direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×D20, C42.127D10, C10.662- (1+4), C5⋊2(D4×Q8), C4⋊3(Q8×D5), (C4×Q8)⋊8D5, (C5×Q8)⋊9D4, C20⋊8(C2×Q8), D10⋊5(C2×Q8), (Q8×C20)⋊10C2, C4.24(C2×D20), C20.56(C2×D4), C4⋊C4.294D10, C20⋊2Q8⋊27C2, (C4×D20).20C2, D10⋊2Q8⋊17C2, (C2×Q8).203D10, C10.18(C22×D4), C2.20(C22×D20), C10.29(C22×Q8), (C4×C20).171C22, (C2×C20).169C23, (C2×C10).119C24, (C2×D20).296C22, C4⋊Dic5.305C22, (Q8×C10).219C22, (C2×Dic5).53C23, C22.140(C23×D5), (C22×D5).188C23, C2.23(D4.10D10), D10⋊C4.100C22, (C2×Dic10).154C22, (C2×Q8×D5)⋊3C2, C2.12(C2×Q8×D5), (C2×C4×D5).80C22, (C5×C4⋊C4).347C22, (C2×C4).583(C22×D5), SmallGroup(320,1247)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 982 in 280 conjugacy classes, 123 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×9], C22, C22 [×8], C5, C2×C4, C2×C4 [×6], C2×C4 [×18], D4 [×4], Q8 [×4], Q8 [×12], C23 [×2], D5 [×4], C10 [×3], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×9], C22×C4 [×6], C2×D4, C2×Q8, C2×Q8 [×14], Dic5 [×6], C20 [×8], C20 [×3], D10 [×4], D10 [×4], C2×C10, C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C4⋊Q8 [×3], C22×Q8 [×2], Dic10 [×12], C4×D5 [×12], D20 [×4], C2×Dic5 [×6], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5 [×2], D4×Q8, C4⋊Dic5 [×9], D10⋊C4 [×6], C4×C20 [×3], C5×C4⋊C4 [×3], C2×Dic10 [×6], C2×C4×D5 [×6], C2×D20, Q8×D5 [×8], Q8×C10, C20⋊2Q8 [×3], C4×D20 [×3], D10⋊2Q8 [×6], Q8×C20, C2×Q8×D5 [×2], Q8×D20
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×6], C24, D10 [×7], C22×D4, C22×Q8, 2- (1+4), D20 [×4], C22×D5 [×7], D4×Q8, C2×D20 [×6], Q8×D5 [×2], C23×D5, C22×D20, C2×Q8×D5, D4.10D10, Q8×D20
Generators and relations
G = < a,b,c,d | a4=c20=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 160 points
Generators in S160
(1 160 117 26)(2 141 118 27)(3 142 119 28)(4 143 120 29)(5 144 101 30)(6 145 102 31)(7 146 103 32)(8 147 104 33)(9 148 105 34)(10 149 106 35)(11 150 107 36)(12 151 108 37)(13 152 109 38)(14 153 110 39)(15 154 111 40)(16 155 112 21)(17 156 113 22)(18 157 114 23)(19 158 115 24)(20 159 116 25)(41 122 85 80)(42 123 86 61)(43 124 87 62)(44 125 88 63)(45 126 89 64)(46 127 90 65)(47 128 91 66)(48 129 92 67)(49 130 93 68)(50 131 94 69)(51 132 95 70)(52 133 96 71)(53 134 97 72)(54 135 98 73)(55 136 99 74)(56 137 100 75)(57 138 81 76)(58 139 82 77)(59 140 83 78)(60 121 84 79)
(1 83 117 59)(2 84 118 60)(3 85 119 41)(4 86 120 42)(5 87 101 43)(6 88 102 44)(7 89 103 45)(8 90 104 46)(9 91 105 47)(10 92 106 48)(11 93 107 49)(12 94 108 50)(13 95 109 51)(14 96 110 52)(15 97 111 53)(16 98 112 54)(17 99 113 55)(18 100 114 56)(19 81 115 57)(20 82 116 58)(21 73 155 135)(22 74 156 136)(23 75 157 137)(24 76 158 138)(25 77 159 139)(26 78 160 140)(27 79 141 121)(28 80 142 122)(29 61 143 123)(30 62 144 124)(31 63 145 125)(32 64 146 126)(33 65 147 127)(34 66 148 128)(35 67 149 129)(36 68 150 130)(37 69 151 131)(38 70 152 132)(39 71 153 133)(40 72 154 134)
(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)(21 25)(22 24)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 51)(42 50)(43 49)(44 48)(45 47)(52 60)(53 59)(54 58)(55 57)(61 69)(62 68)(63 67)(64 66)(70 80)(71 79)(72 78)(73 77)(74 76)(81 99)(82 98)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(101 107)(102 106)(103 105)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)(121 133)(122 132)(123 131)(124 130)(125 129)(126 128)(134 140)(135 139)(136 138)(141 153)(142 152)(143 151)(144 150)(145 149)(146 148)(154 160)(155 159)(156 158)
