metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (Q8×D5)⋊2C4, (C4×D5).5D4, Q8.9(C4×D5), C4.163(D4×D5), C4⋊C4.148D10, Q8⋊C4⋊16D5, (C2×C8).175D10, Q8⋊Dic5⋊4C2, C20.117(C2×D4), C22.77(D4×D5), C10.Q16⋊10C2, C20.48(C22×C4), (C2×Q8).104D10, C20.44D4⋊23C2, (C2×C40).197C22, (C2×C20).242C23, C2.2(Q16⋊D5), Dic10.19(C2×C4), (C2×Dic5).208D4, C2.4(SD16⋊D5), (C22×D5).111D4, C5⋊2(C23.38D4), C4⋊Dic5.90C22, (Q8×C10).25C22, D10.37(C22⋊C4), C10.34(C8.C22), Dic5.26(C22⋊C4), (C2×Dic10).73C22, C4.13(C2×C4×D5), (C2×Q8×D5).3C2, (C4×D5).3(C2×C4), C4⋊C4⋊7D5.1C2, (C5×Q8).18(C2×C4), C2.22(D5×C22⋊C4), (C2×C4×D5).21C22, (C5×Q8⋊C4)⋊19C2, (C2×C10).255(C2×D4), (C2×C8⋊D5).10C2, (C5×C4⋊C4).43C22, C10.62(C2×C22⋊C4), (C2×C5⋊2C8).35C22, (C2×C4).349(C22×D5), SmallGroup(320,429)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (Q8×D5)⋊C4
G = < a,b,c,d,e | a4=c5=d2=e4=1, b2=a2, bab-1=eae-1=a-1, ac=ca, ad=da, bc=cb, bd=db, ebe-1=ab, dcd=c-1, ce=ec, ede-1=a2d >
Subgroups: 558 in 150 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, Q8⋊C4, Q8⋊C4, C42⋊C2, C2×M4(2), C22×Q8, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C23.38D4, C8⋊D5, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5, Q8×D5, Q8×C10, C10.Q16, C20.44D4, Q8⋊Dic5, C5×Q8⋊C4, C4⋊C4⋊7D5, C2×C8⋊D5, C2×Q8×D5, (Q8×D5)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C8.C22, C4×D5, C22×D5, C23.38D4, C2×C4×D5, D4×D5, D5×C22⋊C4, SD16⋊D5, Q16⋊D5, (Q8×D5)⋊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 51 46 56)(42 52 47 57)(43 53 48 58)(44 54 49 59)(45 55 50 60)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)(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)(121 136 126 131)(122 137 127 132)(123 138 128 133)(124 139 129 134)(125 140 130 135)(141 156 146 151)(142 157 147 152)(143 158 148 153)(144 159 149 154)(145 160 150 155)
(1 86 6 81)(2 87 7 82)(3 88 8 83)(4 89 9 84)(5 90 10 85)(11 96 16 91)(12 97 17 92)(13 98 18 93)(14 99 19 94)(15 100 20 95)(21 106 26 101)(22 107 27 102)(23 108 28 103)(24 109 29 104)(25 110 30 105)(31 116 36 111)(32 117 37 112)(33 118 38 113)(34 119 39 114)(35 120 40 115)(41 126 46 121)(42 127 47 122)(43 128 48 123)(44 129 49 124)(45 130 50 125)(51 136 56 131)(52 137 57 132)(53 138 58 133)(54 139 59 134)(55 140 60 135)(61 146 66 141)(62 147 67 142)(63 148 68 143)(64 149 69 144)(65 150 70 145)(71 156 76 151)(72 157 77 152)(73 158 78 153)(74 159 79 154)(75 160 80 155)
(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 20)(12 19)(13 18)(14 17)(15 16)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 45)(42 44)(46 50)(47 49)(51 55)(52 54)(56 60)(57 59)(61 65)(62 64)(66 70)(67 69)(71 75)(72 74)(76 80)(77 79)(81 90)(82 89)(83 88)(84 87)(85 86)(91 100)(92 99)(93 98)(94 97)(95 96)(101 110)(102 109)(103 108)(104 107)(105 106)(111 120)(112 119)(113 118)(114 117)(115 116)(121 125)(122 124)(126 130)(127 129)(131 135)(132 134)(136 140)(137 139)(141 145)(142 144)(146 150)(147 149)(151 155)(152 154)(156 160)(157 159)
(1 61 21 41)(2 62 22 42)(3 63 23 43)(4 64 24 44)(5 65 25 45)(6 66 26 46)(7 67 27 47)(8 68 28 48)(9 69 29 49)(10 70 30 50)(11 71 31 51)(12 72 32 52)(13 73 33 53)(14 74 34 54)(15 75 35 55)(16 76 36 56)(17 77 37 57)(18 78 38 58)(19 79 39 59)(20 80 40 60)(81 151 101 131)(82 152 102 132)(83 153 103 133)(84 154 104 134)(85 155 105 135)(86 156 106 136)(87 157 107 137)(88 158 108 138)(89 159 109 139)(90 160 110 140)(91 146 111 126)(92 147 112 127)(93 148 113 128)(94 149 114 129)(95 150 115 130)(96 141 116 121)(97 142 117 122)(98 143 118 123)(99 144 119 124)(100 145 120 125)
