metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊5Dic10, C42.103D10, C10.132+ (1+4), (C5×D4)⋊6Q8, C20⋊Q8⋊15C2, (C4×D4).11D5, C5⋊2(D4⋊3Q8), C20.42(C2×Q8), C4⋊C4.278D10, (D4×C20).12C2, (C4×Dic10)⋊26C2, (C2×D4).242D10, C20.48D4⋊7C2, (C2×C10).83C24, C4.Dic10⋊14C2, C20.6Q8⋊14C2, (D4×Dic5).12C2, C4.15(C2×Dic10), C10.13(C22×Q8), (C4×C20).146C22, (C2×C20).154C23, C22⋊C4.106D10, (C22×C4).202D10, C4⋊Dic5.37C22, C2.16(D4⋊6D10), C22.1(C2×Dic10), Dic5.35(C4○D4), Dic5.14D4⋊7C2, C23.D5.8C22, (D4×C10).249C22, (C22×C20).77C22, (C4×Dic5).80C22, (C2×Dic5).33C23, C2.15(C22×Dic10), C23.163(C22×D5), C22.111(C23×D5), (C22×C10).153C23, (C2×Dic10).27C22, C10.D4.108C22, (C22×Dic5).91C22, C2.18(D5×C4○D4), (C2×C10).3(C2×Q8), C10.137(C2×C4○D4), (C2×C10.D4)⋊24C2, (C5×C4⋊C4).319C22, (C2×C4).154(C22×D5), (C5×C22⋊C4).104C22, SmallGroup(320,1211)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 694 in 228 conjugacy classes, 113 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×13], C22, C22 [×4], C22 [×4], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×4], Q8 [×4], C23 [×2], C10 [×3], C10 [×4], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×15], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×Q8 [×3], Dic5 [×2], Dic5 [×7], C20 [×2], C20 [×4], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C4⋊C4 [×2], C4×D4, C4×D4 [×2], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, Dic10 [×4], C2×Dic5 [×4], C2×Dic5 [×4], C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×C10 [×2], D4⋊3Q8, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×8], C4⋊Dic5 [×3], C4⋊Dic5 [×2], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×Dic10 [×2], C22×Dic5 [×4], C22×C20 [×2], D4×C10, C4×Dic10, C20.6Q8, Dic5.14D4 [×4], C20⋊Q8, C4.Dic10, C2×C10.D4 [×2], C20.48D4 [×2], D4×Dic5 [×2], D4×C20, D4⋊5Dic10
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22×Q8, C2×C4○D4, 2+ (1+4), Dic10 [×4], C22×D5 [×7], D4⋊3Q8, C2×Dic10 [×6], C23×D5, C22×Dic10, D4⋊6D10, D5×C4○D4, D4⋊5Dic10
Generators and relations
G = < a,b,c,d | a4=b2=c20=1, d2=c10, bab=cac-1=a-1, ad=da, cbc-1=dbd-1=a2b, dcd-1=c-1 >
►On 160 points
Generators in S160
(1 130 117 64)(2 65 118 131)(3 132 119 66)(4 67 120 133)(5 134 101 68)(6 69 102 135)(7 136 103 70)(8 71 104 137)(9 138 105 72)(10 73 106 139)(11 140 107 74)(12 75 108 121)(13 122 109 76)(14 77 110 123)(15 124 111 78)(16 79 112 125)(17 126 113 80)(18 61 114 127)(19 128 115 62)(20 63 116 129)(21 60 156 81)(22 82 157 41)(23 42 158 83)(24 84 159 43)(25 44 160 85)(26 86 141 45)(27 46 142 87)(28 88 143 47)(29 48 144 89)(30 90 145 49)(31 50 146 91)(32 92 147 51)(33 52 148 93)(34 94 149 53)(35 54 150 95)(36 96 151 55)(37 56 152 97)(38 98 153 57)(39 58 154 99)(40 100 155 59)
