direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4⋊C4⋊D5, C4⋊C4⋊41D10, (C2×C10).56C24, C4⋊Dic5⋊54C22, C10⋊3(C42⋊2C2), (C2×C20).617C23, (C4×Dic5)⋊76C22, (C22×C4).181D10, C22.90(C23×D5), C22.78(C4○D20), C10.D4⋊69C22, (C2×Dic5).17C23, (C22×D5).14C23, (C23×D5).34C22, C23.331(C22×D5), D10⋊C4.95C22, C22.75(D4⋊2D5), (C22×C20).360C22, (C22×C10).405C23, C22.37(Q8⋊2D5), (C22×Dic5).83C22, (C2×C4⋊C4)⋊21D5, (C10×C4⋊C4)⋊18C2, C5⋊3(C2×C42⋊2C2), (C2×C4×Dic5)⋊33C2, (C5×C4⋊C4)⋊49C22, (C2×C4⋊Dic5)⋊22C2, C10.23(C2×C4○D4), C2.25(C2×C4○D20), C2.8(C2×Q8⋊2D5), C2.16(C2×D4⋊2D5), (C2×C10.D4)⋊45C2, (C2×C4).144(C22×D5), (C2×D10⋊C4).28C2, (C2×C10).108(C4○D4), SmallGroup(320,1184)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4⋊C4⋊D5
G = < a,b,c,d,e | a2=b4=c4=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=bc2, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 846 in 246 conjugacy classes, 111 normal (31 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊2C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C2×C42⋊2C2, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, C23×D5, C4⋊C4⋊D5, C2×C4×Dic5, C2×C10.D4, C2×C4⋊Dic5, C2×D10⋊C4, C10×C4⋊C4, C2×C4⋊C4⋊D5
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C42⋊2C2, C2×C4○D4, C22×D5, C2×C42⋊2C2, C4○D20, D4⋊2D5, Q8⋊2D5, C23×D5, C4⋊C4⋊D5, C2×C4○D20, C2×D4⋊2D5, C2×Q8⋊2D5, C2×C4⋊C4⋊D5
►On 160 points
Generators in S160
(1 86)(2 87)(3 88)(4 89)(5 90)(6 81)(7 82)(8 83)(9 84)(10 85)(11 96)(12 97)(13 98)(14 99)(15 100)(16 91)(17 92)(18 93)(19 94)(20 95)(21 106)(22 107)(23 108)(24 109)(25 110)(26 101)(27 102)(28 103)(29 104)(30 105)(31 116)(32 117)(33 118)(34 119)(35 120)(36 111)(37 112)(38 113)(39 114)(40 115)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 136)(52 137)(53 138)(54 139)(55 140)(56 131)(57 132)(58 133)(59 134)(60 135)(61 146)(62 147)(63 148)(64 149)(65 150)(66 141)(67 142)(68 143)(69 144)(70 145)(71 156)(72 157)(73 158)(74 159)(75 160)(76 151)(77 152)(78 153)(79 154)(80 155)
(1 116 6 111)(2 117 7 112)(3 118 8 113)(4 119 9 114)(5 120 10 115)(11 106 16 101)(12 107 17 102)(13 108 18 103)(14 109 19 104)(15 110 20 105)(21 91 26 96)(22 92 27 97)(23 93 28 98)(24 94 29 99)(25 95 30 100)(31 81 36 86)(32 82 37 87)(33 83 38 88)(34 84 39 89)(35 85 40 90)(41 151 46 156)(42 152 47 157)(43 153 48 158)(44 154 49 159)(45 155 50 160)(51 141 56 146)(52 142 57 147)(53 143 58 148)(54 144 59 149)(55 145 60 150)(61 136 66 131)(62 137 67 132)(63 138 68 133)(64 139 69 134)(65 140 70 135)(71 126 76 121)(72 127 77 122)(73 128 78 123)(74 129 79 124)(75 130 80 125)
(1 51 11 41)(2 52 12 42)(3 53 13 43)(4 54 14 44)(5 55 15 45)(6 56 16 46)(7 57 17 47)(8 58 18 48)(9 59 19 49)(10 60 20 50)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)(81 131 91 121)(82 132 92 122)(83 133 93 123)(84 134 94 124)(85 135 95 125)(86 136 96 126)(87 137 97 127)(88 138 98 128)(89 139 99 129)(90 140 100 130)(101 151 111 141)(102 152 112 142)(103 153 113 143)(104 154 114 144)(105 155 115 145)(106 156 116 146)(107 157 117 147)(108 158 118 148)(109 159 119 149)(110 160 120 150)