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)(41,51,46,56)(42,52,47,57)(43,53,48,58)(44,54,49,59)(45,55,50,60)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80)(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)(121,136,126,131)(122,137,127,132)(123,138,128,133)(124,139,129,134)(125,140,130,135)(141,156,146,151)(142,157,147,152)(143,158,148,153)(144,159,149,154)(145,160,150,155), (1,86,6,81)(2,87,7,82)(3,88,8,83)(4,89,9,84)(5,90,10,85)(11,96,16,91)(12,97,17,92)(13,98,18,93)(14,99,19,94)(15,100,20,95)(21,106,26,101)(22,107,27,102)(23,108,28,103)(24,109,29,104)(25,110,30,105)(31,116,36,111)(32,117,37,112)(33,118,38,113)(34,119,39,114)(35,120,40,115)(41,126,46,121)(42,127,47,122)(43,128,48,123)(44,129,49,124)(45,130,50,125)(51,136,56,131)(52,137,57,132)(53,138,58,133)(54,139,59,134)(55,140,60,135)(61,146,66,141)(62,147,67,142)(63,148,68,143)(64,149,69,144)(65,150,70,145)(71,156,76,151)(72,157,77,152)(73,158,78,153)(74,159,79,154)(75,160,80,155), (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)(11,20)(12,19)(13,18)(14,17)(15,16)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,45)(42,44)(46,50)(47,49)(51,55)(52,54)(56,60)(57,59)(61,65)(62,64)(66,70)(67,69)(71,75)(72,74)(76,80)(77,79)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(101,110)(102,109)(103,108)(104,107)(105,106)(111,120)(112,119)(113,118)(114,117)(115,116)(121,125)(122,124)(126,130)(127,129)(131,135)(132,134)(136,140)(137,139)(141,145)(142,144)(146,150)(147,149)(151,155)(152,154)(156,160)(157,159), (1,61,21,41)(2,62,22,42)(3,63,23,43)(4,64,24,44)(5,65,25,45)(6,66,26,46)(7,67,27,47)(8,68,28,48)(9,69,29,49)(10,70,30,50)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60)(81,151,101,131)(82,152,102,132)(83,153,103,133)(84,154,104,134)(85,155,105,135)(86,156,106,136)(87,157,107,137)(88,158,108,138)(89,159,109,139)(90,160,110,140)(91,146,111,126)(92,147,112,127)(93,148,113,128)(94,149,114,129)(95,150,115,130)(96,141,116,121)(97,142,117,122)(98,143,118,123)(99,144,119,124)(100,145,120,125)>;
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)(41,51,46,56)(42,52,47,57)(43,53,48,58)(44,54,49,59)(45,55,50,60)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80)(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)(121,136,126,131)(122,137,127,132)(123,138,128,133)(124,139,129,134)(125,140,130,135)(141,156,146,151)(142,157,147,152)(143,158,148,153)(144,159,149,154)(145,160,150,155), (1,86,6,81)(2,87,7,82)(3,88,8,83)(4,89,9,84)(5,90,10,85)(11,96,16,91)(12,97,17,92)(13,98,18,93)(14,99,19,94)(15,100,20,95)(21,106,26,101)(22,107,27,102)(23,108,28,103)(24,109,29,104)(25,110,30,105)(31,116,36,111)(32,117,37,112)(33,118,38,113)(34,119,39,114)(35,120,40,115)(41,126,46,121)(42,127,47,122)(43,128,48,123)(44,129,49,124)(45,130,50,125)(51,136,56,131)(52,137,57,132)(53,138,58,133)(54,139,59,134)(55,140,60,135)(61,146,66,141)(62,147,67,142)(63,148,68,143)(64,149,69,144)(65,150,70,145)(71,156,76,151)(72,157,77,152)(73,158,78,153)(74,159,79,154)(75,160,80,155), (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)(11,20)(12,19)(13,18)(14,17)(15,16)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,45)(42,44)(46,50)(47,49)(51,55)(52,54)(56,60)(57,59)(61,65)(62,64)(66,70)(67,69)(71,75)(72,74)(76,80)(77,79)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(101,110)(102,109)(103,108)(104,107)(105,106)(111,120)(112,119)(113,118)(114,117)(115,116)(121,125)(122,124)(126,130)(127,129)(131,135)(132,134)(136,140)(137,139)(141,145)(142,144)(146,150)(147,149)(151,155)(152,154)(156,160)(157,159), (1,61,21,41)(2,62,22,42)(3,63,23,43)(4,64,24,44)(5,65,25,45)(6,66,26,46)(7,67,27,47)(8,68,28,48)(9,69,29,49)(10,70,30,50)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60)(81,151,101,131)(82,152,102,132)(83,153,103,133)(84,154,104,134)(85,155,105,135)(86,156,106,136)(87,157,107,137)(88,158,108,138)(89,159,109,139)(90,160,110,140)(91,146,111,126)(92,147,112,127)(93,148,113,128)(94,149,114,129)(95,150,115,130)(96,141,116,121)(97,142,117,122)(98,143,118,123)(99,144,119,124)(100,145,120,125) );