(1 140)(2 75)(3 122)(4 77)(5 124)(6 79)(7 126)(8 61)(9 128)(10 63)(11 130)(12 65)(13 132)(14 67)(15 134)(16 69)(17 136)(18 71)(19 138)(20 73)(21 91)(22 51)(23 93)(24 53)(25 95)(26 55)(27 97)(28 57)(29 99)(30 59)(31 81)(32 41)(33 83)(34 43)(35 85)(36 45)(37 87)(38 47)(39 89)(40 49)(42 148)(44 150)(46 152)(48 154)(50 156)(52 158)(54 160)(56 142)(58 144)(60 146)(62 105)(64 107)(66 109)(68 111)(70 113)(72 115)(74 117)(76 119)(78 101)(80 103)(82 147)(84 149)(86 151)(88 153)(90 155)(92 157)(94 159)(96 141)(98 143)(100 145)(102 125)(104 127)(106 129)(108 131)(110 133)(112 135)(114 137)(116 139)(118 121)(120 123)
(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 39 11 29)(2 38 12 28)(3 37 13 27)(4 36 14 26)(5 35 15 25)(6 34 16 24)(7 33 17 23)(8 32 18 22)(9 31 19 21)(10 30 20 40)(41 137 51 127)(42 136 52 126)(43 135 53 125)(44 134 54 124)(45 133 55 123)(46 132 56 122)(47 131 57 121)(48 130 58 140)(49 129 59 139)(50 128 60 138)(61 82 71 92)(62 81 72 91)(63 100 73 90)(64 99 74 89)(65 98 75 88)(66 97 76 87)(67 96 77 86)(68 95 78 85)(69 94 79 84)(70 93 80 83)(101 150 111 160)(102 149 112 159)(103 148 113 158)(104 147 114 157)(105 146 115 156)(106 145 116 155)(107 144 117 154)(108 143 118 153)(109 142 119 152)(110 141 120 151)
G:=sub<Sym(160)| (1,130,117,64)(2,65,118,131)(3,132,119,66)(4,67,120,133)(5,134,101,68)(6,69,102,135)(7,136,103,70)(8,71,104,137)(9,138,105,72)(10,73,106,139)(11,140,107,74)(12,75,108,121)(13,122,109,76)(14,77,110,123)(15,124,111,78)(16,79,112,125)(17,126,113,80)(18,61,114,127)(19,128,115,62)(20,63,116,129)(21,60,156,81)(22,82,157,41)(23,42,158,83)(24,84,159,43)(25,44,160,85)(26,86,141,45)(27,46,142,87)(28,88,143,47)(29,48,144,89)(30,90,145,49)(31,50,146,91)(32,92,147,51)(33,52,148,93)(34,94,149,53)(35,54,150,95)(36,96,151,55)(37,56,152,97)(38,98,153,57)(39,58,154,99)(40,100,155,59), (1,140)(2,75)(3,122)(4,77)(5,124)(6,79)(7,126)(8,61)(9,128)(10,63)(11,130)(12,65)(13,132)(14,67)(15,134)(16,69)(17,136)(18,71)(19,138)(20,73)(21,91)(22,51)(23,93)(24,53)(25,95)(26,55)(27,97)(28,57)(29,99)(30,59)(31,81)(32,41)(33,83)(34,43)(35,85)(36,45)(37,87)(38,47)(39,89)(40,49)(42,148)(44,150)(46,152)(48,154)(50,156)(52,158)(54,160)(56,142)(58,144)(60,146)(62,105)(64,107)(66,109)(68,111)(70,113)(72,115)(74,117)(76,119)(78,101)(80,103)(82,147)(84,149)(86,151)(88,153)(90,155)(92,157)(94,159)(96,141)(98,143)(100,145)(102,125)(104,127)(106,129)(108,131)(110,133)(112,135)(114,137)(116,139)(118,121)(120,123), (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), (1,39,11,29)(2,38,12,28)(3,37,13,27)(4,36,14,26)(5,35,15,25)(6,34,16,24)(7,33,17,23)(8,32,18,22)(9,31,19,21)(10,30,20,40)(41,137,51,127)(42,136,52,126)(43,135,53,125)(44,134,54,124)(45,133,55,123)(46,132,56,122)(47,131,57,121)(48,130,58,140)(49,129,59,139)(50,128,60,138)(61,82,71,92)(62,81,72,91)(63,100,73,90)(64,99,74,89)(65,98,75,88)(66,97,76,87)(67,96,77,86)(68,95,78,85)(69,94,79,84)(70,93,80,83)(101,150,111,160)(102,149,112,159)(103,148,113,158)(104,147,114,157)(105,146,115,156)(106,145,116,155)(107,144,117,154)(108,143,118,153)(109,142,119,152)(110,141,120,151)>;
