metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊5D20, C42.128D10, C10.112- (1+4), (C4×Q8)⋊9D5, (C5×Q8)⋊10D4, (C4×D20)⋊38C2, (Q8×C20)⋊11C2, C5⋊2(Q8⋊5D4), C4.25(C2×D20), C20.57(C2×D4), C4⋊2D20⋊17C2, C4⋊C4.295D10, D10⋊13(C4○D4), D10⋊2Q8⋊18C2, C4.D20⋊20C2, (C2×Q8).204D10, C2.21(C22×D20), C10.19(C22×D4), (C2×C10).120C24, (C2×C20).498C23, (C4×C20).172C22, D10⋊C4.6C22, (C2×D20).224C22, C4⋊Dic5.306C22, (Q8×C10).220C22, (C2×Dic5).54C23, (C22×D5).45C23, C22.141(C23×D5), C2.12(Q8.10D10), (C2×Dic10).155C22, (C2×Q8×D5)⋊4C2, C2.29(D5×C4○D4), (C2×Q8⋊2D5)⋊3C2, (C2×C4×D5).81C22, C10.145(C2×C4○D4), (C5×C4⋊C4).348C22, (C2×C4).168(C22×D5), SmallGroup(320,1248)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1126 in 290 conjugacy classes, 113 normal (22 characteristic)
C1, C2 [×3], C2 [×5], C4 [×6], C4 [×8], C22, C22 [×13], C5, C2×C4, C2×C4 [×6], C2×C4 [×16], D4 [×12], Q8 [×4], Q8 [×6], C23 [×4], D5 [×5], C10 [×3], C42 [×3], C22⋊C4 [×10], C4⋊C4 [×3], C4⋊C4 [×3], C22×C4 [×6], C2×D4 [×6], C2×Q8, C2×Q8 [×7], C4○D4 [×4], Dic5 [×4], C20 [×6], C20 [×4], D10 [×2], D10 [×11], C2×C10, C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, Dic10 [×6], C4×D5 [×12], D20 [×12], C2×Dic5, C2×Dic5 [×3], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5, C22×D5 [×3], Q8⋊5D4, C4⋊Dic5 [×3], D10⋊C4, D10⋊C4 [×9], C4×C20 [×3], C5×C4⋊C4 [×3], C2×Dic10 [×3], C2×C4×D5 [×6], C2×D20 [×6], Q8×D5 [×4], Q8⋊2D5 [×4], Q8×C10, C4×D20 [×3], C4.D20 [×3], C4⋊2D20 [×3], D10⋊2Q8 [×3], Q8×C20, C2×Q8×D5, C2×Q8⋊2D5, Q8⋊5D20
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), D20 [×4], C22×D5 [×7], Q8⋊5D4, C2×D20 [×6], C23×D5, C22×D20, Q8.10D10, D5×C4○D4, Q8⋊5D20
Generators and relations
G = < a,b,c,d | a4=c20=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
►On 160 points
Generators in S160
(1 105 88 126)(2 106 89 127)(3 107 90 128)(4 108 91 129)(5 109 92 130)(6 110 93 131)(7 111 94 132)(8 112 95 133)(9 113 96 134)(10 114 97 135)(11 115 98 136)(12 116 99 137)(13 117 100 138)(14 118 81 139)(15 119 82 140)(16 120 83 121)(17 101 84 122)(18 102 85 123)(19 103 86 124)(20 104 87 125)(21 49 144 71)(22 50 145 72)(23 51 146 73)(24 52 147 74)(25 53 148 75)(26 54 149 76)(27 55 150 77)(28 56 151 78)(29 57 152 79)(30 58 153 80)(31 59 154 61)(32 60 155 62)(33 41 156 63)(34 42 157 64)(35 43 158 65)(36 44 159 66)(37 45 160 67)(38 46 141 68)(39 47 142 69)(40 48 143 70)
(1 49 88 71)(2 72 89 50)(3 51 90 73)(4 74 91 52)(5 53 92 75)(6 76 93 54)(7 55 94 77)(8 78 95 56)(9 57 96 79)(10 80 97 58)(11 59 98 61)(12 62 99 60)(13 41 100 63)(14 64 81 42)(15 43 82 65)(16 66 83 44)(17 45 84 67)(18 68 85 46)(19 47 86 69)(20 70 87 48)(21 126 144 105)(22 106 145 127)(23 128 146 107)(24 108 147 129)(25 130 148 109)(26 110 149 131)(27 132 150 111)(28 112 151 133)(29 134 152 113)(30 114 153 135)(31 136 154 115)(32 116 155 137)(33 138 156 117)(34 118 157 139)(35 140 158 119)(36 120 159 121)(37 122 160 101)(38 102 141 123)(39 124 142 103)(40 104 143 125)
(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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 148)(22 147)(23 146)(24 145)(25 144)(26 143)(27 142)(28 141)(29 160)(30 159)(31 158)(32 157)(33 156)(34 155)(35 154)(36 153)(37 152)(38 151)(39 150)(40 149)(41 63)(42 62)(43 61)(44 80)(45 79)(46 78)(47 77)(48 76)(49 75)(50 74)(51 73)(52 72)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)(81 99)(82 98)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)(114 120)(115 119)(116 118)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(136 140)(137 139)
