metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊2D5⋊2C4, (C4×D5).96D4, C4.165(D4×D5), Q8.10(C4×D5), C4⋊C4.150D10, Q8⋊C4⋊22D5, D20.17(C2×C4), C20.119(C2×D4), Q8⋊Dic5⋊6C2, (C2×C8).209D10, D20⋊5C4⋊23C2, D20⋊6C4⋊10C2, C22.79(D4×D5), C10.47(C4○D8), C20.50(C22×C4), (C2×Q8).106D10, (C22×D5).80D4, C2.2(Q8.D10), (C2×C40).199C22, (C2×C20).244C23, (C2×Dic5).272D4, (C2×D20).65C22, C5⋊3(C23.24D4), C4⋊Dic5.92C22, (Q8×C10).27C22, D10.26(C22⋊C4), C2.4(SD16⋊3D5), Dic5.57(C22⋊C4), (D5×C2×C8)⋊20C2, C4.15(C2×C4×D5), C4⋊C4⋊7D5⋊5C2, (C4×D5).51(C2×C4), (C5×Q8).19(C2×C4), C2.24(D5×C22⋊C4), (C5×Q8⋊C4)⋊21C2, (C2×C10).257(C2×D4), (C5×C4⋊C4).45C22, C10.64(C2×C22⋊C4), (C2×Q8⋊2D5).3C2, (C2×C4×D5).297C22, (C2×C4).351(C22×D5), (C2×C5⋊2C8).227C22, SmallGroup(320,431)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊2D5⋊C4
G = < a,b,c,d,e | a4=c5=d2=e4=1, b2=a2, bab-1=dad=eae-1=a-1, ac=ca, bc=cb, bd=db, ebe-1=a-1b, dcd=c-1, ce=ec, ede-1=a-1d >
Subgroups: 654 in 158 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, D4⋊C4, Q8⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C5⋊2C8, C40, C4×D5, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C22×D5, C23.24D4, C8×D5, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q8⋊2D5, Q8⋊2D5, Q8×C10, D20⋊6C4, D20⋊5C4, Q8⋊Dic5, C5×Q8⋊C4, C4⋊C4⋊7D5, D5×C2×C8, C2×Q8⋊2D5, Q8⋊2D5⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C4○D8, C4×D5, C22×D5, C23.24D4, C2×C4×D5, D4×D5, D5×C22⋊C4, SD16⋊3D5, Q8.D10, Q8⋊2D5⋊C4
►On 160 points
Generators in S160
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)(121 131 126 136)(122 132 127 137)(123 133 128 138)(124 134 129 139)(125 135 130 140)(141 151 146 156)(142 152 147 157)(143 153 148 158)(144 154 149 159)(145 155 150 160)
(1 91 6 96)(2 92 7 97)(3 93 8 98)(4 94 9 99)(5 95 10 100)(11 86 16 81)(12 87 17 82)(13 88 18 83)(14 89 19 84)(15 90 20 85)(21 111 26 116)(22 112 27 117)(23 113 28 118)(24 114 29 119)(25 115 30 120)(31 106 36 101)(32 107 37 102)(33 108 38 103)(34 109 39 104)(35 110 40 105)(41 131 46 136)(42 132 47 137)(43 133 48 138)(44 134 49 139)(45 135 50 140)(51 126 56 121)(52 127 57 122)(53 128 58 123)(54 129 59 124)(55 130 60 125)(61 151 66 156)(62 152 67 157)(63 153 68 158)(64 154 69 159)(65 155 70 160)(71 146 76 141)(72 147 77 142)(73 148 78 143)(74 149 79 144)(75 150 80 145)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 15)(12 14)(16 20)(17 19)(21 25)(22 24)(26 30)(27 29)(31 40)(32 39)(33 38)(34 37)(35 36)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 75)(62 74)(63 73)(64 72)(65 71)(66 80)(67 79)(68 78)(69 77)(70 76)(81 85)(82 84)(86 90)(87 89)(91 100)(92 99)(93 98)(94 97)(95 96)(101 110)(102 109)(103 108)(104 107)(105 106)(111 115)(112 114)(116 120)(117 119)(121 135)(122 134)(123 133)(124 132)(125 131)(126 140)(127 139)(128 138)(129 137)(130 136)(141 160)(142 159)(143 158)(144 157)(145 156)(146 155)(147 154)(148 153)(149 152)(150 151)