G:=Group( (1,130,117,64)(2,65,118,131)(3,132,119,66)(4,67,120,133)(5,134,101,68)(6,69,102,135)(7,136,103,70)(8,71,104,137)(9,138,105,72)(10,73,106,139)(11,140,107,74)(12,75,108,121)(13,122,109,76)(14,77,110,123)(15,124,111,78)(16,79,112,125)(17,126,113,80)(18,61,114,127)(19,128,115,62)(20,63,116,129)(21,60,156,81)(22,82,157,41)(23,42,158,83)(24,84,159,43)(25,44,160,85)(26,86,141,45)(27,46,142,87)(28,88,143,47)(29,48,144,89)(30,90,145,49)(31,50,146,91)(32,92,147,51)(33,52,148,93)(34,94,149,53)(35,54,150,95)(36,96,151,55)(37,56,152,97)(38,98,153,57)(39,58,154,99)(40,100,155,59), (1,140)(2,75)(3,122)(4,77)(5,124)(6,79)(7,126)(8,61)(9,128)(10,63)(11,130)(12,65)(13,132)(14,67)(15,134)(16,69)(17,136)(18,71)(19,138)(20,73)(21,91)(22,51)(23,93)(24,53)(25,95)(26,55)(27,97)(28,57)(29,99)(30,59)(31,81)(32,41)(33,83)(34,43)(35,85)(36,45)(37,87)(38,47)(39,89)(40,49)(42,148)(44,150)(46,152)(48,154)(50,156)(52,158)(54,160)(56,142)(58,144)(60,146)(62,105)(64,107)(66,109)(68,111)(70,113)(72,115)(74,117)(76,119)(78,101)(80,103)(82,147)(84,149)(86,151)(88,153)(90,155)(92,157)(94,159)(96,141)(98,143)(100,145)(102,125)(104,127)(106,129)(108,131)(110,133)(112,135)(114,137)(116,139)(118,121)(120,123), (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), (1,39,11,29)(2,38,12,28)(3,37,13,27)(4,36,14,26)(5,35,15,25)(6,34,16,24)(7,33,17,23)(8,32,18,22)(9,31,19,21)(10,30,20,40)(41,137,51,127)(42,136,52,126)(43,135,53,125)(44,134,54,124)(45,133,55,123)(46,132,56,122)(47,131,57,121)(48,130,58,140)(49,129,59,139)(50,128,60,138)(61,82,71,92)(62,81,72,91)(63,100,73,90)(64,99,74,89)(65,98,75,88)(66,97,76,87)(67,96,77,86)(68,95,78,85)(69,94,79,84)(70,93,80,83)(101,150,111,160)(102,149,112,159)(103,148,113,158)(104,147,114,157)(105,146,115,156)(106,145,116,155)(107,144,117,154)(108,143,118,153)(109,142,119,152)(110,141,120,151) );
G=PermutationGroup([(1,130,117,64),(2,65,118,131),(3,132,119,66),(4,67,120,133),(5,134,101,68),(6,69,102,135),(7,136,103,70),(8,71,104,137),(9,138,105,72),(10,73,106,139),(11,140,107,74),(12,75,108,121),(13,122,109,76),(14,77,110,123),(15,124,111,78),(16,79,112,125),(17,126,113,80),(18,61,114,127),(19,128,115,62),(20,63,116,129),(21,60,156,81),(22,82,157,41),(23,42,158,83),(24,84,159,43),(25,44,160,85),(26,86,141,45),(27,46,142,87),(28,88,143,47),(29,48,144,89),(30,90,145,49),(31,50,146,91),(32,92,147,51),(33,52,148,93),(34,94,149,53),(35,54,150,95),(36,96,151,55),(37,56,152,97),(38,98,153,57),(39,58,154,99),(40,100,155,59)], [(1,140),(2,75),(3,122),(4,77),