metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊2C8⋊4D4, C4⋊D4⋊4D5, C5⋊3(C8⋊2D4), C4⋊C4.59D10, (C2×C20).72D4, C4.171(D4×D5), C20⋊7D4⋊24C2, (C2×D4).39D10, C20.148(C2×D4), D20⋊6C4⋊36C2, D4⋊Dic5⋊16C2, C20.Q8⋊35C2, (C22×C10).85D4, C20.184(C4○D4), C4.60(D4⋊2D5), C10.94(C4⋊D4), C10.91(C8⋊C22), (C2×C20).358C23, (D4×C10).55C22, (C22×C4).121D10, C23.24(C5⋊D4), C2.13(D4⋊D10), (C2×D20).102C22, C4⋊Dic5.143C22, C2.15(Dic5⋊D4), C2.12(D4.D10), (C22×C20).162C22, (C2×D4⋊D5)⋊11C2, (C5×C4⋊D4)⋊4C2, (C2×C10).489(C2×D4), (C2×C4).50(C5⋊D4), (C2×C4.Dic5)⋊11C2, (C5×C4⋊C4).106C22, (C2×C4).458(C22×D5), C22.164(C2×C5⋊D4), (C2×C5⋊2C8).109C22, SmallGroup(320,666)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊D4⋊D5
G = < a,b,c,d,e | a4=b4=c2=d5=e2=1, bab-1=cac=eae=a-1, ad=da, cbc=b-1, bd=db, ebe=ab-1, cd=dc, ece=a-1c, ede=d-1 >
Subgroups: 550 in 130 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), D8, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, C4.Q8, C4⋊D4, C4⋊D4, C2×M4(2), C2×D8, C5⋊2C8, C5⋊2C8, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, C8⋊2D4, C2×C5⋊2C8, C4.Dic5, C4⋊Dic5, D10⋊C4, D4⋊D5, C5×C22⋊C4, C5×C4⋊C4, C2×D20, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C20.Q8, D20⋊6C4, D4⋊Dic5, C2×C4.Dic5, C20⋊7D4, C2×D4⋊D5, C5×C4⋊D4, C4⋊D4⋊D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8⋊C22, C5⋊D4, C22×D5, C8⋊2D4, D4×D5, D4⋊2D5, C2×C5⋊D4, D4.D10, Dic5⋊D4, D4⋊D10, C4⋊D4⋊D5
►On 160 points
Generators in S160
(1 29 9 24)(2 30 10 25)(3 26 6 21)(4 27 7 22)(5 28 8 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 61 46 66)(42 62 47 67)(43 63 48 68)(44 64 49 69)(45 65 50 70)(51 71 56 76)(52 72 57 77)(53 73 58 78)(54 74 59 79)(55 75 60 80)(81 106 86 101)(82 107 87 102)(83 108 88 103)(84 109 89 104)(85 110 90 105)(91 116 96 111)(92 117 97 112)(93 118 98 113)(94 119 99 114)(95 120 100 115)(121 141 126 146)(122 142 127 147)(123 143 128 148)(124 144 129 149)(125 145 130 150)(131 151 136 156)(132 152 137 157)(133 153 138 158)(134 154 139 159)(135 155 140 160)
(1 124 19 139)(2 125 20 140)(3 121 16 136)(4 122 17 137)(5 123 18 138)(6 126 11 131)(7 127 12 132)(8 128 13 133)(9 129 14 134)(10 130 15 135)(21 141 36 156)(22 142 37 157)(23 143 38 158)(24 144 39 159)(25 145 40 160)(26 146 31 151)(27 147 32 152)(28 148 33 153)(29 149 34 154)(30 150 35 155)(41 81 56 96)(42 82 57 97)(43 83 58 98)(44 84 59 99)(45 85 60 100)(46 86 51 91)(47 87 52 92)(48 88 53 93)(49 89 54 94)(50 90 55 95)(61 101 76 116)(62 102 77 117)(63 103 78 118)(64 104 79 119)(65 105 80 120)(66 106 71 111)(67 107 72 112)(68 108 73 113)(69 109 74 114)(70 110 75 115)
(1 44)(2 45)(3 41)(4 42)(5 43)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(81 136)(82 137)(83 138)(84 139)(85 140)(86 131)(87 132)(88 133)(89 134)(90 135)(91 126)(92 127)(93 128)(94 129)(95 130)(96 121)(97 122)(98 123)(99 124)(100 125)(101 156)(102 157)(103 158)(104 159)(105 160)(106 151)(107 152)(108 153)(109 154)(110 155)(111 146)(112 147)(113 148)(114 149)(115 150)(116 141)(117 142)(118 143)(119 144)(120 145)
