direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4○D4×C20, D4⋊6(C2×C20), Q8⋊6(C2×C20), (D4×C20)⋊52C2, (C4×D4)⋊23C10, (C2×C42)⋊8C10, (C4×Q8)⋊18C10, (Q8×C20)⋊38C2, C2.6(C23×C20), C42.87(C2×C10), C10.79(C23×C4), C4.18(C22×C20), C42⋊C2⋊19C10, C20.222(C22×C4), (C2×C10).337C24, (C2×C20).959C23, (C4×C20).371C22, C22.1(C22×C20), (D4×C10).331C22, C22.10(C23×C10), C23.29(C22×C10), (Q8×C10).283C22, (C22×C10).253C23, (C22×C20).595C22, (C2×C4×C20)⋊21C2, (C2×C4)⋊8(C2×C20), (C2×C20)⋊46(C2×C4), (C5×D4)⋊36(C2×C4), (C5×Q8)⋊33(C2×C4), C2.4(C10×C4○D4), C4⋊C4.81(C2×C10), (C10×C4○D4).27C2, (C2×C4○D4).13C10, (C2×D4).77(C2×C10), C10.223(C2×C4○D4), (C2×Q8).71(C2×C10), (C5×C42⋊C2)⋊40C2, C22⋊C4.28(C2×C10), (C5×C4⋊C4).406C22, (C22×C4).99(C2×C10), (C2×C10).133(C22×C4), (C2×C4).134(C22×C10), (C5×C22⋊C4).159C22, SmallGroup(320,1519)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 370 in 310 conjugacy classes, 250 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×12], C4 [×6], C22, C22 [×6], C22 [×6], C5, C2×C4, C2×C4 [×23], C2×C4 [×12], D4 [×12], Q8 [×4], C23 [×3], C10, C10 [×2], C10 [×6], C42, C42 [×9], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], C20 [×12], C20 [×6], C2×C10, C2×C10 [×6], C2×C10 [×6], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, C2×C20, C2×C20 [×23], C2×C20 [×12], C5×D4 [×12], C5×Q8 [×4], C22×C10 [×3], C4×C4○D4, C4×C20, C4×C20 [×9], C5×C22⋊C4 [×6], C5×C4⋊C4 [×6], C22×C20 [×9], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C2×C4×C20 [×3], C5×C42⋊C2 [×3], D4×C20 [×6], Q8×C20 [×2], C10×C4○D4, C4○D4×C20
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C5, C2×C4 [×28], C23 [×15], C10 [×15], C22×C4 [×14], C4○D4 [×4], C24, C20 [×8], C2×C10 [×35], C23×C4, C2×C4○D4 [×2], C2×C20 [×28], C22×C10 [×15], C4×C4○D4, C22×C20 [×14], C5×C4○D4 [×4], C23×C10, C23×C20, C10×C4○D4 [×2], C4○D4×C20
Generators and relations
G = < a,b,c,d | a20=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
►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)
(1 122 156 67)(2 123 157 68)(3 124 158 69)(4 125 159 70)(5 126 160 71)(6 127 141 72)(7 128 142 73)(8 129 143 74)(9 130 144 75)(10 131 145 76)(11 132 146 77)(12 133 147 78)(13 134 148 79)(14 135 149 80)(15 136 150 61)(16 137 151 62)(17 138 152 63)(18 139 153 64)(19 140 154 65)(20 121 155 66)(21 44 95 120)(22 45 96 101)(23 46 97 102)(24 47 98 103)(25 48 99 104)(26 49 100 105)(27 50 81 106)(28 51 82 107)(29 52 83 108)(30 53 84 109)(31 54 85 110)(32 55 86 111)(33 56 87 112)(34 57 88 113)(35 58 89 114)(36 59 90 115)(37 60 91 116)(38 41 92 117)(39 42 93 118)(40 43 94 119)
(1 23 156 97)(2 24 157 98)(3 25 158 99)(4 26 159 100)(5 27 160 81)(6 28 141 82)(7 29 142 83)(8 30 143 84)(9 31 144 85)(10 32 145 86)(11 33 146 87)(12 34 147 88)(13 35 148 89)(14 36 149 90)(15 37 150 91)(16 38 151 92)(17 39 152 93)(18 40 153 94)(19 21 154 95)(20 22 155 96)(41 62 117 137)(42 63 118 138)(43 64 119 139)(44 65 120 140)(45 66 101 121)(46 67 102 122)(47 68 103 123)(48 69 104 124)(49 70 105 125)(50 71 106 126)(51 72 107 127)(52 73 108 128)(53 74 109 129)(54 75 110 130)(55 76 111 131)(56 77 112 132)(57 78 113 133)(58 79 114 134)(59 80 115 135)(60 61 116 136)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 99)(36 100)(37 81)(38 82)(39 83)(40 84)(41 107)(42 108)(43 109)(44 110)(45 111)(46 112)(47 113)(48 114)(49 115)(50 116)(51 117)(52 118)(53 119)(54 120)(55 101)(56 102)(57 103)(58 104)(59 105)(60 106)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