(5,124),(6,79),(7,126),(8,61),(9,128),(10,63),(11,130),(12,65),(13,132),(14,67),(15,134),(16,69),(17,136),(18,71),(19,138),(20,73),(21,91),(22,51),(23,93),(24,53),(25,95),(26,55),(27,97),(28,57),(29,99),(30,59),(31,81),(32,41),(33,83),(34,43),(35,85),(36,45),(37,87),(38,47),(39,89),(40,49),(42,148),(44,150),(46,152),(48,154),(50,156),(52,158),(54,160),(56,142),(58,144),(60,146),(62,105),(64,107),(66,109),(68,111),(70,113),(72,115),(74,117),(76,119),(78,101),(80,103),(82,147),(84,149),(86,151),(88,153),(90,155),(92,157),(94,159),(96,141),(98,143),(100,145),(102,125),(104,127),(106,129),(108,131),(110,133),(112,135),(114,137),(116,139),(118,121),(120,123)], [(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)], [(1,39,11,29),(2,38,12,28),(3,37,13,27),(4,36,14,26),(5,35,15,25),(6,34,16,24),(7,33,17,23),(8,32,18,22),(9,31,19,21),(10,30,20,40),(41,137,51,127),(42,136,52,126),(43,135,53,125),(44,134,54,124),(45,133,55,123),(46,132,56,122),(47,131,57,121),(48,130,58,140),(49,129,59,139),(50,128,60,138),(61,82,71,92),(62,81,72,91),(63,100,73,90),(64,99,74,89),(65,98,75,88),(66,97,76,87),(67,96,77,86),(68,95,78,85),(69,94,79,84),(70,93,80,83),(101,150,111,160),(102,149,112,159),(103,148,113,158),(104,147,114,157),(105,146,115,156),(106,145,116,155),(107,144,117,154),(108,143,118,153),(109,142,119,152),(110,141,120,151)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 21 |
| 0 | 0 | 0 | 0 | 37 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 21 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 34 | 31 | 0 | 0 | 0 | 0 |
| 5 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 8 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 36 | 9 |
| 40 | 11 | 0 | 0 | 0 | 0 |
| 11 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 5 | 0 | 0 |
| 0 | 0 | 23 | 38 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 21 |
| 0 | 0 | 0 | 0 | 37 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,37,0,0,0,0,21,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,21,40],[34,5,0,0,0,0,31,7,0,0,0,0,0,0,1,8,0,0,0,0,40,34,0,0,0,0,0,0,32,36,0,0,0,0,0,9],[40,11,0,0,0,0,11,1,0,0,0,0,0,0,3,23,0,0,0,0,5,38,0,0,0,0,0,0,1,37,0,0,0,0,21,40] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | Dic10 | 2+ (1+4) | D4⋊6D10 | D5×C4○D4 |
| kernel | D4⋊5Dic10 | C4×Dic10 | C20.6Q8 | Dic5.14D4 | C20⋊Q8 | C4.Dic10 | C2×C10.D4 | C20.48D4 | D4×Dic5 | D4×C20 | C5×D4 | C4×D4 | Dic5 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
D_4\rtimes_5Dic_{10} % in TeX
G:=Group("D4:5Dic10"); // GroupNames label
G:=SmallGroup(320,1211);
// by ID
G=gap.SmallGroup(320,1211);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,675,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=1,d^2=c^10,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