direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D10.Q8, (C8×D5).4C4, C8.24(C2×F5), (C2×C8).11F5, (C2×C40).11C4, C40.25(C2×C4), (C4×D5).85D4, C4.15(C4⋊F5), C20.22(C4⋊C4), (C4×D5).27Q8, D10.13(C2×Q8), C10⋊2(C8.C4), D10.31(C4⋊C4), C4.39(C22×F5), C4.F5.8C22, C20.79(C22×C4), Dic5.32(C2×D4), (C4×D5).79C23, (C8×D5).55C22, C22.25(C4⋊F5), (C22×D5).19Q8, Dic5.32(C4⋊C4), (C2×Dic5).176D4, C5⋊2(C2×C8.C4), (D5×C2×C8).21C2, C2.18(C2×C4⋊F5), C10.15(C2×C4⋊C4), (C2×C5⋊2C8).25C4, C5⋊2C8.49(C2×C4), (C4×D5).88(C2×C4), (C2×C4).140(C2×F5), (C2×C10).23(C4⋊C4), (C2×C4.F5).10C2, (C2×C20).129(C2×C4), (C2×C4×D5).397C22, SmallGroup(320,1061)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D10.Q8
G = < a,b,c,d,e | a2=b10=c2=1, d4=b5, e2=b-1cd2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b3, cd=dc, ece-1=b7c, ede-1=b5d3 >
Subgroups: 346 in 106 conjugacy classes, 52 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C8.C4, C22×C8, C2×M4(2), C5⋊2C8, C40, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C8.C4, C8×D5, C2×C5⋊2C8, C2×C40, C4.F5, C4.F5, C2×C5⋊C8, C2×C4×D5, D10.Q8, D5×C2×C8, C2×C4.F5, C2×D10.Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, F5, C8.C4, C2×C4⋊C4, C2×F5, C2×C8.C4, C4⋊F5, C22×F5, D10.Q8, C2×C4⋊F5, C2×D10.Q8
►On 160 points
Generators in S160
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 31)(10 32)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 21)(41 77)(42 78)(43 79)(44 80)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 61)(81 119)(82 120)(83 111)(84 112)(85 113)(86 114)(87 115)(88 116)(89 117)(90 118)(91 104)(92 105)(93 106)(94 107)(95 108)(96 109)(97 110)(98 101)(99 102)(100 103)(121 159)(122 160)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 144)(132 145)(133 146)(134 147)(135 148)(136 149)(137 150)(138 141)(139 142)(140 143)
(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 32)(2 31)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 23)(12 22)(13 21)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 24)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 72)(48 71)(49 80)(50 79)(51 63)(52 62)(53 61)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)(81 113)(82 112)(83 111)(84 120)(85 119)(86 118)(87 117)(88 116)(89 115)(90 114)(91 108)(92 107)(93 106)(94 105)(95 104)(96 103)(97 102)(98 101)(99 110)(100 109)(121 153)(122 152)(123 151)(124 160)(125 159)(126 158)(127 157)(128 156)(129 155)(130 154)(131 148)(132 147)(133 146)(134 145)(135 144)(136 143)(137 142)(138 141)(139 150)(140 149)
(1 57 17 42 6 52 12 47)(2 58 18 43 7 53 13 48)(3 59 19 44 8 54 14 49)(4 60 20 45 9 55 15 50)(5 51 11 46 10 56 16 41)(21 71 31 66 26 76 36 61)(22 72 32 67 27 77 37 62)(23 73 33 68 28 78 38 63)(24 74 34 69 29 79 39 64)(25 75 35 70 30 80 40 65)(81 121 96 136 86 126 91 131)(82 122 97 137 87 127 92 132)(83 123 98 138 88 128 93 133)(84 124 99 139 89 129 94 134)(85 125 100 140 90 130 95 135)(101 141 116 156 106 146 111 151)(102 142 117 157 107 147 112 152)(103 143 118 158 108 148 113 153)(104 144 119 159 109 149 114 154)(105 145 120 160 110 150 115 155)