(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 5)(2 4)(6 10)(7 9)(11 15)(12 14)(16 20)(17 19)(21 35)(22 34)(23 33)(24 32)(25 31)(26 40)(27 39)(28 38)(29 37)(30 36)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(81 85)(82 84)(86 90)(87 89)(91 95)(92 94)(96 100)(97 99)(101 115)(102 114)(103 113)(104 112)(105 111)(106 120)(107 119)(108 118)(109 117)(110 116)(121 130)(122 129)(123 128)(124 127)(125 126)(131 140)(132 139)(133 138)(134 137)(135 136)(141 160)(142 159)(143 158)(144 157)(145 156)(146 155)(147 154)(148 153)(149 152)(150 151)
G:=sub<Sym(160)| (1,86)(2,87)(3,88)(4,89)(5,90)(6,81)(7,82)(8,83)(9,84)(10,85)(11,96)(12,97)(13,98)(14,99)(15,100)(16,91)(17,92)(18,93)(19,94)(20,95)(21,106)(22,107)(23,108)(24,109)(25,110)(26,101)(27,102)(28,103)(29,104)(30,105)(31,116)(32,117)(33,118)(34,119)(35,120)(36,111)(37,112)(38,113)(39,114)(40,115)(41,126)(42,127)(43,128)(44,129)(45,130)(46,121)(47,122)(48,123)(49,124)(50,125)(51,136)(52,137)(53,138)(54,139)(55,140)(56,131)(57,132)(58,133)(59,134)(60,135)(61,146)(62,147)(63,148)(64,149)(65,150)(66,141)(67,142)(68,143)(69,144)(70,145)(71,156)(72,157)(73,158)(74,159)(75,160)(76,151)(77,152)(78,153)(79,154)(80,155), (1,116,6,111)(2,117,7,112)(3,118,8,113)(4,119,9,114)(5,120,10,115)(11,106,16,101)(12,107,17,102)(13,108,18,103)(14,109,19,104)(15,110,20,105)(21,91,26,96)(22,92,27,97)(23,93,28,98)(24,94,29,99)(25,95,30,100)(31,81,36,86)(32,82,37,87)(33,83,38,88)(34,84,39,89)(35,85,40,90)(41,151,46,156)(42,152,47,157)(43,153,48,158)(44,154,49,159)(45,155,50,160)(51,141,56,146)(52,142,57,147)(53,143,58,148)(54,144,59,149)(55,145,60,150)(61,136,66,131)(62,137,67,132)(63,138,68,133)(64,139,69,134)(65,140,70,135)(71,126,76,121)(72,127,77,122)(73,128,78,123)(74,129,79,124)(75,130,80,125), (1,51,11,41)(2,52,12,42)(3,53,13,43)(4,54,14,44)(5,55,15,45)(6,56,16,46)(7,57,17,47)(8,58,18,48)(9,59,19,49)(10,60,20,50)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70)(81,131,91,121)(82,132,92,122)(83,133,93,123)(84,134,94,124)(85,135,95,125)(86,136,96,126)(87,137,97,127)(88,138,98,128)(89,139,99,129)(90,140,100,130)(101,151,111,141)(102,152,112,142)(103,153,113,143)(104,154,114,144)(105,155,115,145)(106,156,116,146)(107,157,117,147)(108,158,118,148)(109,159,119,149)(110,160,120,150), (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), (1,5)(2,4)(6,10)(7,9)(11,15)(12,14)(16,20)(17,19)(21,35)(22,34)(23,33)(24,32)(25,31)(26,40)(27,39)(28,38)(29,37)(30,36)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,85)(82,84)(86,90)(87,89)(91,95)(92,94)(96,100)(97,99)(101,115)(102,114)(103,113)(104,112)(105,111)(106,120)(107,119)(108,118)(109,117)(110,116)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,160)(142,159)(143,158)(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151)>;