(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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(41 61)(42 65)(43 64)(44 63)(45 62)(46 66)(47 70)(48 69)(49 68)(50 67)(51 71)(52 75)(53 74)(54 73)(55 72)(56 76)(57 80)(58 79)(59 78)(60 77)(81 91)(82 95)(83 94)(84 93)(85 92)(86 96)(87 100)(88 99)(89 98)(90 97)(101 116)(102 120)(103 119)(104 118)(105 117)(106 111)(107 115)(108 114)(109 113)(110 112)(121 151)(122 155)(123 154)(124 153)(125 152)(126 156)(127 160)(128 159)(129 158)(130 157)(131 141)(132 145)(133 144)(134 143)(135 142)(136 146)(137 150)(138 149)(139 148)(140 147)
G:=sub<Sym(160)| (1,29,9,24)(2,30,10,25)(3,26,6,21)(4,27,7,22)(5,28,8,23)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,61,46,66)(42,62,47,67)(43,63,48,68)(44,64,49,69)(45,65,50,70)(51,71,56,76)(52,72,57,77)(53,73,58,78)(54,74,59,79)(55,75,60,80)(81,106,86,101)(82,107,87,102)(83,108,88,103)(84,109,89,104)(85,110,90,105)(91,116,96,111)(92,117,97,112)(93,118,98,113)(94,119,99,114)(95,120,100,115)(121,141,126,146)(122,142,127,147)(123,143,128,148)(124,144,129,149)(125,145,130,150)(131,151,136,156)(132,152,137,157)(133,153,138,158)(134,154,139,159)(135,155,140,160), (1,124,19,139)(2,125,20,140)(3,121,16,136)(4,122,17,137)(5,123,18,138)(6,126,11,131)(7,127,12,132)(8,128,13,133)(9,129,14,134)(10,130,15,135)(21,141,36,156)(22,142,37,157)(23,143,38,158)(24,144,39,159)(25,145,40,160)(26,146,31,151)(27,147,32,152)(28,148,33,153)(29,149,34,154)(30,150,35,155)(41,81,56,96)(42,82,57,97)(43,83,58,98)(44,84,59,99)(45,85,60,100)(46,86,51,91)(47,87,52,92)(48,88,53,93)(49,89,54,94)(50,90,55,95)(61,101,76,116)(62,102,77,117)(63,103,78,118)(64,104,79,119)(65,105,80,120)(66,106,71,111)(67,107,72,112)(68,108,73,113)(69,109,74,114)(70,110,75,115), (1,44)(2,45)(3,41)(4,42)(5,43)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(81,136)(82,137)(83,138)(84,139)(85,140)(86,131)(87,132)(88,133)(89,134)(90,135)(91,126)(92,127)(93,128)(94,129)(95,130)(96,121)(97,122)(98,123)(99,124)(100,125)(101,156)(102,157)(103,158)(104,159)(105,160)(106,151)(107,152)(108,153)(109,154)(110,155)(111,146)(112,147)(113,148)(114,149)(115,150)(116,141)(117,142)(118,143)(119,144)(120,145), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(41,61)(42,65)(43,64)(44,63)(45,62)(46,66)(47,70)(48,69)(49,68)(50,67)(51,71)(52,75)(53,74)(54,73)(55,72)(56,76)(57,80)(58,79)(59,78)(60,77)(81,91)(82,95)(83,94)(84,93)(85,92)(86,96)(87,100)(88,99)(89,98)(90,97)(101,116)(102,120)(103,119)(104,118)(105,117)(106,111)(107,115)(108,114)(109,113)(110,112)(121,151)(122,155)(123,154)(124,153)(125,152)(126,156)(127,160)(128,159)(129,158)(130,157)(131,141)(132,145)(133,144)(134,143)(135,142)(136,146)(137,150)(138,149)(139,148)(140,147)>;