(1 104 28 86 6 109 23 81)(2 101 27 89 7 106 22 84)(3 108 26 82 8 103 21 87)(4 105 25 85 9 110 30 90)(5 102 24 88 10 107 29 83)(11 112 34 98 16 117 39 93)(12 119 33 91 17 114 38 96)(13 116 32 94 18 111 37 99)(14 113 31 97 19 118 36 92)(15 120 40 100 20 115 35 95)(41 142 64 128 46 147 69 123)(42 149 63 121 47 144 68 126)(43 146 62 124 48 141 67 129)(44 143 61 127 49 148 66 122)(45 150 70 130 50 145 65 125)(51 152 74 138 56 157 79 133)(52 159 73 131 57 154 78 136)(53 156 72 134 58 151 77 139)(54 153 71 137 59 158 76 132)(55 160 80 140 60 155 75 135)
G:=sub<Sym(160)| (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,31)(10,32)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,21)(41,77)(42,78)(43,79)(44,80)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,61)(81,119)(82,120)(83,111)(84,112)(85,113)(86,114)(87,115)(88,116)(89,117)(90,118)(91,104)(92,105)(93,106)(94,107)(95,108)(96,109)(97,110)(98,101)(99,102)(100,103)(121,159)(122,160)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,144)(132,145)(133,146)(134,147)(135,148)(136,149)(137,150)(138,141)(139,142)(140,143), (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), (1,32)(2,31)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,23)(12,22)(13,21)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,80)(50,79)(51,63)(52,62)(53,61)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(81,113)(82,112)(83,111)(84,120)(85,119)(86,118)(87,117)(88,116)(89,115)(90,114)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,110)(100,109)(121,153)(122,152)(123,151)(124,160)(125,159)(126,158)(127,157)(128,156)(129,155)(130,154)(131,148)(132,147)(133,146)(134,145)(135,144)(136,143)(137,142)(138,141)(139,150)(140,149), (1,57,17,42,6,52,12,47)(2,58,18,43,7,53,13,48)(3,59,19,44,8,54,14,49)(4,60,20,45,9,55,15,50)(5,51,11,46,10,56,16,41)(21,71,31,66,26,76,36,61)(22,72,32,67,27,77,37,62)(23,73,33,68,28,78,38,63)(24,74,34,69,29,79,39,64)(25,75,35,70,30,80,40,65)(81,121,96,136,86,126,91,131)(82,122,97,137,87,127,92,132)(83,123,98,138,88,128,93,133)(84,124,99,139,89,129,94,134)(85,125,100,140,90,130,95,135)(101,141,116,156,106,146,111,151)(102,142,117,157,107,147,112,152)(103,143,118,158,108,148,113,153)(104,144,119,159,109,149,114,154)(105,145,120,160,110,150,115,155), (1,104,28,86,6,109,23,81)(2,101,27,89,7,106,22,84)(3,108,26,82,8,103,21,87)(4,105,25,85,9,110,30,90)(5,102,24,88,10,107,29,83)(11,112,34,98,16,117,39,93)(12,119,33,91,17,114,38,96)(13,116,32,94,18,111,37,99)(14,113,31,97,19,118,36,92)(15,120,40,100,20,115,35,95)(41,142,64,128,46,147,69,123)(42,149,63,121,47,144,68,126)(43,146,62,124,48,141,67,129)(44,143,61,127,49,148,66,122)(45,150,70,130,50,145,65,125)(51,152,74,138,56,157,79,133)(52,159,73,131,57,154,78,136)(53,156,72,134,58,151,77,139)(54,153,71,137,59,158,76,132)(55,160,80,140,60,155,75,135)>;
