metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊23D4, C42.110D10, C10.602- (1+4), (C4×D4)⋊14D5, (D4×C20)⋊16C2, C20⋊7D4⋊8C2, C5⋊1(Q8⋊5D4), C4.141(D4×D5), C4⋊C4.283D10, D10⋊Q8⋊7C2, C20.347(C2×D4), (C4×Dic10)⋊31C2, (C2×D4).215D10, C4.D20⋊17C2, C22⋊1(C4○D20), (C2×C10).96C24, Dic5.41(C2×D4), C10.51(C22×D4), Dic5⋊4D4⋊47C2, Dic5⋊D4⋊26C2, C20.48D4⋊21C2, (C4×C20).153C22, (C2×C20).784C23, C22⋊C4.111D10, Dic5.5D4⋊6C2, (C22×Dic10)⋊9C2, (C22×C4).209D10, C23.96(C22×D5), (D4×C10).306C22, (C2×D20).219C22, C4⋊Dic5.298C22, (C22×D5).31C23, C22.121(C23×D5), C23.D5.13C22, D10⋊C4.66C22, (C22×C20).108C22, (C22×C10).166C23, (C2×Dic5).214C23, (C4×Dic5).224C22, C2.17(D4.10D10), (C2×Dic10).248C22, C10.D4.154C22, (C22×Dic5).96C22, C2.24(C2×D4×D5), (C2×C4○D20)⋊9C2, (C2×C10)⋊3(C4○D4), C10.43(C2×C4○D4), C2.47(C2×C4○D20), (C2×C4×D5).250C22, (C5×C4⋊C4).327C22, (C2×C4).159(C22×D5), (C2×C5⋊D4).122C22, (C5×C22⋊C4).123C22, SmallGroup(320,1224)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1030 in 290 conjugacy classes, 107 normal (51 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×11], C5, C2×C4 [×5], C2×C4 [×18], D4 [×12], Q8 [×10], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×3], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×5], C2×Q8 [×8], C4○D4 [×4], Dic5 [×4], Dic5 [×4], C20 [×2], C20 [×4], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×5], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, Dic10 [×4], Dic10 [×6], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×5], C2×C20 [×4], C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], Q8⋊5D4, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5, D10⋊C4 [×6], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×Dic10 [×2], C2×Dic10 [×4], C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C4×Dic10, C4.D20, Dic5⋊4D4 [×2], Dic5.5D4 [×2], D10⋊Q8 [×2], C20.48D4, C20⋊7D4, Dic5⋊D4 [×2], D4×C20, C22×Dic10, C2×C4○D20, Dic10⋊23D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2- (1+4), C22×D5 [×7], Q8⋊5D4, C4○D20 [×2], D4×D5 [×2], C23×D5, C2×C4○D20, C2×D4×D5, D4.10D10, Dic10⋊23D4
Generators and relations
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, ac=ca, ad=da, cbc-1=a10b, bd=db, dcd=c-1 >
►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)
(1 159 11 149)(2 158 12 148)(3 157 13 147)(4 156 14 146)(5 155 15 145)(6 154 16 144)(7 153 17 143)(8 152 18 142)(9 151 19 141)(10 150 20 160)(21 47 31 57)(22 46 32 56)(23 45 33 55)(24 44 34 54)(25 43 35 53)(26 42 36 52)(27 41 37 51)(28 60 38 50)(29 59 39 49)(30 58 40 48)(61 112 71 102)(62 111 72 101)(63 110 73 120)(64 109 74 119)(65 108 75 118)(66 107 76 117)(67 106 77 116)(68 105 78 115)(69 104 79 114)(70 103 80 113)(81 122 91 132)(82 121 92 131)(83 140 93 130)(84 139 94 129)(85 138 95 128)(86 137 96 127)(87 136 97 126)(88 135 98 125)(89 134 99 124)(90 133 100 123)
(1 117 33 121)(2 118 34 122)(3 119 35 123)(4 120 36 124)(5 101 37 125)(6 102 38 126)(7 103 39 127)(8 104 40 128)(9 105 21 129)(10 106 22 130)(11 107 23 131)(12 108 24 132)(13 109 25 133)(14 110 26 134)(15 111 27 135)(16 112 28 136)(17 113 29 137)(18 114 30 138)(19 115 31 139)(20 116 32 140)(41 88 145 62)(42 89 146 63)(43 90 147 64)(44 91 148 65)(45 92 149 66)(46 93 150 67)(47 94 151 68)(48 95 152 69)(49 96 153 70)(50 97 154 71)(51 98 155 72)(52 99 156 73)(53 100 157 74)(54 81 158 75)(55 82 159 76)(56 83 160 77)(57 84 141 78)(58 85 142 79)(59 86 143 80)(60 87 144 61)