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), (1,122,156,67)(2,123,157,68)(3,124,158,69)(4,125,159,70)(5,126,160,71)(6,127,141,72)(7,128,142,73)(8,129,143,74)(9,130,144,75)(10,131,145,76)(11,132,146,77)(12,133,147,78)(13,134,148,79)(14,135,149,80)(15,136,150,61)(16,137,151,62)(17,138,152,63)(18,139,153,64)(19,140,154,65)(20,121,155,66)(21,44,95,120)(22,45,96,101)(23,46,97,102)(24,47,98,103)(25,48,99,104)(26,49,100,105)(27,50,81,106)(28,51,82,107)(29,52,83,108)(30,53,84,109)(31,54,85,110)(32,55,86,111)(33,56,87,112)(34,57,88,113)(35,58,89,114)(36,59,90,115)(37,60,91,116)(38,41,92,117)(39,42,93,118)(40,43,94,119), (1,23,156,97)(2,24,157,98)(3,25,158,99)(4,26,159,100)(5,27,160,81)(6,28,141,82)(7,29,142,83)(8,30,143,84)(9,31,144,85)(10,32,145,86)(11,33,146,87)(12,34,147,88)(13,35,148,89)(14,36,149,90)(15,37,150,91)(16,38,151,92)(17,39,152,93)(18,40,153,94)(19,21,154,95)(20,22,155,96)(41,62,117,137)(42,63,118,138)(43,64,119,139)(44,65,120,140)(45,66,101,121)(46,67,102,122)(47,68,103,123)(48,69,104,124)(49,70,105,125)(50,71,106,126)(51,72,107,127)(52,73,108,128)(53,74,109,129)(54,75,110,130)(55,76,111,131)(56,77,112,132)(57,78,113,133)(58,79,114,134)(59,80,115,135)(60,61,116,136), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,81)(38,82)(39,83)(40,84)(41,107)(42,108)(43,109)(44,110)(45,111)(46,112)(47,113)(48,114)(49,115)(50,116)(51,117)(52,118)(53,119)(54,120)(55,101)(56,102)(57,103)(58,104)(59,105)(60,106)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160)>;
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), (1,122,156,67)(2,123,157,68)(3,124,158,69)(4,125,159,70)(5,126,160,71)(6,127,141,72)(7,128,142,73)(8,129,143,74)(9,130,144,75)(10,131,145,76)(11,132,146,77)(12,133,147,78)(13,134,148,79)(14,135,149,80)(15,136,150,61)(16,137,151,62)(17,138,152,63)(18,139,153,64)(19,140,154,65)(20,121,155,66)(21,44,95,120)(22,45,96,101)(23,46,97,102)(24,47,98,103)(25,48,99,104)(26,49,100,105)(27,50,81,106)(28,51,82,107)(29,52,83,108)(30,53,84,109)(31,54,85,110)(32,55,86,111)(33,56,87,112)(34,57,88,113)(35,58,89,114)(36,59,90,115)(37,60,91,116)(38,41,92,117)(39,42,93,118)(40,43,94,119), (1,23,156,97)(2,24,157,98)(3,25,158,99)(4,26,159,100)(5,27,160,81)(6,28,141,82)(7,29,142,83)(8,30,143,84)(9,31,144,85)(10,32,145,86)(11,33,146,87)(12,34,147,88)(13,35,148,89)(14,36,149,90)(15,37,150,91)(16,38,151,92)(17,39,152,93)(18,40,153,94)(19,21,154,95)(20,22,155,96)(41,62,117,137)(42,63,118,138)(43,64,119,139)(44,65,120,140)(45,66,101,121)(46,67,102,122)(47,68,103,123)(48,69,104,124)(49,70,105,125)(50,71,106,126)(51,72,107,127)(52,73,108,128)(53,74,109,129)(54,75,110,130)(55,76,111,131)(56,77,112,132)(57,78,113,133)(58,79,114,134)(59,80,115,135)(60,61,116,136), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,81)(38,82)(39,83)(40,84)(41,107)(42,108)(43,109)(44,110)(45,111)(46,112)(47,113)(48,114)(49,115)(50,116)(51,117)(52,118)(53,119)(54,120)(55,101)(56,102)(57,103)(58,104)(59,105)(60,106)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160) );
