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), (C2×D20).219C22, (D4×C10).306C22, 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
Generators and relations for Dic10⋊23D4
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 >
Subgroups: 1030 in 290 conjugacy classes, 107 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, Dic10, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, Q8⋊5D4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C4×Dic10, C4.D20, Dic5⋊4D4, Dic5.5D4, D10⋊Q8, C20.48D4, C20⋊7D4, Dic5⋊D4, D4×C20, C22×Dic10, C2×C4○D20, Dic10⋊23D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2- 1+4, C22×D5, Q8⋊5D4, C4○D20, D4×D5, C23×D5, C2×C4○D20, C2×D4×D5, D4.10D10, Dic10⋊23D4
►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)
(1 132 11 122)(2 131 12 121)(3 130 13 140)(4 129 14 139)(5 128 15 138)(6 127 16 137)(7 126 17 136)(8 125 18 135)(9 124 19 134)(10 123 20 133)(21 98 31 88)(22 97 32 87)(23 96 33 86)(24 95 34 85)(25 94 35 84)(26 93 36 83)(27 92 37 82)(28 91 38 81)(29 90 39 100)(30 89 40 99)(41 101 51 111)(42 120 52 110)(43 119 53 109)(44 118 54 108)(45 117 55 107)(46 116 56 106)(47 115 57 105)(48 114 58 104)(49 113 59 103)(50 112 60 102)(61 142 71 152)(62 141 72 151)(63 160 73 150)(64 159 74 149)(65 158 75 148)(66 157 76 147)(67 156 77 146)(68 155 78 145)(69 154 79 144)(70 153 80 143)
(1 106 64 25)(2 107 65 26)(3 108 66 27)(4 109 67 28)(5 110 68 29)(6 111 69 30)(7 112 70 31)(8 113 71 32)(9 114 72 33)(10 115 73 34)(11 116 74 35)(12 117 75 36)(13 118 76 37)(14 119 77 38)(15 120 78 39)(16 101 79 40)(17 102 80 21)(18 103 61 22)(19 104 62 23)(20 105 63 24)(41 144 89 137)(42 145 90 138)(43 146 91 139)(44 147 92 140)(45 148 93 121)(46 149 94 122)(47 150 95 123)(48 151 96 124)(49 152 97 125)(50 153 98 126)(51 154 99 127)(52 155 100 128)(53 156 81 129)(54 157 82 130)(55 158 83 131)(56 159 84 132)(57 160 85 133)(58 141 86 134)(59 142 87 135)(60 143 88 136)
(1 84)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 91)(9 92)(10 93)(11 94)(12 95)(13 96)(14 97)(15 98)(16 99)(17 100)(18 81)(19 82)(20 83)(21 128)(22 129)(23 130)(24 131)(25 132)(26 133)(27 134)(28 135)(29 136)(30 137)(31 138)(32 139)(33 140)(34 121)(35 122)(36 123)(37 124)(38 125)(39 126)(40 127)(41 69)(42 70)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 79)(52 80)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)(101 154)(102 155)(103 156)(104 157)(105 158)(106 159)(107 160)(108 141)(109 142)(110 143)(111 144)(112 145)(113 146)(114 147)(115 148)(116 149)(117 150)(118 151)(119 152)(120 153)
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), (1,132,11,122)(2,131,12,121)(3,130,13,140)(4,129,14,139)(5,128,15,138)(6,127,16,137)(7,126,17,136)(8,125,18,135)(9,124,19,134)(10,123,20,133)(21,98,31,88)(22,97,32,87)(23,96,33,86)(24,95,34,85)(25,94,35,84)(26,93,36,83)(27,92,37,82)(28,91,38,81)(29,90,39,100)(30,89,40,99)(41,101,51,111)(42,120,52,110)(43,119,53,109)(44,118,54,108)(45,117,55,107)(46,116,56,106)(47,115,57,105)(48,114,58,104)(49,113,59,103)(50,112,60,102)(61,142,71,152)(62,141,72,151)(63,160,73,150)(64,159,74,149)(65,158,75,148)(66,157,76,147)(67,156,77,146)(68,155,78,145)(69,154,79,144)(70,153,80,143), (1,106,64,25)(2,107,65,26)(3,108,66,27)(4,109,67,28)(5,110,68,29)(6,111,69,30)(7,112,70,31)(8,113,71,32)(9,114,72,33)(10,115,73,34)(11,116,74,35)(12,117,75,36)(13,118,76,37)(14,119,77,38)(15,120,78,39)(16,101,79,40)(17,102,80,21)(18,103,61,22)(19,104,62,23)(20,105,63,24)(41,144,89,137)(42,145,90,138)(43,146,91,139)(44,147,92,140)(45,148,93,121)(46,149,94,122)(47,150,95,123)(48,151,96,124)(49,152,97,125)(50,153,98,126)(51,154,99,127)(52,155,100,128)(53,156,81,129)(54,157,82,130)(55,158,83,131)(56,159,84,132)(57,160,85,133)(58,141,86,134)(59,142,87,135)(60,143,88,136), (1,84)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,97)(15,98)(16,99)(17,100)(18,81)(19,82)(20,83)(21,128)(22,129)(23,130)(24,131)(25,132)(26,133)(27,134)(28,135)(29,136)(30,137)(31,138)(32,139)(33,140)(34,121)(35,122)(36,123)(37,124)(38,125)(39,126)(40,127)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(101,154)(102,155)(103,156)(104,157)(105,158)(106,159)(107,160)(108,141)(109,142)(110,143)(111,144)(112,145)(113,146)(114,147)(115,148)(116,149)(117,150)(118,151)(119,152)(120,153)>;