G:=Group( (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,31)(10,32)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,21)(41,77)(42,78)(43,79)(44,80)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,61)(81,119)(82,120)(83,111)(84,112)(85,113)(86,114)(87,115)(88,116)(89,117)(90,118)(91,104)(92,105)(93,106)(94,107)(95,108)(96,109)(97,110)(98,101)(99,102)(100,103)(121,159)(122,160)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,144)(132,145)(133,146)(134,147)(135,148)(136,149)(137,150)(138,141)(139,142)(140,143), (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), (1,32)(2,31)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,23)(12,22)(13,21)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,80)(50,79)(51,63)(52,62)(53,61)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(81,113)(82,112)(83,111)(84,120)(85,119)(86,118)(87,117)(88,116)(89,115)(90,114)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,110)(100,109)(121,153)(122,152)(123,151)(124,160)(125,159)(126,158)(127,157)(128,156)(129,155)(130,154)(131,148)(132,147)(133,146)(134,145)(135,144)(136,143)(137,142)(138,141)(139,150)(140,149), (1,57,17,42,6,52,12,47)(2,58,18,43,7,53,13,48)(3,59,19,44,8,54,14,49)(4,60,20,45,9,55,15,50)(5,51,11,46,10,56,16,41)(21,71,31,66,26,76,36,61)(22,72,32,67,27,77,37,62)(23,73,33,68,28,78,38,63)(24,74,34,69,29,79,39,64)(25,75,35,70,30,80,40,65)(81,121,96,136,86,126,91,131)(82,122,97,137,87,127,92,132)(83,123,98,138,88,128,93,133)(84,124,99,139,89,129,94,134)(85,125,100,140,90,130,95,135)(101,141,116,156,106,146,111,151)(102,142,117,157,107,147,112,152)(103,143,118,158,108,148,113,153)(104,144,119,159,109,149,114,154)(105,145,120,160,110,150,115,155), (1,104,28,86,6,109,23,81)(2,101,27,89,7,106,22,84)(3,108,26,82,8,103,21,87)(4,105,25,85,9,110,30,90)(5,102,24,88,10,107,29,83)(11,112,34,98,16,117,39,93)(12,119,33,91,17,114,38,96)(13,116,32,94,18,111,37,99)(14,113,31,97,19,118,36,92)(15,120,40,100,20,115,35,95)(41,142,64,128,46,147,69,123)(42,149,63,121,47,144,68,126)(43,146,62,124,48,141,67,129)(44,143,61,127,49,148,66,122)(45,150,70,130,50,145,65,125)(51,152,74,138,56,157,79,133)(52,159,73,131,57,154,78,136)(53,156,72,134,58,151,77,139)(54,153,71,137,59,158,76,132)(55,160,80,140,60,155,75,135) );
G=PermutationGroup([[(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,31),(10,32),(11,22),(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,21),(41,77),(42,78),(43,79),(44,80),(45,71),(46,72),(47,73),(48,74),(49,75),(50,76),(51,62),(52,63),(53,64),(54,65),(55,66),(56,67),(57,68),(58,69),(59,70),(60,61),(81,119),(82,120),(83,111),(84,112),(85,113),(86,114),(87,115),(88,116),(89,117),(90,118),(91,104),(92,105),(93,106),(94,107),(95,108),(96,109),(97,110),(98,101),(99,102),(100,103),(121,159),(122,160),(123,151),(124,152),(125,153),(126,154),(127,155),(128,156),(129,157),(130,158),(131,144),(132,145),(133,146),(134,147),(135,148),(136,149),(137,150),(138,141),(139,142),(140,143)], [(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)], [(1,32),(2,31),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,33),(11,23),(12,22),(13,21),(14,30),(15,29),(16,28),(17,27),(18,26),(19,25),(20,24),(41,78),(42,77),(43,76),(44,75),(45,74),(46,73),(47,72),(48,71),(49,80),(50,79),(51,63),(52,62),(53,61),(54,70),(55,69),(56,68),(57,67),(58,66),(59,65),(60,64),(81,113),(82,112),(83,111),(84,120),(85,119),(86,118),(87,117),(88