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 81)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 61)(39 62)(40 63)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 101)(60 102)(121 159)(122 160)(123 141)(124 142)(125 143)(126 144)(127 145)(128 146)(129 147)(130 148)(131 149)(132 150)(133 151)(134 152)(135 153)(136 154)(137 155)(138 156)(139 157)(140 158)
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), (1,159,11,149)(2,158,12,148)(3,157,13,147)(4,156,14,146)(5,155,15,145)(6,154,16,144)(7,153,17,143)(8,152,18,142)(9,151,19,141)(10,150,20,160)(21,47,31,57)(22,46,32,56)(23,45,33,55)(24,44,34,54)(25,43,35,53)(26,42,36,52)(27,41,37,51)(28,60,38,50)(29,59,39,49)(30,58,40,48)(61,112,71,102)(62,111,72,101)(63,110,73,120)(64,109,74,119)(65,108,75,118)(66,107,76,117)(67,106,77,116)(68,105,78,115)(69,104,79,114)(70,103,80,113)(81,122,91,132)(82,121,92,131)(83,140,93,130)(84,139,94,129)(85,138,95,128)(86,137,96,127)(87,136,97,126)(88,135,98,125)(89,134,99,124)(90,133,100,123), (1,117,33,121)(2,118,34,122)(3,119,35,123)(4,120,36,124)(5,101,37,125)(6,102,38,126)(7,103,39,127)(8,104,40,128)(9,105,21,129)(10,106,22,130)(11,107,23,131)(12,108,24,132)(13,109,25,133)(14,110,26,134)(15,111,27,135)(16,112,28,136)(17,113,29,137)(18,114,30,138)(19,115,31,139)(20,116,32,140)(41,88,145,62)(42,89,146,63)(43,90,147,64)(44,91,148,65)(45,92,149,66)(46,93,150,67)(47,94,151,68)(48,95,152,69)(49,96,153,70)(50,97,154,71)(51,98,155,72)(52,99,156,73)(53,100,157,74)(54,81,158,75)(55,82,159,76)(56,83,160,77)(57,84,141,78)(58,85,142,79)(59,86,143,80)(60,87,144,61), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,81)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,61)(39,62)(40,63)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,101)(60,102)(121,159)(122,160)(123,141)(124,142)(125,143)(126,144)(127,145)(128,146)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,153)(136,154)(137,155)(138,156)(139,157)(140,158)>;
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), (1,159,11,149)(2,158,12,148)(3,157,13,147)(4,156,14,146)(5,155,15,145)(6,154,16,144)(7,153,17,143)(8,152,18,142)(9,151,19,141)(10,150,20,160)(21,47,31,57)(22,46,32,56)(23,45,33,55)(24,44,34,54)(25,43,35,53)(26,42,36,52)(27,41,37,51)(28,60,38,50)(29,59,39,49)(30,58,40,48)(61,112,71,102)(62,111,72,101)(63,110,73,120)(64,109,74,119)(65,108,75,118)(66,107,76,117)(67,106,77,116)(68,105,78,115)(69,104,79,114)(70,103,80,113)(81,122,91,132)(82,121,92,131)(83,140,93,130)(84,139,94,129)(85,138,95,128)(86,137,96,127)(87,136,97,126)(88,135,98,125)(89,134,99,124)(90,133,100,123), (1,117,33,121)(2,118,34,122)(3,119,35,123)(4,120,36,124)(5,101,37,125)(6,102,38,126)(7,103,39,127)(8,104,40,128)(9,105,21,129)(10,106,22,130)(11,107,23,131)(12,108,24,132)(13,109,25,133)(14,110,26,134)(15,111,27,135)(16,112,28,136)(17,113,29,137)(18,114,30,138)(19,115,31,139)(20,116,32,140)(41,88,145,62)(42,89,146,63)(43,90,147,64)(44,91,148,65)(45,92,149,66)(46,93,150,67)(47,94,151,68)(48,95,152,69)(49,96,153,70)(50,97,154,71)(51,98,155,72)(52,99,156,73)(53,100,157,74)(54,81,158,75)(55,82,159,76)(56,83,160,77)(57,84,141,78)(58,85,142,79)(59,86,143,80)(60,87,144,61), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,81)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,61)(39,62)(40,63)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,101)(60,102)(121,159)(122,160)(123,141)(124,142)(125,143)(126,144)(127,145)(128,146)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,153)(136,154)(137,155)(138,156)(139,157)(140,158) );