(1 156 36 121)(2 157 37 122)(3 158 38 123)(4 159 39 124)(5 160 40 125)(6 151 31 126)(7 152 32 127)(8 153 33 128)(9 154 34 129)(10 155 35 130)(11 141 21 131)(12 142 22 132)(13 143 23 133)(14 144 24 134)(15 145 25 135)(16 146 26 136)(17 147 27 137)(18 148 28 138)(19 149 29 139)(20 150 30 140)(41 91 76 101)(42 92 77 102)(43 93 78 103)(44 94 79 104)(45 95 80 105)(46 96 71 106)(47 97 72 107)(48 98 73 108)(49 99 74 109)(50 100 75 110)(51 86 61 111)(52 87 62 112)(53 88 63 113)(54 89 64 114)(55 90 65 115)(56 81 66 116)(57 82 67 117)(58 83 68 118)(59 84 69 119)(60 85 70 120)
G:=sub<Sym(160)| (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75)(81,91,86,96)(82,92,87,97)(83,93,88,98)(84,94,89,99)(85,95,90,100)(101,111,106,116)(102,112,107,117)(103,113,108,118)(104,114,109,119)(105,115,110,120)(121,131,126,136)(122,132,127,137)(123,133,128,138)(124,134,129,139)(125,135,130,140)(141,151,146,156)(142,152,147,157)(143,153,148,158)(144,154,149,159)(145,155,150,160), (1,91,6,96)(2,92,7,97)(3,93,8,98)(4,94,9,99)(5,95,10,100)(11,86,16,81)(12,87,17,82)(13,88,18,83)(14,89,19,84)(15,90,20,85)(21,111,26,116)(22,112,27,117)(23,113,28,118)(24,114,29,119)(25,115,30,120)(31,106,36,101)(32,107,37,102)(33,108,38,103)(34,109,39,104)(35,110,40,105)(41,131,46,136)(42,132,47,137)(43,133,48,138)(44,134,49,139)(45,135,50,140)(51,126,56,121)(52,127,57,122)(53,128,58,123)(54,129,59,124)(55,130,60,125)(61,151,66,156)(62,152,67,157)(63,153,68,158)(64,154,69,159)(65,155,70,160)(71,146,76,141)(72,147,77,142)(73,148,78,143)(74,149,79,144)(75,150,80,145), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,15)(12,14)(16,20)(17,19)(21,25)(22,24)(26,30)(27,29)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,75)(62,74)(63,73)(64,72)(65,71)(66,80)(67,79)(68,78)(69,77)(70,76)(81,85)(82,84)(86,90)(87,89)(91,100)(92,99)(93,98)(94,97)(95,96)(101,110)(102,109)(103,108)(104,107)(105,106)(111,115)(112,114)(116,120)(117,119)(121,135)(122,134)(123,133)(124,132)(125,131)(126,140)(127,139)(128,138)(129,137)(130,136)(141,160)(142,159)(143,158)(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151), (1,156,36,121)(2,157,37,122)(3,158,38,123)(4,159,39,124)(5,160,40,125)(6,151,31,126)(7,152,32,127)(8,153,33,128)(9,154,34,129)(10,155,35,130)(11,141,21,131)(12,142,22,132)(13,143,23,133)(14,144,24,134)(15,145,25,135)(16,146,26,136)(17,147,27,137)(18,148,28,138)(19,149,29,139)(20,150,30,140)(41,91,76,101)(42,92,77,102)(43,93,78,103)(44,94,79,104)(45,95,80,105)(46,96,71,106)(47,97,72,107)(48,98,73,108)(49,99,74,109)(50,100,75,110)(51,86,61,111)(52,87,62,112)(53,88,63,113)(54,89,64,114)(55,90,65,115)(56,81,66,116)(57,82,67,117)(58,83,68,118)(59,84,69,119)(60,85,70,120)>;