G:=Group( (1,86)(2,87)(3,88)(4,89)(5,90)(6,81)(7,82)(8,83)(9,84)(10,85)(11,96)(12,97)(13,98)(14,99)(15,100)(16,91)(17,92)(18,93)(19,94)(20,95)(21,106)(22,107)(23,108)(24,109)(25,110)(26,101)(27,102)(28,103)(29,104)(30,105)(31,116)(32,117)(33,118)(34,119)(35,120)(36,111)(37,112)(38,113)(39,114)(40,115)(41,126)(42,127)(43,128)(44,129)(45,130)(46,121)(47,122)(48,123)(49,124)(50,125)(51,136)(52,137)(53,138)(54,139)(55,140)(56,131)(57,132)(58,133)(59,134)(60,135)(61,146)(62,147)(63,148)(64,149)(65,150)(66,141)(67,142)(68,143)(69,144)(70,145)(71,156)(72,157)(73,158)(74,159)(75,160)(76,151)(77,152)(78,153)(79,154)(80,155), (1,116,6,111)(2,117,7,112)(3,118,8,113)(4,119,9,114)(5,120,10,115)(11,106,16,101)(12,107,17,102)(13,108,18,103)(14,109,19,104)(15,110,20,105)(21,91,26,96)(22,92,27,97)(23,93,28,98)(24,94,29,99)(25,95,30,100)(31,81,36,86)(32,82,37,87)(33,83,38,88)(34,84,39,89)(35,85,40,90)(41,151,46,156)(42,152,47,157)(43,153,48,158)(44,154,49,159)(45,155,50,160)(51,141,56,146)(52,142,57,147)(53,143,58,148)(54,144,59,149)(55,145,60,150)(61,136,66,131)(62,137,67,132)(63,138,68,133)(64,139,69,134)(65,140,70,135)(71,126,76,121)(72,127,77,122)(73,128,78,123)(74,129,79,124)(75,130,80,125), (1,51,11,41)(2,52,12,42)(3,53,13,43)(4,54,14,44)(5,55,15,45)(6,56,16,46)(7,57,17,47)(8,58,18,48)(9,59,19,49)(10,60,20,50)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70)(81,131,91,121)(82,132,92,122)(83,133,93,123)(84,134,94,124)(85,135,95,125)(86,136,96,126)(87,137,97,127)(88,138,98,128)(89,139,99,129)(90,140,100,130)(101,151,111,141)(102,152,112,142)(103,153,113,143)(104,154,114,144)(105,155,115,145)(106,156,116,146)(107,157,117,147)(108,158,118,148)(109,159,119,149)(110,160,120,150), (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), (1,5)(2,4)(6,10)(7,9)(11,15)(12,14)(16,20)(17,19)(21,35)(22,34)(23,33)(24,32)(25,31)(26,40)(27,39)(28,38)(29,37)(30,36)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,85)(82,84)(86,90)(87,89)(91,95)(92,94)(96,100)(97,99)(101,115)(102,114)(103,113)(104,112)(105,111)(106,120)(107,119)(108,118)(109,117)(110,116)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,160)(142,159)(143,158)(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151) );
G=PermutationGroup([[(1,86),(2,87),(3,88),(4,89),(5,90),(6,81),(7,82),(8,83),(9,84),(10,85),(11,96),(12,97),(13,98),(14,99),(15,100),(16,91),(17,92),(18,93),(19,94),(20,95),(21,106),(22,107),(23,108),(24,109),(25,110),(26,101),(27,102),(28,103),(29,104),(30,105),(31,116),(32,117),(33,118),(34,119),(35,120),(36,111),(37,112),(38,113),(39,114),(40,115),(41,126),(42,127),(43,128),(44,129),(45,130),(46,121),(47,122),(48,123),(49,124),(50,125),(51,136),(52,137),(53,138),(54,139),(55,140),(56,131),(57,132),(58,133),(59,134),(60,135),(61,146),(62,147),(63,148),(64,149),(65,150),(66,141),(67,142),(68,143),(69,144),(70,145),(71,156),(72,157),(73,158),(74,159),(75,160),(76,151),(77,152),(78,153),(79,154),(80,155)], [(1,116,6,111),(2,117,7,112),(3,118,8,113),(4,119,9,114),(5,120,10,115),(11,106,16,101),(12,107,17,102),(13,108,18,103),(14,109,19,104),(15,110,20