metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊6D20, C42.129D10, C10.1102+ (1+4), (C5×Q8)⋊11D4, (C4×Q8)⋊10D5, (C4×D20)⋊39C2, (Q8×C20)⋊12C2, C5⋊2(Q8⋊6D4), C4.26(C2×D20), C20.58(C2×D4), C20⋊17(C4○D4), C4⋊2D20⋊18C2, C4⋊D20⋊13C2, C4⋊C4.296D10, C4⋊3(Q8⋊2D5), (C2×Q8).205D10, C2.22(C22×D20), C10.20(C22×D4), (C2×C20).170C23, (C2×C10).121C24, (C4×C20).173C22, (C2×D20).30C22, C2.22(D4⋊8D10), C4⋊Dic5.399C22, (Q8×C10).221C22, (C22×D5).46C23, C22.142(C23×D5), (C2×Dic5).225C23, D10⋊C4.101C22, (C2×Q8⋊2D5)⋊4C2, (C2×C4×D5).82C22, C10.112(C2×C4○D4), C2.11(C2×Q8⋊2D5), (C5×C4⋊C4).349C22, (C2×C4).734(C22×D5), SmallGroup(320,1249)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1366 in 312 conjugacy classes, 115 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×5], C22, C22 [×18], C5, C2×C4, C2×C4 [×6], C2×C4 [×14], D4 [×24], Q8 [×4], C23 [×6], D5 [×6], C10 [×3], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4, C22×C4 [×6], C2×D4 [×15], C2×Q8, C4○D4 [×8], Dic5 [×2], C20 [×8], C20 [×3], D10 [×18], C2×C10, C4×D4 [×3], C4×Q8, C4⋊D4 [×6], C4⋊1D4 [×3], C2×C4○D4 [×2], C4×D5 [×12], D20 [×24], C2×Dic5 [×2], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5 [×6], Q8⋊6D4, C4⋊Dic5, D10⋊C4 [×6], C4×C20 [×3], C5×C4⋊C4 [×3], C2×C4×D5 [×6], C2×D20 [×15], Q8⋊2D5 [×8], Q8×C10, C4×D20 [×3], C4⋊D20 [×3], C4⋊2D20 [×6], Q8×C20, C2×Q8⋊2D5 [×2], Q8⋊6D20
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⋊6D4, C2×D20 [×6], Q8⋊2D5 [×2], C23×D5, C22×D20, C2×Q8⋊2D5, D4⋊8D10, Q8⋊6D20
Generators and relations
G = < a,b,c,d | a4=c20=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >
►On 160 points
Generators in S160
(1 123 96 47)(2 124 97 48)(3 125 98 49)(4 126 99 50)(5 127 100 51)(6 128 81 52)(7 129 82 53)(8 130 83 54)(9 131 84 55)(10 132 85 56)(11 133 86 57)(12 134 87 58)(13 135 88 59)(14 136 89 60)(15 137 90 41)(16 138 91 42)(17 139 92 43)(18 140 93 44)(19 121 94 45)(20 122 95 46)(21 75 153 115)(22 76 154 116)(23 77 155 117)(24 78 156 118)(25 79 157 119)(26 80 158 120)(27 61 159 101)(28 62 160 102)(29 63 141 103)(30 64 142 104)(31 65 143 105)(32 66 144 106)(33 67 145 107)(34 68 146 108)(35 69 147 109)(36 70 148 110)(37 71 149 111)(38 72 150 112)(39 73 151 113)(40 74 152 114)
(1 76 96 116)(2 77 97 117)(3 78 98 118)(4 79 99 119)(5 80 100 120)(6 61 81 101)(7 62 82 102)(8 63 83 103)(9 64 84 104)(10 65 85 105)(11 66 86 106)(12 67 87 107)(13 68 88 108)(14 69 89 109)(15 70 90 110)(16 71 91 111)(17 72 92 112)(18 73 93 113)(19 74 94 114)(20 75 95 115)(21 46 153 122)(22 47 154 123)(23 48 155 124)(24 49 156 125)(25 50 157 126)(26 51 158 127)(27 52 159 128)(28 53 160 129)(29 54 141 130)(30 55 142 131)(31 56 143 132)(32 57 144 133)(33 58 145 134)(34 59 146 135)(35 60 147 136)(36 41 148 137)(37 42 149 138)(38 43 150 139)(39 44 151 140)(40 45 152 121)
(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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 60)(18 59)(19 58)(20 57)(21 106)(22 105)(23 104)(24 103)(25 102)(26 101)(27 120)(28 119)(29 118)(30 117)(31 116)(32 115)(33 114)(34 113)(35 112)(36 111)(37 110)(38 109)(39 108)(40 107)(61 158)(62 157)(63 156)(64 155)(65 154)(66 153)(67 152)(68 151)(69 150)(70 149)(71 148)(72 147)(73 146)(74 145)(75 144)(76 143)(77 142)(78 141)(79 160)(80 159)(81 127)(82 126)(83 125)(84 124)(85 123)(86 122)(87 121)(88 140)(89 139)(90 138)(91 137)(92 136)(93 135)(94 134)(95 133)(96 132)(97 131)(98 130)(99 129)(100 128)