G:=sub<Sym(160)| (1,160,117,26)(2,141,118,27)(3,142,119,28)(4,143,120,29)(5,144,101,30)(6,145,102,31)(7,146,103,32)(8,147,104,33)(9,148,105,34)(10,149,106,35)(11,150,107,36)(12,151,108,37)(13,152,109,38)(14,153,110,39)(15,154,111,40)(16,155,112,21)(17,156,113,22)(18,157,114,23)(19,158,115,24)(20,159,116,25)(41,122,85,80)(42,123,86,61)(43,124,87,62)(44,125,88,63)(45,126,89,64)(46,127,90,65)(47,128,91,66)(48,129,92,67)(49,130,93,68)(50,131,94,69)(51,132,95,70)(52,133,96,71)(53,134,97,72)(54,135,98,73)(55,136,99,74)(56,137,100,75)(57,138,81,76)(58,139,82,77)(59,140,83,78)(60,121,84,79), (1,83,117,59)(2,84,118,60)(3,85,119,41)(4,86,120,42)(5,87,101,43)(6,88,102,44)(7,89,103,45)(8,90,104,46)(9,91,105,47)(10,92,106,48)(11,93,107,49)(12,94,108,50)(13,95,109,51)(14,96,110,52)(15,97,111,53)(16,98,112,54)(17,99,113,55)(18,100,114,56)(19,81,115,57)(20,82,116,58)(21,73,155,135)(22,74,156,136)(23,75,157,137)(24,76,158,138)(25,77,159,139)(26,78,160,140)(27,79,141,121)(28,80,142,122)(29,61,143,123)(30,62,144,124)(31,63,145,125)(32,64,146,126)(33,65,147,127)(34,66,148,128)(35,67,149,129)(36,68,150,130)(37,69,151,131)(38,70,152,132)(39,71,153,133)(40,72,154,134), (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)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(61,69)(62,68)(63,67)(64,66)(70,80)(71,79)(72,78)(73,77)(74,76)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(101,107)(102,106)(103,105)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(134,140)(135,139)(136,138)(141,153)(142,152)(143,151)(144,150)(145,149)(146,148)(154,160)(155,159)(156,158)>;
G:=Group( (1,160,117,26)(2,141,118,27)(3,142,119,28)(4,143,120,29)(5,144,101,30)(6,145,102,31)(7,146,103,32)(8,147,104,33)(9,148,105,34)(10,149,106,35)(11,150,107,36)(12,151,108,37)(13,152,109,38)(14,153,110,39)(15,154,111,40)(16,155,112,21)(17,156,113,22)(18,157,114,23)(19,158,115,24)(20,159,116,25)(41,122,85,80)(42,123,86,61)(43,124,87,62)(44,125,88,63)(45,126,89,64)(46,127,90,65)(47,128,91,66)(48,129,92,67)(49,130,93,68)(50,131,94,69)(51,132,95,70)(52,133,96,71)(53,134,97,72)(54,135,98,73)(55,136,99,74)(56,137,100,75)(57,138,81,76)(58,139,82,77)(59,140,83,78)(60,121,84,79), (1,83,117,59)(2,84,118,60)(3,85,119,41)(4,86,120,42)(5,87,101,43)(6,88,102,44)(7,89,103,45)(8,90,104,46)(9,91,105,47)(10,92,106,48)(11,93,107,49)(12,94,108,50)(13,95,109,51)(14,96,110,52)(15,97,111,53)(16,98,112,54)(17,99,113,55)(18,100,114,56)(19,81,115,57)(20,82,116,58)(21,73,155,135)(22,74,156,136)(23,75,157,137)(24,76,158,138)(25,77,159,139)(26,78,160,140)(27,79,141,121)(28,80,142,122)(29,61,143,123)(30,62,144,124)(31,63,145,125)(32,64,146,126)(33,65,147,127)(34,66,148,128)(35,67,149,129)(36,68,150,130)(37,69,151,131)(38,70,152,132)(39,71,153,133)(40,72,154,134), (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)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(61,69)(62,68)(63,67)(64,66)(70,80)(71,79)(72,78)(73,77)(74,76)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(101,107)(102,106)(103,105)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(134,140)(135,139)(136,138)(141,153)(142,152)(143,151)(144,150)(145,149)(146,148)(154,160)(155,159)(156,158) );