G:=sub<Sym(160)| (1,105,88,126)(2,106,89,127)(3,107,90,128)(4,108,91,129)(5,109,92,130)(6,110,93,131)(7,111,94,132)(8,112,95,133)(9,113,96,134)(10,114,97,135)(11,115,98,136)(12,116,99,137)(13,117,100,138)(14,118,81,139)(15,119,82,140)(16,120,83,121)(17,101,84,122)(18,102,85,123)(19,103,86,124)(20,104,87,125)(21,49,144,71)(22,50,145,72)(23,51,146,73)(24,52,147,74)(25,53,148,75)(26,54,149,76)(27,55,150,77)(28,56,151,78)(29,57,152,79)(30,58,153,80)(31,59,154,61)(32,60,155,62)(33,41,156,63)(34,42,157,64)(35,43,158,65)(36,44,159,66)(37,45,160,67)(38,46,141,68)(39,47,142,69)(40,48,143,70), (1,49,88,71)(2,72,89,50)(3,51,90,73)(4,74,91,52)(5,53,92,75)(6,76,93,54)(7,55,94,77)(8,78,95,56)(9,57,96,79)(10,80,97,58)(11,59,98,61)(12,62,99,60)(13,41,100,63)(14,64,81,42)(15,43,82,65)(16,66,83,44)(17,45,84,67)(18,68,85,46)(19,47,86,69)(20,70,87,48)(21,126,144,105)(22,106,145,127)(23,128,146,107)(24,108,147,129)(25,130,148,109)(26,110,149,131)(27,132,150,111)(28,112,151,133)(29,134,152,113)(30,114,153,135)(31,136,154,115)(32,116,155,137)(33,138,156,117)(34,118,157,139)(35,140,158,119)(36,120,159,121)(37,122,160,101)(38,102,141,123)(39,124,142,103)(40,104,143,125), (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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,148)(22,147)(23,146)(24,145)(25,144)(26,143)(27,142)(28,141)(29,160)(30,159)(31,158)(32,157)(33,156)(34,155)(35,154)(36,153)(37,152)(38,151)(39,150)(40,149)(41,63)(42,62)(43,61)(44,80)(45,79)(46,78)(47,77)(48,76)(49,75)(50,74)(51,73)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108)(114,120)(115,119)(116,118)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,140)(137,139)>;
G:=Group( (1,105,88,126)(2,106,89,127)(3,107,90,128)(4,108,91,129)(5,109,92,130)(6,110,93,131)(7,111,94,132)(8,112,95,133)(9,113,96,134)(10,114,97,135)(11,115,98,136)(12,116,99,137)(13,117,100,138)(14,118,81,139)(15,119,82,140)(16,120,83,121)(17,101,84,122)(18,102,85,123)(19,103,86,124)(20,104,87,125)(21,49,144,71)(22,50,145,72)(23,51,146,73)(24,52,147,74)(25,53,148,75)(26,54,149,76)(27,55,150,77)(28,56,151,78)(29,57,152,79)(30,58,153,80)(31,59,154,61)(32,60,155,62)(33,41,156,63)(34,42,157,64)(35,43,158,65)(36,44,159,66)(37,45,160,67)(38,46,141,68)(39,47,142,69)(40,48,143,70), (1,49,88,71)(2,72,89,50)(3,51,90,73)(4,74,91,52)(5,53,92,75)(6,76,93,54)(7,55,94,77)(8,78,95,56)(9,57,96,79)(10,80,97,58)(11,59,98,61)(12,62,99,60)(13,41,100,63)(14,64,81,42)(15,43,82,65)(16,66,83,44)(17,45,84,67)(18,68,85,46)(19,47,86,69)(20,70,87,48)(21,126,144,105)(22,106,145,127)(23,128,146,107)(24,108,147,129)(25,130,148,109)(26,110,149,131)(27,132,150,111)(28,112,151,133)(29,134,152,113)(30,114,153,135)(31,136,154,115)(32,116,155,137)(33,138,156,117)(34,118,157,139)(35,140,158,119)(36,120,159,121)(37,122,160,101)(38,102,141,123)(39,124,142,103)(40,104,143,125), (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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,148)(22,147)(23,146)(24,145)(25,144)(26,143)(27,142)(28,141)(29,160)(30,159)(31,158)(32,157)(33,156)(34,155)(35,154)(36,153)(37,152)(38,151)(39,150)(40,149)(41,63)(42,62)(43,61)(44,80)(45,79)(46,78)(47,77)(48,76)(49,75)(50,74)(51,73)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108)(114,120)(115,119)(116,118)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,140)(137,139) );