G:=Group( (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75)(81,91,86,96)(82,92,87,97)(83,93,88,98)(84,94,89,99)(85,95,90,100)(101,111,106,116)(102,112,107,117)(103,113,108,118)(104,114,109,119)(105,115,110,120)(121,131,126,136)(122,132,127,137)(123,133,128,138)(124,134,129,139)(125,135,130,140)(141,151,146,156)(142,152,147,157)(143,153,148,158)(144,154,149,159)(145,155,150,160), (1,91,6,96)(2,92,7,97)(3,93,8,98)(4,94,9,99)(5,95,10,100)(11,86,16,81)(12,87,17,82)(13,88,18,83)(14,89,19,84)(15,90,20,85)(21,111,26,116)(22,112,27,117)(23,113,28,118)(24,114,29,119)(25,115,30,120)(31,106,36,101)(32,107,37,102)(33,108,38,103)(34,109,39,104)(35,110,40,105)(41,131,46,136)(42,132,47,137)(43,133,48,138)(44,134,49,139)(45,135,50,140)(51,126,56,121)(52,127,57,122)(53,128,58,123)(54,129,59,124)(55,130,60,125)(61,151,66,156)(62,152,67,157)(63,153,68,158)(64,154,69,159)(65,155,70,160)(71,146,76,141)(72,147,77,142)(73,148,78,143)(74,149,79,144)(75,150,80,145), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,15)(12,14)(16,20)(17,19)(21,25)(22,24)(26,30)(27,29)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,75)(62,74)(63,73)(64,72)(65,71)(66,80)(67,79)(68,78)(69,77)(70,76)(81,85)(82,84)(86,90)(87,89)(91,100)(92,99)(93,98)(94,97)(95,96)(101,110)(102,109)(103,108)(104,107)(105,106)(111,115)(112,114)(116,120)(117,119)(121,135)(122,134)(123,133)(124,132)(125,131)(126,140)(127,139)(128,138)(129,137)(130,136)(141,160)(142,159)(143,158)(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151), (1,156,36,121)(2,157,37,122)(3,158,38,123)(4,159,39,124)(5,160,40,125)(6,151,31,126)(7,152,32,127)(8,153,33,128)(9,154,34,129)(10,155,35,130)(11,141,21,131)(12,142,22,132)(13,143,23,133)(14,144,24,134)(15,145,25,135)(16,146,26,136)(17,147,27,137)(18,148,28,138)(19,149,29,139)(20,150,30,140)(41,91,76,101)(42,92,77,102)(43,93,78,103)(44,94,79,104)(45,95,80,105)(46,96,71,106)(47,97,72,107)(48,98,73,108)(49,99,74,109)(50,100,75,110)(51,86,61,111)(52,87,62,112)(53,88,63,113)(54,89,64,114)(55,90,65,115)(56,81,66,116)(57,82,67,117)(58,83,68,118)(59,84,69,119)(60,85,70,120) );
G=PermutationGroup([[(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35),(41,56,46,51),(42,57,47,52),(43,58,48,53),(44,59,49,54),(45,60,50,55),(61,76,66,71),(62,77,67,72),(63,78,68,73),(64,79,69,74),(65,80,70,75),(81,91,86,96),(82,92,87,97),(83,93,88,98),(84,94,89,99),(85,95,90,100),(101,111,106,116),(102,112,107,117),(103,113,108,118),(104,114,109,119),(105,115,110,120),(121,131,126,136),(122,132,127,137),(123,133,128,138),(124,134,129,139),(125,135,130,140),(141,151,146,156),(142,152,147,157),(143,153,148,158),(144,154,149,159),(145,155,150,160)], [(1,91,6,96),(2,92,7,97),(3,93,8,98),(4,94,9,99),(5,95,10,100),(11,86,16,81),(12,87,17,82),(13,88,18,83),(14,89,19,84),(15,90,20,85),(21,111,26,116),(22,112,27,117),(23,113,28,118),(24,114,29,119),(25,115,30,120),(31,106,36,101),(32,107,37,102),(33,108,38,103),(34,109,39,104),(35,110,40,105),(41,131,46,136),(42