G:=sub<Sym(160)| (1,123,96,47)(2,124,97,48)(3,125,98,49)(4,126,99,50)(5,127,100,51)(6,128,81,52)(7,129,82,53)(8,130,83,54)(9,131,84,55)(10,132,85,56)(11,133,86,57)(12,134,87,58)(13,135,88,59)(14,136,89,60)(15,137,90,41)(16,138,91,42)(17,139,92,43)(18,140,93,44)(19,121,94,45)(20,122,95,46)(21,75,153,115)(22,76,154,116)(23,77,155,117)(24,78,156,118)(25,79,157,119)(26,80,158,120)(27,61,159,101)(28,62,160,102)(29,63,141,103)(30,64,142,104)(31,65,143,105)(32,66,144,106)(33,67,145,107)(34,68,146,108)(35,69,147,109)(36,70,148,110)(37,71,149,111)(38,72,150,112)(39,73,151,113)(40,74,152,114), (1,76,96,116)(2,77,97,117)(3,78,98,118)(4,79,99,119)(5,80,100,120)(6,61,81,101)(7,62,82,102)(8,63,83,103)(9,64,84,104)(10,65,85,105)(11,66,86,106)(12,67,87,107)(13,68,88,108)(14,69,89,109)(15,70,90,110)(16,71,91,111)(17,72,92,112)(18,73,93,113)(19,74,94,114)(20,75,95,115)(21,46,153,122)(22,47,154,123)(23,48,155,124)(24,49,156,125)(25,50,157,126)(26,51,158,127)(27,52,159,128)(28,53,160,129)(29,54,141,130)(30,55,142,131)(31,56,143,132)(32,57,144,133)(33,58,145,134)(34,59,146,135)(35,60,147,136)(36,41,148,137)(37,42,149,138)(38,43,150,139)(39,44,151,140)(40,45,152,121), (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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,60)(18,59)(19,58)(20,57)(21,106)(22,105)(23,104)(24,103)(25,102)(26,101)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,110)(38,109)(39,108)(40,107)(61,158)(62,157)(63,156)(64,155)(65,154)(66,153)(67,152)(68,151)(69,150)(70,149)(71,148)(72,147)(73,146)(74,145)(75,144)(76,143)(77,142)(78,141)(79,160)(80,159)(81,127)(82,126)(83,125)(84,124)(85,123)(86,122)(87,121)(88,140)(89,139)(90,138)(91,137)(92,136)(93,135)(94,134)(95,133)(96,132)(97,131)(98,130)(99,129)(100,128)>;
G:=Group( (1,123,96,47)(2,124,97,48)(3,125,98,49)(4,126,99,50)(5,127,100,51)(6,128,81,52)(7,129,82,53)(8,130,83,54)(9,131,84,55)(10,132,85,56)(11,133,86,57)(12,134,87,58)(13,135,88,59)(14,136,89,60)(15,137,90,41)(16,138,91,42)(17,139,92,43)(18,140,93,44)(19,121,94,45)(20,122,95,46)(21,75,153,115)(22,76,154,116)(23,77,155,117)(24,78,156,118)(25,79,157,119)(26,80,158,120)(27,61,159,101)(28,62,160,102)(29,63,141,103)(30,64,142,104)(31,65,143,105)(32,66,144,106)(33,67,145,107)(34,68,146,108)(35,69,147,109)(36,70,148,110)(37,71,149,111)(38,72,150,112)(39,73,151,113)(40,74,152,114), (1,76,96,116)(2,77,97,117)(3,78,98,118)(4,79,99,119)(5,80,100,120)(6,61,81,101)(7,62,82,102)(8,63,83,103)(9,64,84,104)(10,65,85,105)(11,66,86,106)(12,67,87,107)(13,68,88,108)(14,69,89,109)(15,70,90,110)(16,71,91,111)(17,72,92,112)(18,73,93,113)(19,74,94,114)(20,75,95,115)(21,46,153,122)(22,47,154,123)(23,48,155,124)(24,49,156,125)(25,50,157,126)(26,51,158,127)(27,52,159,128)(28,53,160,129)(29,54,141,130)(30,55,142,131)(31,56,143,132)(32,57,144,133)(33,58,145,134)(34,59,146,135)(35,60,147,136)(36,41,148,137)(37,42,149,138)(38,43,150,139)(39,44,151,140)(40,45,152,121), (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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,60)(18,59)(19,58)(20,57)(21,106)(22,105)(23,104)(24,103)(25,102)(26,101)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,110)(38,109)(39,108)(40,107)(61,158)(62,157)(63,156)(64,155)(65,154)(66,153)(67,152)(68,151)(69,150)(70,149)(71,148)(72,147)(73,146)(74,145)(75,144)(76,143)(77,142)(78,141)(79,160)(80,159)(81,127)(82,126)(83,125)(84,124)(85,123)(86,122)(87,121)(88,140)(89,139)(90,138)(91,137)(92,136)(93,135)(94,134)(95,133)(96,132)(97,131)(98,130)(99,129)(100,128) );