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)], [(1,159,11,149),(2,158,12,148),(3,157,13,147),(4,156,14,146),(5,155,15,145),(6,154,16,144),(7,153,17,143),(8,152,18,142),(9,151,19,141),(10,150,20,160),(21,47,31,57),(22,46,32,56),(23,45,33,55),(24,44,34,54),(25,43,35,53),(26,42,36,52),(27,41,37,51),(28,60,38,50),(29,59,39,49),(30,58,40,48),(61,112,71,102),(62,111,72,101),(63,110,73,120),(64,109,74,119),(65,108,75,118),(66,107,76,117),(67,106,77,116),(68,105,78,115),(69,104,79,114),(70,103,80,113),(81,122,91,132),(82,121,92,131),(83,140,93,130),(84,139,94,129),(85,138,95,128),(86,137,96,127),(87,136,97,126),(88,135,98,125),(89,134,99,124),(90,133,100,123)], [(1,117,33,121),(2,118,34,122),(3,119,35,123),(4,120,36,124),(5,101,37,125),(6,102,38,126),(7,103,39,127),(8,104,40,128),(9,105,21,129),(10,106,22,130),(11,107,23,131),(12,108,24,132),(13,109,25,133),(14,110,26,134),(15,111,27,135),(16,112,28,136),(17,113,29,137),(18,114,30,138),(19,115,31,139),(20,116,32,140),(41,88,145,62),(42,89,146,63),(43,90,147,64),(44,91,148,65),(45,92,149,66),(46,93,150,67),(47,94,151,68),(48,95,152,69),(49,96,153,70),(50,97,154,71),(51,98,155,72),(52,99,156,73),(53,100,157,74),(54,81,158,75),(55,82,159,76),(56,83,160,77),(57,84,141,78),(58,85,142,79),(59,86,143,80),(60,87,144,61)], [(1,82),(2,83),(3,84),(4,85),(5,86),(6,87),(7,88),(8,89),(9,90),(10,91),(11,92),(12,93),(13,94),(14,95),(15,96),(16,97),(17,98),(18,99),(19,100),(20,81),(21,64),(22,65),(23,66),(24,67),(25,68),(26,69),(27,70),(28,71),(29,72),(30,73),(31,74),(32,75),(33,76),(34,77),(35,78),(36,79),(37,80),(38,61),(39,62),(40,63),(41,103),(42,104),(43,105),(44,106),(45,107),(46,108),(47,109),(48,110),(49,111),(50,112),(51,113),(52,114),(53,115),(54,116),(55,117),(56,118),(57,119),(58,120),(59,101),(60,102),(121,159),(122,160),(123,141),(124,142),(125,143),(126,144),(127,145),(128,146),(129,147),(130,148),(131,149),(132,150),(133,151),(134,152),(135,153),(136,154),(137,155),(138,156),(139,157),(140,158)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 11 | 16 | 0 | 0 |
| 39 | 27 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 38 | 20 | 0 | 0 |
| 20 | 3 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 24 | 34 | 0 | 0 |
| 6 | 17 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 40 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [11,39,0,0,16,27,0,0,0,0,1,0,0,0,0,1],[38,20,0,0,20,3,0,0,0,0,40,0,0,0,0,40],[24,6,0,0,34,17,0,0,0,0,1,40,0,0,2,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,39,1] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2- (1+4) | D4×D5 | D4.10D10 |
| kernel | Dic10⋊23D4 | C4×Dic10 | C4.D20 | Dic5⋊4D4 | Dic5.5D4 | D10⋊Q8 | C20.48D4 | C20⋊7D4 | Dic5⋊D4 | D4×C20 | C22×Dic10 | C2×C4○D20 | Dic10 | C4×D4 | C2×C10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
Dic_{10}\rtimes_{23}D_4 % in TeX
G:=Group("Dic10:23D4"); // GroupNames label
G:=SmallGroup(320,1224);
// by ID
G=gap.SmallGroup(320,1224);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,100,675,570,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