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)], [(1,122,156,67),(2,123,157,68),(3,124,158,69),(4,125,159,70),(5,126,160,71),(6,127,141,72),(7,128,142,73),(8,129,143,74),(9,130,144,75),(10,131,145,76),(11,132,146,77),(12,133,147,78),(13,134,148,79),(14,135,149,80),(15,136,150,61),(16,137,151,62),(17,138,152,63),(18,139,153,64),(19,140,154,65),(20,121,155,66),(21,44,95,120),(22,45,96,101),(23,46,97,102),(24,47,98,103),(25,48,99,104),(26,49,100,105),(27,50,81,106),(28,51,82,107),(29,52,83,108),(30,53,84,109),(31,54,85,110),(32,55,86,111),(33,56,87,112),(34,57,88,113),(35,58,89,114),(36,59,90,115),(37,60,91,116),(38,41,92,117),(39,42,93,118),(40,43,94,119)], [(1,23,156,97),(2,24,157,98),(3,25,158,99),(4,26,159,100),(5,27,160,81),(6,28,141,82),(7,29,142,83),(8,30,143,84),(9,31,144,85),(10,32,145,86),(11,33,146,87),(12,34,147,88),(13,35,148,89),(14,36,149,90),(15,37,150,91),(16,38,151,92),(17,39,152,93),(18,40,153,94),(19,21,154,95),(20,22,155,96),(41,62,117,137),(42,63,118,138),(43,64,119,139),(44,65,120,140),(45,66,101,121),(46,67,102,122),(47,68,103,123),(48,69,104,124),(49,70,105,125),(50,71,106,126),(51,72,107,127),(52,73,108,128),(53,74,109,129),(54,75,110,130),(55,76,111,131),(56,77,112,132),(57,78,113,133),(58,79,114,134),(59,80,115,135),(60,61,116,136)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,85),(22,86),(23,87),(24,88),(25,89),(26,90),(27,91),(28,92),(29,93),(30,94),(31,95),(32,96),(33,97),(34,98),(35,99),(36,100),(37,81),(38,82),(39,83),(40,84),(41,107),(42,108),(43,109),(44,110),(45,111),(46,112),(47,113),(48,114),(49,115),(50,116),(51,117),(52,118),(53,119),(54,120),(55,101),(56,102),(57,103),(58,104),(59,105),(60,106),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(141,151),(142,152),(143,153),(144,154),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160)])
Matrix representation ►G ⊆ GL3(𝔽41) generated by
| 32 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 1 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 0 | 32 |
| 40 | 0 | 0 |
| 0 | 40 | 39 |
| 0 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 40 | 40 |
G:=sub<GL(3,GF(41))| [32,0,0,0,10,0,0,0,10],[1,0,0,0,32,0,0,0,32],[40,0,0,0,40,1,0,39,1],[1,0,0,0,1,40,0,0,40] >;
200 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4L | 4M | ··· | 4AD | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10AJ | 20A | ··· | 20AV | 20AW | ··· | 20DP |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | C4○D4 | C5×C4○D4 |
| kernel | C4○D4×C20 | C2×C4×C20 | C5×C42⋊C2 | D4×C20 | Q8×C20 | C10×C4○D4 | C5×C4○D4 | C4×C4○D4 | C2×C42 | C42⋊C2 | C4×D4 | C4×Q8 | C2×C4○D4 | C4○D4 | C20 | C4 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 16 | 4 | 12 | 12 | 24 | 8 | 4 | 64 | 8 | 32 |
In GAP, Magma, Sage, TeX
C_4\circ D_4\times C_{20} % in TeX
G:=Group("C4oD4xC20"); // GroupNames label
G:=SmallGroup(320,1519);
// by ID
G=gap.SmallGroup(320,1519);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,856,304]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