G=PermutationGroup([(1,123,96,47),(2,124,97,48),(3,125,98,49),(4,126,99,50),(5,127,100,51),(6,128,81,52),(7,129,82,53),(8,130,83,54),(9,131,84,55),(10,132,85,56),(11,133,86,57),(12,134,87,58),(13,135,88,59),(14,136,89,60),(15,137,90,41),(16,138,91,42),(17,139,92,43),(18,140,93,44),(19,121,94,45),(20,122,95,46),(21,75,153,115),(22,76,154,116),(23,77,155,117),(24,78,156,118),(25,79,157,119),(26,80,158,120),(27,61,159,101),(28,62,160,102),(29,63,141,103),(30,64,142,104),(31,65,143,105),(32,66,144,106),(33,67,145,107),(34,68,146,108),(35,69,147,109),(36,70,148,110),(37,71,149,111),(38,72,150,112),(39,73,151,113),(40,74,152,114)], [(1,76,96,116),(2,77,97,117),(3,78,98,118),(4,79,99,119),(5,80,100,120),(6,61,81,101),(7,62,82,102),(8,63,83,103),(9,64,84,104),(10,65,85,105),(11,66,86,106),(12,67,87,107),(13,68,88,108),(14,69,89,109),(15,70,90,110),(16,71,91,111),(17,72,92,112),(18,73,93,113),(19,74,94,114),(20,75,95,115),(21,46,153,122),(22,47,154,123),(23,48,155,124),(24,49,156,125),(25,50,157,126),(26,51,158,127),(27,52,159,128),(28,53,160,129),(29,54,141,130),(30,55,142,131),(31,56,143,132),(32,57,144,133),(33,58,145,134),(34,59,146,135),(35,60,147,136),(36,41,148,137),(37,42,149,138),(38,43,150,139),(39,44,151,140),(40,45,152,121)], [(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)], [(1,56),(2,55),(3,54),(4,53),(5,52),(6,51),(7,50),(8,49),(9,48),(10,47),(11,46),(12,45),(13,44),(14,43),(15,42),(16,41),(17,60),(18,59),(19,58),(20,57),(21,106),(22,105),(23,104),(24,103),(25,102),(26,101),(27,120),(28,119),(29,118),(30,117),(31,116),(32,115),(33,114),(34,113),(35,112),(36,111),(37,110),(38,109),(39,108),(40,107),(61,158),(62,157),(63,156),(64,155),(65,154),(66,153),(67,152),(68,151),(69,150),(70,149),(71,148),(72,147),(73,146),(74,145),(75,144),(76,143),(77,142),(78,141),(79,160),(80,159),(81,127),(82,126),(83,125),(84,124),(85,123),(86,122),(87,121),(88,140),(89,139),(90,138),(91,137),(92,136),(93,135),(94,134),(95,133),(96,132),(97,131),(98,130),(99,129),(100,128)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 21 |
| 0 | 0 | 37 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 25 |
| 0 | 0 | 0 | 9 |
| 27 | 30 | 0 | 0 |
| 11 | 32 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 30 | 27 | 0 | 0 |
| 32 | 11 | 0 | 0 |
| 0 | 0 | 40 | 21 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,37,0,0,21,1],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,25,9],[27,11,0,0,30,32,0,0,0,0,1,0,0,0,0,1],[30,32,0,0,27,11,0,0,0,0,40,0,0,0,21,1] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | ··· | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D20 | 2+ (1+4) | Q8⋊2D5 | D4⋊8D10 |
| kernel | Q8⋊6D20 | C4×D20 | C4⋊D20 | C4⋊2D20 | Q8×C20 | C2×Q8⋊2D5 | C5×Q8 | C4×Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | Q8 | C10 | C4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 4 | 2 | 4 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
Q_8\rtimes_6D_{20} % in TeX
G:=Group("Q8:6D20"); // GroupNames label
G:=SmallGroup(320,1249);
// by ID
G=gap.SmallGroup(320,1249);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,184,675,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^20=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