,100),(98,99),(117,120),(118,119),(121,154),(122,153),(123,152),(124,151),(125,150),(126,149),(127,148),(128,147),(129,146),(130,145),(131,144),(132,143),(133,142),(134,141),(135,140),(136,139),(137,138),(155,160),(156,159),(157,158)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 33 | 14 | 15 | 25 |
| 27 | 8 | 16 | 26 |
| 15 | 25 | 8 | 27 |
| 16 | 26 | 14 | 33 |
| 35 | 3 | 0 | 0 |
| 38 | 15 | 0 | 0 |
| 0 | 0 | 35 | 3 |
| 0 | 0 | 38 | 15 |
| 20 | 39 | 0 | 0 |
| 15 | 21 | 0 | 0 |
| 0 | 0 | 20 | 39 |
| 0 | 0 | 15 | 21 |
G:=sub<GL(4,GF(41))| [0,0,40,0,0,0,0,40,1,0,0,0,0,1,0,0],[33,27,15,16,14,8,25,26,15,16,8,14,25,26,27,33],[35,38,0,0,3,15,0,0,0,0,35,38,0,0,3,15],[20,15,0,0,39,21,0,0,0,0,20,15,0,0,39,21] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | 10B | 10C | ··· | 10H | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 40A | ··· | 40H | 40I | ··· | 40T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 20 | ··· | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | D20 | D20 | Q8○D8 | D4.13D20 |
| kernel | D4.13D20 | D40⋊7C2 | C2×Dic20 | C8.D10 | C5×C8○D4 | D4.10D10 | C5×D4 | C5×Q8 | C8○D4 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C5 | C1 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 3 | 1 | 2 | 6 | 6 | 2 | 12 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
D_4._{13}D_{20} % in TeX
G:=Group("D4.13D20"); // GroupNames label
G:=SmallGroup(320,1425);
// by ID
G=gap.SmallGroup(320,1425);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,675,192,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^20=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=a^2*c^19>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