,105),(21,91,26,96),(22,92,27,97),(23,93,28,98),(24,94,29,99),(25,95,30,100),(31,81,36,86),(32,82,37,87),(33,83,38,88),(34,84,39,89),(35,85,40,90),(41,151,46,156),(42,152,47,157),(43,153,48,158),(44,154,49,159),(45,155,50,160),(51,141,56,146),(52,142,57,147),(53,143,58,148),(54,144,59,149),(55,145,60,150),(61,136,66,131),(62,137,67,132),(63,138,68,133),(64,139,69,134),(65,140,70,135),(71,126,76,121),(72,127,77,122),(73,128,78,123),(74,129,79,124),(75,130,80,125)], [(1,51,11,41),(2,52,12,42),(3,53,13,43),(4,54,14,44),(5,55,15,45),(6,56,16,46),(7,57,17,47),(8,58,18,48),(9,59,19,49),(10,60,20,50),(21,71,31,61),(22,72,32,62),(23,73,33,63),(24,74,34,64),(25,75,35,65),(26,76,36,66),(27,77,37,67),(28,78,38,68),(29,79,39,69),(30,80,40,70),(81,131,91,121),(82,132,92,122),(83,133,93,123),(84,134,94,124),(85,135,95,125),(86,136,96,126),(87,137,97,127),(88,138,98,128),(89,139,99,129),(90,140,100,130),(101,151,111,141),(102,152,112,142),(103,153,113,143),(104,154,114,144),(105,155,115,145),(106,156,116,146),(107,157,117,147),(108,158,118,148),(109,159,119,149),(110,160,120,150)], [(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)], [(1,5),(2,4),(6,10),(7,9),(11,15),(12,14),(16,20),(17,19),(21,35),(22,34),(23,33),(24,32),(25,31),(26,40),(27,39),(28,38),(29,37),(30,36),(41,50),(42,49),(43,48),(44,47),(45,46),(51,60),(52,59),(53,58),(54,57),(55,56),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(81,85),(82,84),(86,90),(87,89),(91,95),(92,94),(96,100),(97,99),(101,115),(102,114),(103,113),(104,112),(105,111),(106,120),(107,119),(108,118),(109,117),(110,116),(121,130),(122,129),(123,128),(124,127),(125,126),(131,140),(132,139),(133,138),(134,137),(135,136),(141,160),(142,159),(143,158),(144,157),(145,156),(146,155),(147,154),(148,153),(149,152),(150,151)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | C4○D20 | D4⋊2D5 | Q8⋊2D5 |
| kernel | C2×C4⋊C4⋊D5 | C4⋊C4⋊D5 | C2×C4×Dic5 | C2×C10.D4 | C2×C4⋊Dic5 | C2×D10⋊C4 | C10×C4⋊C4 | C2×C4⋊C4 | C2×C10 | C4⋊C4 | C22×C4 | C22 | C22 | C22 |
| # reps | 1 | 8 | 1 | 1 | 1 | 3 | 1 | 2 | 12 | 8 | 6 | 16 | 4 | 4 |
Matrix representation of C2×C4⋊C4⋊D5 ►in GL6(𝔽41)
| 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 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 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 39 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,32,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,39,0,0,0,0,0,40,0,0,0,0,0,0,34,7,0,0,0,0,40,7,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
C2×C4⋊C4⋊D5 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes C_4\rtimes D_5
% in TeX
G:=Group("C2xC4:C4:D5"); // GroupNames label
G:=SmallGroup(320,1184);
// by ID
G=gap.SmallGroup(320,1184);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,100,675,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^5=e^2=1,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,e*b*e=b*c^2,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations