direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8.F5, Dic5.19C24, C5⋊C8.3C23, C10⋊2(C8○D4), D5⋊C8⋊7C22, (Q8×C10).9C4, Q8.11(C2×F5), (C2×Q8).10F5, C4.F5⋊9C22, D20.11(C2×C4), (C2×D20).14C4, Q8⋊2D5.2C4, C4.28(C22×F5), C2.12(C23×F5), C20.28(C22×C4), C10.11(C23×C4), (C4×D5).51C23, D10.4(C22×C4), C22.58(C22×F5), Q8⋊2D5.15C22, Dic5.47(C22×C4), (C2×Dic5).364C23, C5⋊2(C2×C8○D4), (C2×D5⋊C8)⋊6C2, (C2×C4.F5)⋊7C2, (C2×C4).92(C2×F5), (C2×C20).72(C2×C4), (C4×D5).33(C2×C4), (C2×C5⋊C8).45C22, (C5×Q8).11(C2×C4), (C2×C4×D5).218C22, (C2×Q8⋊2D5).13C2, (C22×D5).62(C2×C4), (C2×C10).101(C22×C4), SmallGroup(320,1597)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8.F5
G = < a,b,c,d,e | a2=b4=d5=1, c2=e4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >
Subgroups: 794 in 266 conjugacy classes, 140 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, D10, C2×C10, C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C5⋊C8, C4×D5, D20, C2×Dic5, C2×C20, C5×Q8, C22×D5, C2×C8○D4, D5⋊C8, C4.F5, C2×C5⋊C8, C2×C5⋊C8, C2×C4×D5, C2×D20, Q8⋊2D5, Q8×C10, C2×D5⋊C8, C2×C4.F5, Q8.F5, C2×Q8⋊2D5, C2×Q8.F5
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, F5, C8○D4, C23×C4, C2×F5, C2×C8○D4, C22×F5, Q8.F5, C23×F5, C2×Q8.F5
►On 160 points
Generators in S160
(1 111)(2 112)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 145)(18 146)(19 147)(20 148)(21 149)(22 150)(23 151)(24 152)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(49 159)(50 160)(51 153)(52 154)(53 155)(54 156)(55 157)(56 158)(57 125)(58 126)(59 127)(60 128)(61 121)(62 122)(63 123)(64 124)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 97)(72 98)(73 81)(74 82)(75 83)(76 84)(77 85)(78 86)(79 87)(80 88)(89 113)(90 114)(91 115)(92 116)(93 117)(94 118)(95 119)(96 120)(129 137)(130 138)(131 139)(132 140)(133 141)(134 142)(135 143)(136 144)
(1 109 5 105)(2 110 6 106)(3 111 7 107)(4 112 8 108)(9 119 13 115)(10 120 14 116)(11 113 15 117)(12 114 16 118)(17 67 21 71)(18 68 22 72)(19 69 23 65)(20 70 24 66)(25 135 29 131)(26 136 30 132)(27 129 31 133)(28 130 32 134)(33 137 37 141)(34 138 38 142)(35 139 39 143)(36 140 40 144)(41 95 45 91)(42 96 46 92)(43 89 47 93)(44 90 48 94)(49 153 53 157)(50 154 54 158)(51 155 55 159)(52 156 56 160)(57 87 61 83)(58 88 62 84)(59 81 63 85)(60 82 64 86)(73 123 77 127)(74 124 78 128)(75 125 79 121)(76 126 80 122)(97 145 101 149)(98 146 102 150)(99 147 103 151)(100 148 104 152)
(1 155 5 159)(2 156 6 160)(3 157 7 153)(4 158 8 154)(9 71 13 67)(10 72 14 68)(11 65 15 69)(12 66 16 70)(17 119 21 115)(18 120 22 116)(19 113 23 117)(20 114 24 118)(25 125 29 121)(26 126 30 122)(27 127 31 123)(28 128 32 124)(33 59 37 63)(34 60 38 64)(35 61 39 57)(36 62 40 58)(41 97 45 101)(42 98 46 102)(43 99 47 103)(44 100 48 104)(49 111 53 107)(50 112 54 108)(51 105 55 109)(52 106 56 110)(73 129 77 133)(74 130 78 134)(75 131 79 135)(76 132 80 136)(81 137 85 141)(82 138 86 142)(83 139 87 143)(84 140 88 144)(89 151 93 147)(90 152 94 148)(91 145 95 149)(92 146 96 150)
(1 97 35 133 19)(2 134 98 20 36)(3 21 135 37 99)(4 38 22 100 136)(5 101 39 129 23)(6 130 102 24 40)(7 17 131 33 103)(8 34 18 104 132)(9 125 85 93 49)(10 94 126 50 86)(11 51 95 87 127)(12 88 52 128 96)(13 121 81 89 53)(14 90 122 54 82)(15 55 91 83 123)(16 84 56 124 92)(25 137 151 107 67)(26 108 138 68 152)(27 69 109 145 139)(28 146 70 140 110)(29 141 147 111 71)(30 112 142 72 148)(31 65 105 149 143)(32 150 66 144 106)(41 57 77 117 159)(42 118 58 160 78)(43 153 119 79 59)(44 80 154 60 120)(45 61 73 113 155)(46 114 62 156 74)(47 157 115 75 63)(48 76 158 64 116)
(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)
G:=sub<Sym(160)| (1,111)(2,112)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,152)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(49,159)(50,160)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,125)(58,126)(59,127)(60,128)(61,121)(62,122)(63,123)(64,124)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,97)(72,98)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(89,113)(90,114)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144), (1,109,5,105)(2,110,6,106)(3,111,7,107)(4,112,8,108)(9,119,13,115)(10,120,14,116)(11,113,15,117)(12,114,16,118)(17,67,21,71)(18,68,22,72)(19,69,23,65)(20,70,24,66)(25,135,29,131)(26,136,30,132)(27,129,31,133)(28,130,32,134)(33,137,37,141)(34,138,38,142)(35,139,39,143)(36,140,40,144)(41,95,45,91)(42,96,46,92)(43,89,47,93)(44,90,48,94)(49,153,53,157)(50,154,54,158)(51,155,55,159)(52,156,56,160)(57,87,61,83)(58,88,62,84)(59,81,63,85)(60,82,64,86)(73,123,77,127)(74,124,78,128)(75,125,79,121)(76,126,80,122)(97,145,101,149)(98,146,102,150)(99,147,103,151)(100,148,104,152), (1,155,5,159)(2,156,6,160)(3,157,7,153)(4,158,8,154)(9,71,13,67)(10,72,14,68)(11,65,15,69)(12,66,16,70)(17,119,21,115)(18,120,22,116)(19,113,23,117)(20,114,24,118)(25,125,29,121)(26,126,30,122)(27,127,31,123)(28,128,32,124)(33,59,37,63)(34,60,38,64)(35,61,39,57)(36,62,40,58)(41,97,45,101)(42,98,46,102)(43,99,47,103)(44,100,48,104)(49,111,53,107)(50,112,54,108)(51,105,55,109)(52,106,56,110)(73,129,77,133)(74,130,78,134)(75,131,79,135)(76,132,80,136)(81,137,85,141)(82,138,86,142)(83,139,87,143)(84,140,88,144)(89,151,93,147)(90,152,94,148)(91,145,95,149)(92,146,96,150), (1,97,35,133,19)(2,134,98,20,36)(3,21,135,37,99)(4,38,22,100,136)(5,101,39,129,23)(6,130,102,24,40)(7,17,131,33,103)(8,34,18,104,132)(9,125,85,93,49)(10,94,126,50,86)(11,51,95,87,127)(12,88,52,128,96)(13,121,81,89,53)(14,90,122,54,82)(15,55,91,83,123)(16,84,56,124,92)(25,137,151,107,67)(26,108,138,68,152)(27,69,109,145,139)(28,146,70,140,110)(29,141,147,111,71)(30,112,142,72,148)(31,65,105,149,143)(32,150,66,144,106)(41,57,77,117,159)(42,118,58,160,78)(43,153,119,79,59)(44,80,154,60,120)(45,61,73,113,155)(46,114,62,156,74)(47,157,115,75,63)(48,76,158,64,116), (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)>;
G:=Group( (1,111)(2,112)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,152)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(49,159)(50,160)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,125)(58,126)(59,127)(60,128)(61,121)(62,122)(63,123)(64,124)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,97)(72,98)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(89,113)(90,114)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144), (1,109,5,105)(2,110,6,106)(3,111,7,107)(4,112,8,108)(9,119,13,115)(10,120,14,116)(11,113,15,117)(12,114,16,118)(17,67,21,71)(18,68,22,72)(19,69,23,65)(20,70,24,66)(25,135,29,131)(26,136,30,132)(27,129,31,133)(28,130,32,134)(33,137,37,141)(34,138,38,142)(35,139,39,143)(36,140,40,144)(41,95,45,91)(42,96,46,92)(43,89,47,93)(44,90,48,94)(49,153,53,157)(50,154,54,158)(51,155,55,159)(52,156,56,160)(57,87,61,83)(58,88,62,84)(59,81,63,85)(60,82,64,86)(73,123,77,127)(74,124,78,128)(75,125,79,121)(76,126,80,122)(97,145,101,149)(98,146,102,150)(99,147,103,151)(100,148,104,152), (1,155,5,159)(2,156,6,160)(3,157,7,153)(4,158,8,154)(9,71,13,67)(10,72,14,68)(11,65,15,69)(12,66,16,70)(17,119,21,115)(18,120,22,116)(19,113,23,117)(20,114,24,118)(25,125,29,121)(26,126,30,122)(27,127,31,123)(28,128,32,124)(33,59,37,63)(34,60,38,64)(35,61,39,57)(36,62,40,58)(41,97,45,101)(42,98,46,102)(43,99,47,103)(44,100,48,104)(49,111,53,107)(50,112,54,108)(51,105,55,109)(52,106,56,110)(73,129,77,133)(74,130,78,134)(75,131,79,135)(76,132,80,136)(81,137,85,141)(82,138,86,142)(83,139,87,143)(84,140,88,144)(89,151,93,147)(90,152,94,148)(91,145,95,149)(92,146,96,150), (1,97,35,133,19)(2,134,98,20,36)(3,21,135,37,99)(4,38,22,100,136)(5,101,39,129,23)(6,130,102,24,40)(7,17,131,33,103)(8,34,18,104,132)(9,125,85,93,49)(10,94,126,50,86)(11,51,95,87,127)(12,88,52,128,96)(13,121,81,89,53)(14,90,122,54,82)(15,55,91,83,123)(16,84,56,124,92)(25,137,151,107,67)(26,108,138,68,152)(27,69,109,145,139)(28,146,70,140,110)(29,141,147,111,71)(30,112,142,72,148)(31,65,105,149,143)(32,150,66,144,106)(41,57,77,117,159)(42,118,58,160,78)(43,153,119,79,59)(44,80,154,60,120)(45,61,73,113,155)(46,114,62,156,74)(47,157,115,75,63)(48,76,158,64,116), (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) );
G=PermutationGroup([[(1,111),(2,112),(3,105),(4,106),(5,107),(6,108),(7,109),(8,110),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,145),(18,146),(19,147),(20,148),(21,149),(22,150),(23,151),(24,152),(25,39),(26,40),(27,33),(28,34),(29,35),(30,36),(31,37),(32,38),(49,159),(50,160),(51,153),(52,154),(53,155),(54,156),(55,157),(56,158),(57,125),(58,126),(59,127),(60,128),(61,121),(62,122),(63,123),(64,124),(65,99),(66,100),(67,101),(68,102),(69,103),(70,104),(71,97),(72,98),(73,81),(74,82),(75,83),(76,84),(77,85),(78,86),(79,87),(80,88),(89,113),(90,114),(91,115),(92,116),(93,117),(94,118),(95,119),(96,120),(129,137),(130,138),(131,139),(132,140),(133,141),(134,142),(135,143),(136,144)], [(1,109,5,105),(2,110,6,106),(3,111,7,107),(4,112,8,108),(9,119,13,115),(10,120,14,116),(11,113,15,117),(12,114,16,118),(17,67,21,71),(18,68,22,72),(19,69,23,65),(20,70,24,66),(25,135,29,131),(26,136,30,132),(27,129,31,133),(28,130,32,134),(33,137,37,141),(34,138,38,142),(35,139,39,143),(36,140,40,144),(41,95,45,91),(42,96,46,92),(43,89,47,93),(44,90,48,94),(49,153,53,157),(50,154,54,158),(51,155,55,159),(52,156,56,160),(57,87,61,83),(58,88,62,84),(59,81,63,85),(60,82,64,86),(73,123,77,127),(74,124,78,128),(75,125,79,121),(76,126,80,122),(97,145,101,149),(98,146,102,150),(99,147,103,151),(100,148,104,152)], [(1,155,5,159),(2,156,6,160),(3,157,7,153),(4,158,8,154),(9,71,13,67),(10,72,14,68),(11,65,15,69),(12,66,16,70),(17,119,21,115),(18,120,22,116),(19,113,23,117),(20,114,24,118),(25,125,29,121),(26,126,30,122),(27,127,31,123),(28,128,32,124),(33,59,37,63),(34,60,38,64),(35,61,39,57),(36,62,40,58),(41,97,45,101),(42,98,46,102),(43,99,47,103),(44,100,48,104),(49,111,53,107),(50,112,54,108),(51,105,55,109),(52,106,56,110),(73,129,77,133),(74,130,78,134),(75,131,79,135),(76,132,80,136),(81,137,85,141),(82,138,86,142),(83,139,87,143),(84,140,88,144),(89,151,93,147),(90,152,94,148),(91,145,95,149),(92,146,96,150)], [(1,97,35,133,19),(2,134,98,20,36),(3,21,135,37,99),(4,38,22,100,136),(5,101,39,129,23),(6,130,102,24,40),(7,17,131,33,103),(8,34,18,104,132),(9,125,85,93,49),(10,94,126,50,86),(11,51,95,87,127),(12,88,52,128,96),(13,121,81,89,53),(14,90,122,54,82),(15,55,91,83,123),(16,84,56,124,92),(25,137,151,107,67),(26,108,138,68,152),(27,69,109,145,139),(28,146,70,140,110),(29,141,147,111,71),(30,112,142,72,148),(31,65,105,149,143),(32,150,66,144,106),(41,57,77,117,159),(42,118,58,160,78),(43,153,119,79,59),(44,80,154,60,120),(45,61,73,113,155),(46,114,62,156,74),(47,157,115,75,63),(48,76,158,64,116)], [(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)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5 | 8A | ··· | 8H | 8I | ··· | 8T | 10A | 10B | 10C | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | ··· | 10 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 4 | 4 | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8○D4 | F5 | C2×F5 | C2×F5 | Q8.F5 |
| kernel | C2×Q8.F5 | C2×D5⋊C8 | C2×C4.F5 | Q8.F5 | C2×Q8⋊2D5 | C2×D20 | Q8⋊2D5 | Q8×C10 | C10 | C2×Q8 | C2×C4 | Q8 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 6 | 8 | 2 | 8 | 1 | 3 | 4 | 2 |
Matrix representation of C2×Q8.F5 ►in GL8(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 0 | 22 | 22 |
| 0 | 0 | 0 | 0 | 22 | 22 | 0 | 38 |
| 0 | 0 | 0 | 0 | 19 | 16 | 19 | 0 |
| 0 | 0 | 0 | 0 | 3 | 25 | 25 | 3 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[32,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,32,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,38,22,19,3,0,0,0,0,0,22,16,25,0,0,0,0,22,0,19,25,0,0,0,0,22,38,0,3] >;
C2×Q8.F5 in GAP, Magma, Sage, TeX
C_2\times Q_8.F_5
% in TeX
G:=Group("C2xQ8.F5"); // GroupNames label
G:=SmallGroup(320,1597);
// by ID
G=gap.SmallGroup(320,1597);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,297,136,102,6278,818]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^5=1,c^2=e^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations