metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊10Q8, C42.130D10, C10.1112+ (1+4), (C4×Q8)⋊12D5, C4.50(Q8×D5), (Q8×C20)⋊14C2, C5⋊4(D4⋊3Q8), C4⋊C4.326D10, D10⋊3Q8⋊8C2, C20⋊2Q8⋊28C2, (C4×D20).21C2, D10.20(C2×Q8), C20.108(C2×Q8), D10⋊Q8⋊11C2, C4.67(C4○D20), (C2×Q8).178D10, C20.6Q8⋊18C2, C20.118(C4○D4), C10.31(C22×Q8), (C2×C20).590C23, (C4×C20).175C22, (C2×C10).123C24, C2.23(D4⋊8D10), (C2×D20).297C22, C4⋊Dic5.202C22, (Q8×C10).223C22, (C2×Dic5).55C23, C22.144(C23×D5), (C2×Dic10).33C22, C10.D4.69C22, (C22×D5).190C23, D10⋊C4.103C22, (D5×C4⋊C4)⋊18C2, C2.14(C2×Q8×D5), C10.55(C2×C4○D4), C2.62(C2×C4○D20), (C2×C4×D5).83C22, (C5×C4⋊C4).351C22, (C2×C4).169(C22×D5), SmallGroup(320,1251)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 790 in 228 conjugacy classes, 107 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×11], C22, C22 [×8], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×4], Q8 [×4], C23 [×2], D5 [×4], C10 [×3], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×13], C22×C4 [×6], C2×D4, C2×Q8, C2×Q8 [×2], Dic5 [×6], C20 [×4], C20 [×5], D10 [×4], D10 [×4], C2×C10, C2×C4⋊C4 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, Dic10 [×2], C4×D5 [×8], D20 [×4], C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5 [×2], D4⋊3Q8, C10.D4 [×8], C4⋊Dic5, C4⋊Dic5 [×4], D10⋊C4 [×6], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×C4×D5 [×6], C2×D20, Q8×C10, C20⋊2Q8, C20.6Q8 [×2], C4×D20, C4×D20 [×2], D5×C4⋊C4 [×2], D10⋊Q8 [×4], D10⋊3Q8 [×2], Q8×C20, D20⋊10Q8
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), C22×D5 [×7], D4⋊3Q8, C4○D20 [×2], Q8×D5 [×2], C23×D5, C2×C4○D20, C2×Q8×D5, D4⋊8D10, D20⋊10Q8
Generators and relations
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a10b, bd=db, dcd-1=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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(74 80)(75 79)(76 78)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(101 103)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)(122 140)(123 139)(124 138)(125 137)(126 136)(127 135)(128 134)(129 133)(130 132)(141 143)(144 160)(145 159)(146 158)(147 157)(148 156)(149 155)(150 154)(151 153)
(1 39 84 59)(2 40 85 60)(3 21 86 41)(4 22 87 42)(5 23 88 43)(6 24 89 44)(7 25 90 45)(8 26 91 46)(9 27 92 47)(10 28 93 48)(11 29 94 49)(12 30 95 50)(13 31 96 51)(14 32 97 52)(15 33 98 53)(16 34 99 54)(17 35 100 55)(18 36 81 56)(19 37 82 57)(20 38 83 58)(61 140 146 101)(62 121 147 102)(63 122 148 103)(64 123 149 104)(65 124 150 105)(66 125 151 106)(67 126 152 107)(68 127 153 108)(69 128 154 109)(70 129 155 110)(71 130 156 111)(72 131 157 112)(73 132 158 113)(74 133 159 114)(75 134 160 115)(76 135 141 116)(77 136 142 117)(78 137 143 118)(79 138 144 119)(80 139 145 120)
(1 115 84 134)(2 116 85 135)(3 117 86 136)(4 118 87 137)(5 119 88 138)(6 120 89 139)(7 101 90 140)(8 102 91 121)(9 103 92 122)(10 104 93 123)(11 105 94 124)(12 106 95 125)(13 107 96 126)(14 108 97 127)(15 109 98 128)(16 110 99 129)(17 111 100 130)(18 112 81 131)(19 113 82 132)(20 114 83 133)(21 142 41 77)(22 143 42 78)(23 144 43 79)(24 145 44 80)(25 146 45 61)(26 147 46 62)(27 148 47 63)(28 149 48 64)(29 150 49 65)(30 151 50 66)(31 152 51 67)(32 153 52 68)(33 154 53 69)(34 155 54 70)(35 156 55 71)(36 157 56 72)(37 158 57 73)(38 159 58 74)(39 160 59 75)(40 141 60 76)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(42,60)(43,59)(44,58)(45,57)(46,56)(47,55)(48,54)(49,53)(50,52)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(101,103)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)(141,143)(144,160)(145,159)(146,158)(147,157)(148,156)(149,155)(150,154)(151,153), (1,39,84,59)(2,40,85,60)(3,21,86,41)(4,22,87,42)(5,23,88,43)(6,24,89,44)(7,25,90,45)(8,26,91,46)(9,27,92,47)(10,28,93,48)(11,29,94,49)(12,30,95,50)(13,31,96,51)(14,32,97,52)(15,33,98,53)(16,34,99,54)(17,35,100,55)(18,36,81,56)(19,37,82,57)(20,38,83,58)(61,140,146,101)(62,121,147,102)(63,122,148,103)(64,123,149,104)(65,124,150,105)(66,125,151,106)(67,126,152,107)(68,127,153,108)(69,128,154,109)(70,129,155,110)(71,130,156,111)(72,131,157,112)(73,132,158,113)(74,133,159,114)(75,134,160,115)(76,135,141,116)(77,136,142,117)(78,137,143,118)(79,138,144,119)(80,139,145,120), (1,115,84,134)(2,116,85,135)(3,117,86,136)(4,118,87,137)(5,119,88,138)(6,120,89,139)(7,101,90,140)(8,102,91,121)(9,103,92,122)(10,104,93,123)(11,105,94,124)(12,106,95,125)(13,107,96,126)(14,108,97,127)(15,109,98,128)(16,110,99,129)(17,111,100,130)(18,112,81,131)(19,113,82,132)(20,114,83,133)(21,142,41,77)(22,143,42,78)(23,144,43,79)(24,145,44,80)(25,146,45,61)(26,147,46,62)(27,148,47,63)(28,149,48,64)(29,150,49,65)(30,151,50,66)(31,152,51,67)(32,153,52,68)(33,154,53,69)(34,155,54,70)(35,156,55,71)(36,157,56,72)(37,158,57,73)(38,159,58,74)(39,160,59,75)(40,141,60,76)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(42,60)(43,59)(44,58)(45,57)(46,56)(47,55)(48,54)(49,53)(50,52)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(101,103)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)(141,143)(144,160)(145,159)(146,158)(147,157)(148,156)(149,155)(150,154)(151,153), (1,39,84,59)(2,40,85,60)(3,21,86,41)(4,22,87,42)(5,23,88,43)(6,24,89,44)(7,25,90,45)(8,26,91,46)(9,27,92,47)(10,28,93,48)(11,29,94,49)(12,30,95,50)(13,31,96,51)(14,32,97,52)(15,33,98,53)(16,34,99,54)(17,35,100,55)(18,36,81,56)(19,37,82,57)(20,38,83,58)(61,140,146,101)(62,121,147,102)(63,122,148,103)(64,123,149,104)(65,124,150,105)(66,125,151,106)(67,126,152,107)(68,127,153,108)(69,128,154,109)(70,129,155,110)(71,130,156,111)(72,131,157,112)(73,132,158,113)(74,133,159,114)(75,134,160,115)(76,135,141,116)(77,136,142,117)(78,137,143,118)(79,138,144,119)(80,139,145,120), (1,115,84,134)(2,116,85,135)(3,117,86,136)(4,118,87,137)(5,119,88,138)(6,120,89,139)(7,101,90,140)(8,102,91,121)(9,103,92,122)(10,104,93,123)(11,105,94,124)(12,106,95,125)(13,107,96,126)(14,108,97,127)(15,109,98,128)(16,110,99,129)(17,111,100,130)(18,112,81,131)(19,113,82,132)(20,114,83,133)(21,142,41,77)(22,143,42,78)(23,144,43,79)(24,145,44,80)(25,146,45,61)(26,147,46,62)(27,148,47,63)(28,149,48,64)(29,150,49,65)(30,151,50,66)(31,152,51,67)(32,153,52,68)(33,154,53,69)(34,155,54,70)(35,156,55,71)(36,157,56,72)(37,158,57,73)(38,159,58,74)(39,160,59,75)(40,141,60,76) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(42,60),(43,59),(44,58),(45,57),(46,56),(47,55),(48,54),(49,53),(50,52),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68),(74,80),(75,79),(76,78),(82,100),(83,99),(84,98),(85,97),(86,96),(87,95),(88,94),(89,93),(90,92),(101,103),(104,120),(105,119),(106,118),(107,117),(108,116),(109,115),(110,114),(111,113),(122,140),(123,139),(124,138),(125,137),(126,136),(127,135),(128,134),(129,133),(130,132),(141,143),(144,160),(145,159),(146,158),(147,157),(148,156),(149,155),(150,154),(151,153)], [(1,39,84,59),(2,40,85,60),(3,21,86,41),(4,22,87,42),(5,23,88,43),(6,24,89,44),(7,25,90,45),(8,26,91,46),(9,27,92,47),(10,28,93,48),(11,29,94,49),(12,30,95,50),(13,31,96,51),(14,32,97,52),(15,33,98,53),(16,34,99,54),(17,35,100,55),(18,36,81,56),(19,37,82,57),(20,38,83,58),(61,140,146,101),(62,121,147,102),(63,122,148,103),(64,123,149,104),(65,124,150,105),(66,125,151,106),(67,126,152,107),(68,127,153,108),(69,128,154,109),(70,129,155,110),(71,130,156,111),(72,131,157,112),(73,132,158,113),(74,133,159,114),(75,134,160,115),(76,135,141,116),(77,136,142,117),(78,137,143,118),(79,138,144,119),(80,139,145,120)], [(1,115,84,134),(2,116,85,135),(3,117,86,136),(4,118,87,137),(5,119,88,138),(6,120,89,139),(7,101,90,140),(8,102,91,121),(9,103,92,122),(10,104,93,123),(11,105,94,124),(12,106,95,125),(13,107,96,126),(14,108,97,127),(15,109,98,128),(16,110,99,129),(17,111,100,130),(18,112,81,131),(19,113,82,132),(20,114,83,133),(21,142,41,77),(22,143,42,78),(23,144,43,79),(24,145,44,80),(25,146,45,61),(26,147,46,62),(27,148,47,63),(28,149,48,64),(29,150,49,65),(30,151,50,66),(31,152,51,67),(32,153,52,68),(33,154,53,69),(34,155,54,70),(35,156,55,71),(36,157,56,72),(37,158,57,73),(38,159,58,74),(39,160,59,75),(40,141,60,76)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 8 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 33 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 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 | 3 | 20 |
| 0 | 0 | 0 | 0 | 20 | 38 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,8,0,0,0,0,40,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,33,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,20,0,0,0,0,20,38] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | 2+ (1+4) | Q8×D5 | D4⋊8D10 |
| kernel | D20⋊10Q8 | C20⋊2Q8 | C20.6Q8 | C4×D20 | D5×C4⋊C4 | D10⋊Q8 | D10⋊3Q8 | Q8×C20 | D20 | C4×Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 2 | 3 | 2 | 4 | 2 | 1 | 4 | 2 | 4 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
D_{20}\rtimes_{10}Q_8 % in TeX
G:=Group("D20:10Q8"); // GroupNames label
G:=SmallGroup(320,1251);
// by ID
G=gap.SmallGroup(320,1251);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,185,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