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), (1,132,11,122)(2,131,12,121)(3,130,13,140)(4,129,14,139)(5,128,15,138)(6,127,16,137)(7,126,17,136)(8,125,18,135)(9,124,19,134)(10,123,20,133)(21,98,31,88)(22,97,32,87)(23,96,33,86)(24,95,34,85)(25,94,35,84)(26,93,36,83)(27,92,37,82)(28,91,38,81)(29,90,39,100)(30,89,40,99)(41,101,51,111)(42,120,52,110)(43,119,53,109)(44,118,54,108)(45,117,55,107)(46,116,56,106)(47,115,57,105)(48,114,58,104)(49,113,59,103)(50,112,60,102)(61,142,71,152)(62,141,72,151)(63,160,73,150)(64,159,74,149)(65,158,75,148)(66,157,76,147)(67,156,77,146)(68,155,78,145)(69,154,79,144)(70,153,80,143), (1,106,64,25)(2,107,65,26)(3,108,66,27)(4,109,67,28)(5,110,68,29)(6,111,69,30)(7,112,70,31)(8,113,71,32)(9,114,72,33)(10,115,73,34)(11,116,74,35)(12,117,75,36)(13,118,76,37)(14,119,77,38)(15,120,78,39)(16,101,79,40)(17,102,80,21)(18,103,61,22)(19,104,62,23)(20,105,63,24)(41,144,89,137)(42,145,90,138)(43,146,91,139)(44,147,92,140)(45,148,93,121)(46,149,94,122)(47,150,95,123)(48,151,96,124)(49,152,97,125)(50,153,98,126)(51,154,99,127)(52,155,100,128)(53,156,81,129)(54,157,82,130)(55,158,83,131)(56,159,84,132)(57,160,85,133)(58,141,86,134)(59,142,87,135)(60,143,88,136), (1,84)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,97)(15,98)(16,99)(17,100)(18,81)(19,82)(20,83)(21,128)(22,129)(23,130)(24,131)(25,132)(26,133)(27,134)(28,135)(29,136)(30,137)(31,138)(32,139)(33,140)(34,121)(35,122)(36,123)(37,124)(38,125)(39,126)(40,127)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(101,154)(102,155)(103,156)(104,157)(105,158)(106,159)(107,160)(108,141)(109,142)(110,143)(111,144)(112,145)(113,146)(114,147)(115,148)(116,149)(117,150)(118,151)(119,152)(120,153) );
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)], [(1,132,11,122),(2,131,12,121),(3,130,13,140),(4,129,14,139),(5,128,15,138),(6,127,16,137),(7,126,17,136),(8,125,18,135),(9,124,19,134),(10,123,20,133),(21,98,31,88),(22,97,32,87),(23,96,33,86),(24,95,34,85),(25,94,35,84),(26,93,36,83),(27,92,37,82),(28,91,38,81),(29,90,39,100),(30,89,40,99),(41,101,51,111),(42,120,52,110),(43,119,53,109),(44,118,54,108),(45,117,55,107),(46,116,56,106),(47,115,57,105),(48,114,58,104),(49,113,59,103),(50,112,60,102),(61,142,71,152),(62,141,72,151),(63,160,73,150),(64,159,74,149),(65,158,75,148),(66,157,76,147),(67,156,77,146),(68,155,78,145),(69,154,79,144),(70,153,80,143)], [(1,106,64,25),(2,107,65,26),(3,108,66,27),(4,109,67,28),(5,110,68,29),(6,111,69,30),(7,112,70,31),(8,113,71,32),(9,114,72,33),(10,115,73,34),(11,116,74,35),(12,117,75,36),(13,118,76,37),(14,119,77,38),(15,120,78,39),(16,101,79,40),(17,102,80,21),(18,103,61,22),(19,104,62,23),(20,105,63,24),(41,144,89,137),(42,145,90,138),(43,146,91,139),(44,147,92,140),(45,148,93,121),(46,149,94,122),(47,150,95,123),(48,151,96,124),(49,152,97,125),(50,153,98,126),(51,154,99,127),(52,155,100,128),(53,156,81,129),(54,157,82,130),(55,158,83,131),(56,159,84,132),(57,160,85,133),(58,141,86,134),(59,142,87,135),(60,143,88,136)], [(1,84),(2,85),(3,86),(4,87),(5,88),(6,89),(7,90),(8,91),(9,92),(10,93),(11,94),(12,95),(13,96),(14,97),(15,98),(16,99),(17,100),(18,81),(19,82),(20,83),(21,128),(22,129),(23,130),(24,131),(25,132),(26,133),(27,134),(28,135),(29,136),(30,137),(31,138),(32,139),(33,140),(34,121),(35,122),(36,123),(37,124),(38,125),(39,126),(40,127),(41,69),(42,70),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(50,78),(51,79),(52,80),(53,61),(54,62),(55,63),(56,64),(57,65),(58,66),(59,67),(60,68),(101,154),(102,155),(103,156),(104,157),(105,158),(106,159),(107,160),(108,141),(109,142),(110,143),(111,144),(112,145),(113,146),(114,147),(115,148),(116,149),(117,150),(118,151),(119,152),(120,153)]])
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 |
Matrix representation of Dic10⋊23D4 ►in 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] >;
Dic10⋊23D4 in GAP, Magma, Sage, TeX
{\rm 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