G=PermutationGroup([(1,105,88,126),(2,106,89,127),(3,107,90,128),(4,108,91,129),(5,109,92,130),(6,110,93,131),(7,111,94,132),(8,112,95,133),(9,113,96,134),(10,114,97,135),(11,115,98,136),(12,116,99,137),(13,117,100,138),(14,118,81,139),(15,119,82,140),(16,120,83,121),(17,101,84,122),(18,102,85,123),(19,103,86,124),(20,104,87,125),(21,49,144,71),(22,50,145,72),(23,51,146,73),(24,52,147,74),(25,53,148,75),(26,54,149,76),(27,55,150,77),(28,56,151,78),(29,57,152,79),(30,58,153,80),(31,59,154,61),(32,60,155,62),(33,41,156,63),(34,42,157,64),(35,43,158,65),(36,44,159,66),(37,45,160,67),(38,46,141,68),(39,47,142,69),(40,48,143,70)], [(1,49,88,71),(2,72,89,50),(3,51,90,73),(4,74,91,52),(5,53,92,75),(6,76,93,54),(7,55,94,77),(8,78,95,56),(9,57,96,79),(10,80,97,58),(11,59,98,61),(12,62,99,60),(13,41,100,63),(14,64,81,42),(15,43,82,65),(16,66,83,44),(17,45,84,67),(18,68,85,46),(19,47,86,69),(20,70,87,48),(21,126,144,105),(22,106,145,127),(23,128,146,107),(24,108,147,129),(25,130,148,109),(26,110,149,131),(27,132,150,111),(28,112,151,133),(29,134,152,113),(30,114,153,135),(31,136,154,115),(32,116,155,137),(33,138,156,117),(34,118,157,139),(35,140,158,119),(36,120,159,121),(37,122,160,101),(38,102,141,123),(39,124,142,103),(40,104,143,125)], [(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)], [(1,5),(2,4),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(21,148),(22,147),(23,146),(24,145),(25,144),(26,143),(27,142),(28,141),(29,160),(30,159),(31,158),(32,157),(33,156),(34,155),(35,154),(36,153),(37,152),(38,151),(39,150),(40,149),(41,63),(42,62),(43,61),(44,80),(45,79),(46,78),(47,77),(48,76),(49,75),(50,74),(51,73),(52,72),(53,71),(54,70),(55,69),(56,68),(57,67),(58,66),(59,65),(60,64),(81,99),(82,98),(83,97),(84,96),(85,95),(86,94),(87,93),(88,92),(89,91),(101,113),(102,112),(103,111),(104,110),(105,109),(106,108),(114,120),(115,119),(116,118),(121,135),(122,134),(123,133),(124,132),(125,131),(126,130),(127,129),(136,140),(137,139)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 28 | 32 |
| 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 | 29 | 37 |
| 0 | 0 | 0 | 0 | 26 | 12 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 5 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 35 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 5 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 35 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,28,0,0,0,0,0,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,29,26,0,0,0,0,37,12],[1,40,0,0,0,0,2,40,0,0,0,0,0,0,40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,35,0,0,0,0,0,40],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,40,5,0,0,0,0,0,1,0,0,0,0,0,0,1,35,0,0,0,0,0,40] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 20 | 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 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D20 | 2- (1+4) | Q8.10D10 | D5×C4○D4 |
| kernel | Q8⋊5D20 | C4×D20 | C4.D20 | C4⋊2D20 | D10⋊2Q8 | Q8×C20 | C2×Q8×D5 | C2×Q8⋊2D5 | C5×Q8 | C4×Q8 | D10 | C42 | C4⋊C4 | C2×Q8 | Q8 | C10 | C2 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 4 | 2 | 4 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
Q_8\rtimes_5D_{20} % in TeX
G:=Group("Q8:5D20"); // GroupNames label
G:=SmallGroup(320,1248);
// by ID
G=gap.SmallGroup(320,1248);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,675,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^20=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