,116),(89,115),(90,114),(91,108),(92,107),(93,106),(94,105),(95,104),(96,103),(97,102),(98,101),(99,110),(100,109),(121,153),(122,152),(123,151),(124,160),(125,159),(126,158),(127,157),(128,156),(129,155),(130,154),(131,148),(132,147),(133,146),(134,145),(135,144),(136,143),(137,142),(138,141),(139,150),(140,149)], [(1,57,17,42,6,52,12,47),(2,58,18,43,7,53,13,48),(3,59,19,44,8,54,14,49),(4,60,20,45,9,55,15,50),(5,51,11,46,10,56,16,41),(21,71,31,66,26,76,36,61),(22,72,32,67,27,77,37,62),(23,73,33,68,28,78,38,63),(24,74,34,69,29,79,39,64),(25,75,35,70,30,80,40,65),(81,121,96,136,86,126,91,131),(82,122,97,137,87,127,92,132),(83,123,98,138,88,128,93,133),(84,124,99,139,89,129,94,134),(85,125,100,140,90,130,95,135),(101,141,116,156,106,146,111,151),(102,142,117,157,107,147,112,152),(103,143,118,158,108,148,113,153),(104,144,119,159,109,149,114,154),(105,145,120,160,110,150,115,155)], [(1,104,28,86,6,109,23,81),(2,101,27,89,7,106,22,84),(3,108,26,82,8,103,21,87),(4,105,25,85,9,110,30,90),(5,102,24,88,10,107,29,83),(11,112,34,98,16,117,39,93),(12,119,33,91,17,114,38,96),(13,116,32,94,18,111,37,99),(14,113,31,97,19,118,36,92),(15,120,40,100,20,115,35,95),(41,142,64,128,46,147,69,123),(42,149,63,121,47,144,68,126),(43,146,62,124,48,141,67,129),(44,143,61,127,49,148,66,122),(45,150,70,130,50,145,65,125),(51,152,74,138,56,157,79,133),(52,159,73,131,57,154,78,136),(53,156,72,134,58,151,77,139),(54,153,71,137,59,158,76,132),(55,160,80,140,60,155,75,135)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | ··· | 8P | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 5 | 5 | 5 | 5 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D4 | Q8 | C8.C4 | F5 | C2×F5 | C2×F5 | C4⋊F5 | C4⋊F5 | D10.Q8 |
| kernel | C2×D10.Q8 | D10.Q8 | D5×C2×C8 | C2×C4.F5 | C8×D5 | C2×C5⋊2C8 | C2×C40 | C4×D5 | C4×D5 | C2×Dic5 | C22×D5 | C10 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 2 | 2 | 8 |
Matrix representation of C2×D10.Q8 ►in 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 4 | 4 |
| 0 | 0 | 37 | 27 | 37 | 0 |
| 0 | 0 | 0 | 37 | 27 | 37 |
| 0 | 0 | 4 | 4 | 0 | 31 |
| 15 | 26 | 0 | 0 | 0 | 0 |
| 26 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 24 | 32 | 40 |
| 0 | 0 | 8 | 16 | 17 | 40 |
| 0 | 0 | 1 | 24 | 25 | 33 |
| 0 | 0 | 1 | 9 | 17 | 18 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,40,0,0,0,0,1,0,0,0,40,0,1,0,0,0,0,40,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,0,0,40,0,1,0,0,0,40,1,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,31,37,0,4,0,0,0,27,37,4,0,0,4,37,27,0,0,0,4,0,37,31],[15,26,0,0,0,0,26,26,0,0,0,0,0,0,23,8,1,1,0,0,24,16,24,9,0,0,32,17,25,17,0,0,40,40,33,18] >;
C2×D10.Q8 in GAP, Magma, Sage, TeX
C_2\times D_{10}.Q_8 % in TeX
G:=Group("C2xD10.Q8"); // GroupNames label
G:=SmallGroup(320,1061);
// by ID
G=gap.SmallGroup(320,1061);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,268,136,1684,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^10=c^2=1,d^4=b^5,e^2=b^-1*c*d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=b^3,c*d=d*c,e*c*e^-1=b^7*c,e*d*e^-1=b^5*d^3>;
// generators/relations