,132,47,137),(43,133,48,138),(44,134,49,139),(45,135,50,140),(51,126,56,121),(52,127,57,122),(53,128,58,123),(54,129,59,124),(55,130,60,125),(61,151,66,156),(62,152,67,157),(63,153,68,158),(64,154,69,159),(65,155,70,160),(71,146,76,141),(72,147,77,142),(73,148,78,143),(74,149,79,144),(75,150,80,145)], [(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,15),(12,14),(16,20),(17,19),(21,25),(22,24),(26,30),(27,29),(31,40),(32,39),(33,38),(34,37),(35,36),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(61,75),(62,74),(63,73),(64,72),(65,71),(66,80),(67,79),(68,78),(69,77),(70,76),(81,85),(82,84),(86,90),(87,89),(91,100),(92,99),(93,98),(94,97),(95,96),(101,110),(102,109),(103,108),(104,107),(105,106),(111,115),(112,114),(116,120),(117,119),(121,135),(122,134),(123,133),(124,132),(125,131),(126,140),(127,139),(128,138),(129,137),(130,136),(141,160),(142,159),(143,158),(144,157),(145,156),(146,155),(147,154),(148,153),(149,152),(150,151)], [(1,156,36,121),(2,157,37,122),(3,158,38,123),(4,159,39,124),(5,160,40,125),(6,151,31,126),(7,152,32,127),(8,153,33,128),(9,154,34,129),(10,155,35,130),(11,141,21,131),(12,142,22,132),(13,143,23,133),(14,144,24,134),(15,145,25,135),(16,146,26,136),(17,147,27,137),(18,148,28,138),(19,149,29,139),(20,150,30,140),(41,91,76,101),(42,92,77,102),(43,93,78,103),(44,94,79,104),(45,95,80,105),(46,96,71,106),(47,97,72,107),(48,98,73,108),(49,99,74,109),(50,100,75,110),(51,86,61,111),(52,87,62,112),(53,88,63,113),(54,89,64,114),(55,90,65,115),(56,81,66,116),(57,82,67,117),(58,83,68,118),(59,84,69,119),(60,85,70,120)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C4○D8 | C4×D5 | D4×D5 | D4×D5 | SD16⋊3D5 | Q8.D10 |
| kernel | Q8⋊2D5⋊C4 | D20⋊6C4 | D20⋊5C4 | Q8⋊Dic5 | C5×Q8⋊C4 | C4⋊C4⋊7D5 | D5×C2×C8 | C2×Q8⋊2D5 | Q8⋊2D5 | C4×D5 | C2×Dic5 | C22×D5 | Q8⋊C4 | C4⋊C4 | C2×C8 | C2×Q8 | C10 | Q8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of Q8⋊2D5⋊C4 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 32 | 0 |
| 6 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 40 | 0 | 0 |
| 35 | 35 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 26 | 26 |
| 0 | 0 | 26 | 15 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,32,0,0,32,0],[6,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[6,35,0,0,40,35,0,0,0,0,0,1,0,0,1,0],[9,0,0,0,0,9,0,0,0,0,26,26,0,0,26,15] >;
Q8⋊2D5⋊C4 in GAP, Magma, Sage, TeX
Q_8\rtimes_2D_5\rtimes C_4
% in TeX
G:=Group("Q8:2D5:C4"); // GroupNames label
G:=SmallGroup(320,431);
// by ID
G=gap.SmallGroup(320,431);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,219,58,570,136,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^5=d^2=e^4=1,b^2=a^2,b*a*b^-1=d*a*d=e*a*e^-1=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,e*b*e^-1=a^-1*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=a^-1*d>;
// generators/relations