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),(41,51,46,56),(42,52,47,57),(43,53,48,58),(44,54,49,59),(45,55,50,60),(61,71,66,76),(62,72,67,77),(63,73,68,78),(64,74,69,79),(65,75,70,80),(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),(121,136,126,131),(122,137,127,132),(123,138,128,133),(124,139,129,134),(125,140,130,135),(141,156,146,151),(142,157,147,152),(143,158,148,153),(144,159,149,154),(145,160,150,155)], [(1,86,6,81),(2,87,7,82),(3,88,8,83),(4,89,9,84),(5,90,10,85),(11,96,16,91),(12,97,17,92),(13,98,18,93),(14,99,19,94),(15,100,20,95),(21,106,26,101),(22,107,27,102),(23,108,28,103),(24,109,29,104),(25,110,30,105),(31,116,36,111),(32,117,37,112),(33,118,38,113),(34,119,39,114),(35,120,40,115),(41,126,46,121),(42,127,47,122),(43,128,48,123),(44,129,49,124),(45,130,50,125),(51,136,56,131),(52,137,57,132),(53,138,58,133),(54,139,59,134),(55,140,60,135),(61,146,66,141),(62,147,67,142),(63,148,68,143),(64,149,69,144),(65,150,70,145),(71,156,76,151),(72,157,77,152),(73,158,78,153),(74,159,79,154),(75,160,80,155)], [(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),(11,20),(12,19),(13,18),(14,17),(15,16),(21,30),(22,29),(23,28),(24,27),(25,26),(31,40),(32,39),(33,38),(34,37),(35,36),(41,45),(42,44),(46,50),(47,49),(51,55),(52,54),(56,60),(57,59),(61,65),(62,64),(66,70),(67,69),(71,75),(72,74),(76,80),(77,79),(81,90),(82,89),(83,88),(84,87),(85,86),(91,100),(92,99),(93,98),(94,97),(95,96),(101,110),(102,109),(103,108),(104,107),(105,106),(111,120),(112,119),(113,118),(114,117),(115,116),(121,125),(122,124),(126,130),(127,129),(131,135),(132,134),(136,140),(137,139),(141,145),(142,144),(146,150),(147,149),(151,155),(152,154),(156,160),(157,159)], [(1,61,21,41),(2,62,22,42),(3,63,23,43),(4,64,24,44),(5,65,25,45),(6,66,26,46),(7,67,27,47),(8,68,28,48),(9,69,29,49),(10,70,30,50),(11,71,31,51),(12,72,32,52),(13,73,33,53),(14,74,34,54),(15,75,35,55),(16,76,36,56),(17,77,37,57),(18,78,38,58),(19,79,39,59),(20,80,40,60),(81,151,101,131),(82,152,102,132),(83,153,103,133),(84,154,104,134),(85,155,105,135),(86,156,106,136),(87,157,107,137),(88,158,108,138),(89,159,109,139),(90,160,110,140),(91,146,111,126),(92,147,112,127),(93,148,113,128),(94,149,114,129),(95,150,115,130),(96,141,116,121),(97,142,117,122),(98,143,118,123),(99,144,119,124),(100,145,120,125)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C4×D5 | C8.C22 | D4×D5 | D4×D5 | SD16⋊D5 | Q16⋊D5 |
| kernel | (Q8×D5)⋊C4 | C10.Q16 | C20.44D4 | Q8⋊Dic5 | C5×Q8⋊C4 | C4⋊C4⋊7D5 | C2×C8⋊D5 | C2×Q8×D5 | Q8×D5 | C4×D5 | C2×Dic5 | C22×D5 | Q8⋊C4 | C4⋊C4 | C2×C8 | C2×Q8 | Q8 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of (Q8×D5)⋊C4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 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 | 34 | 27 | 0 | 0 |
| 0 | 0 | 27 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 34 |
| 0 | 0 | 0 | 0 | 34 | 27 |
| 7 | 40 | 0 | 0 | 0 | 0 |
| 8 | 40 | 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 | 1 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,27,0,0,0,0,27,7,0,0,0,0,0,0,14,34,0,0,0,0,34,27],[7,8,0,0,0,0,40,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,7,0,0,0,0,6,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
(Q8×D5)⋊C4 in GAP, Magma, Sage, TeX
(Q_8\times D_5)\rtimes C_4
% in TeX
G:=Group("(Q8xD5):C4"); // GroupNames label
G:=SmallGroup(320,429);
// by ID
G=gap.SmallGroup(320,429);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,58,851,438,102,12550]);
// 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=e*a*e^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,e*b*e^-1=a*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=a^2*d>;
// generators/relations