G=PermutationGroup([(1,160,117,26),(2,141,118,27),(3,142,119,28),(4,143,120,29),(5,144,101,30),(6,145,102,31),(7,146,103,32),(8,147,104,33),(9,148,105,34),(10,149,106,35),(11,150,107,36),(12,151,108,37),(13,152,109,38),(14,153,110,39),(15,154,111,40),(16,155,112,21),(17,156,113,22),(18,157,114,23),(19,158,115,24),(20,159,116,25),(41,122,85,80),(42,123,86,61),(43,124,87,62),(44,125,88,63),(45,126,89,64),(46,127,90,65),(47,128,91,66),(48,129,92,67),(49,130,93,68),(50,131,94,69),(51,132,95,70),(52,133,96,71),(53,134,97,72),(54,135,98,73),(55,136,99,74),(56,137,100,75),(57,138,81,76),(58,139,82,77),(59,140,83,78),(60,121,84,79)], [(1,83,117,59),(2,84,118,60),(3,85,119,41),(4,86,120,42),(5,87,101,43),(6,88,102,44),(7,89,103,45),(8,90,104,46),(9,91,105,47),(10,92,106,48),(11,93,107,49),(12,94,108,50),(13,95,109,51),(14,96,110,52),(15,97,111,53),(16,98,112,54),(17,99,113,55),(18,100,114,56),(19,81,115,57),(20,82,116,58),(21,73,155,135),(22,74,156,136),(23,75,157,137),(24,76,158,138),(25,77,159,139),(26,78,160,140),(27,79,141,121),(28,80,142,122),(29,61,143,123),(30,62,144,124),(31,63,145,125),(32,64,146,126),(33,65,147,127),(34,66,148,128),(35,67,149,129),(36,68,150,130),(37,69,151,131),(38,70,152,132),(39,71,153,133),(40,72,154,134)], [(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),(21,25),(22,24),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(41,51),(42,50),(43,49),(44,48),(45,47),(52,60),(53,59),(54,58),(55,57),(61,69),(62,68),(63,67),(64,66),(70,80),(71,79),(72,78),(73,77),(74,76),(81,99),(82,98),(83,97),(84,96),(85,95),(86,94),(87,93),(88,92),(89,91),(101,107),(102,106),(103,105),(108,120),(109,119),(110,118),(111,117),(112,116),(113,115),(121,133),(122,132),(123,131),(124,130),(125,129),(126,128),(134,140),(135,139),(136,138),(141,153),(142,152),(143,151),(144,150),(145,149),(146,148),(154,160),(155,159),(156,158)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 40 |
| 0 | 0 | 0 | 0 | 40 | 11 |
| 37 | 5 | 0 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 36 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 23 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,40,0,0,0,0,40,11],[37,13,0,0,0,0,5,4,0,0,0,0,0,0,6,36,0,0,0,0,40,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,23,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | + | - | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D5 | D10 | D10 | D10 | D20 | 2- (1+4) | Q8×D5 | D4.10D10 |
| kernel | Q8×D20 | C20⋊2Q8 | C4×D20 | D10⋊2Q8 | Q8×C20 | C2×Q8×D5 | D20 | C5×Q8 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | Q8 | C10 | C4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 4 | 4 | 2 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
Q_8\times D_{20} % in TeX
G:=Group("Q8xD20"); // GroupNames label
G:=SmallGroup(320,1247);
// by ID
G=gap.SmallGroup(320,1247);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,675,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^20=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