G:=Group( (1,29,9,24)(2,30,10,25)(3,26,6,21)(4,27,7,22)(5,28,8,23)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,61,46,66)(42,62,47,67)(43,63,48,68)(44,64,49,69)(45,65,50,70)(51,71,56,76)(52,72,57,77)(53,73,58,78)(54,74,59,79)(55,75,60,80)(81,106,86,101)(82,107,87,102)(83,108,88,103)(84,109,89,104)(85,110,90,105)(91,116,96,111)(92,117,97,112)(93,118,98,113)(94,119,99,114)(95,120,100,115)(121,141,126,146)(122,142,127,147)(123,143,128,148)(124,144,129,149)(125,145,130,150)(131,151,136,156)(132,152,137,157)(133,153,138,158)(134,154,139,159)(135,155,140,160), (1,124,19,139)(2,125,20,140)(3,121,16,136)(4,122,17,137)(5,123,18,138)(6,126,11,131)(7,127,12,132)(8,128,13,133)(9,129,14,134)(10,130,15,135)(21,141,36,156)(22,142,37,157)(23,143,38,158)(24,144,39,159)(25,145,40,160)(26,146,31,151)(27,147,32,152)(28,148,33,153)(29,149,34,154)(30,150,35,155)(41,81,56,96)(42,82,57,97)(43,83,58,98)(44,84,59,99)(45,85,60,100)(46,86,51,91)(47,87,52,92)(48,88,53,93)(49,89,54,94)(50,90,55,95)(61,101,76,116)(62,102,77,117)(63,103,78,118)(64,104,79,119)(65,105,80,120)(66,106,71,111)(67,107,72,112)(68,108,73,113)(69,109,74,114)(70,110,75,115), (1,44)(2,45)(3,41)(4,42)(5,43)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(81,136)(82,137)(83,138)(84,139)(85,140)(86,131)(87,132)(88,133)(89,134)(90,135)(91,126)(92,127)(93,128)(94,129)(95,130)(96,121)(97,122)(98,123)(99,124)(100,125)(101,156)(102,157)(103,158)(104,159)(105,160)(106,151)(107,152)(108,153)(109,154)(110,155)(111,146)(112,147)(113,148)(114,149)(115,150)(116,141)(117,142)(118,143)(119,144)(120,145), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(41,61)(42,65)(43,64)(44,63)(45,62)(46,66)(47,70)(48,69)(49,68)(50,67)(51,71)(52,75)(53,74)(54,73)(55,72)(56,76)(57,80)(58,79)(59,78)(60,77)(81,91)(82,95)(83,94)(84,93)(85,92)(86,96)(87,100)(88,99)(89,98)(90,97)(101,116)(102,120)(103,119)(104,118)(105,117)(106,111)(107,115)(108,114)(109,113)(110,112)(121,151)(122,155)(123,154)(124,153)(125,152)(126,156)(127,160)(128,159)(129,158)(130,157)(131,141)(132,145)(133,144)(134,143)(135,142)(136,146)(137,150)(138,149)(139,148)(140,147) );
G=PermutationGroup([[(1,29,9,24),(2,30,10,25),(3,26,6,21),(4,27,7,22),(5,28,8,23),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35),(41,61,46,66),(42,62,47,67),(43,63,48,68),(44,64,49,69),(45,65,50,70),(51,71,56,76),(52,72,57,77),(53,73,58,78),(54,74,59,79),(55,75,60,80),(81,106,86,101),(82,107,87,102),(83,108,88,103),(84,109,89,104),(85,110,90,105),(91,116,96,111),(92,117,97,112),(93,118,98,113),(94,119,99,114),(95,120,100,115),(121,141,126,146),(122,142,127,147),(123,143,128,148),(124,144,129,149),(125,145,130,150),(131,151,136,156),(132,152,137,157),(133,153,138,158),(134,154,139,159),(135,155,140,160)], [(1,124,19,139),(2,125,20,140),(3,121,16,136),(4,122,17,137),(5,123,18,138),(6,126,11,131),(7,127,12,132),(8,128,13,133),(9,129,14,134),(10,130,15,135),(21,141,36,156),(22,142,37,157),(23,143,38,158),(24,144,39,159),(25,145,40,160),(26,146,31,151),(27,147,32,152),(28,148,33,153),(29,149,34,154),(30,150,35,155),(41,81,56,96),(42,82,57,97),(43,83,58,98),(44,84,59,99),(45,85,60,100),(46,86,51,91),(47,87,52,92),(48,88,53,93),(49,89,54,94),(50,90,55,95),(61,101,76,116),(62,102,77,117),(63,103,78,118),(64,104,79,119),(65,105,80,120),(66,106,71,111),(67,107,72,112),(68,108,73,113),(69,109,74,114),(70,110,75,115)], [(1,44),(2,45),(3,41),(4,42),(5,43),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(81,136),(82,137),(83,138),(84,139),(85,140),(86,131),(87,132),(88,133),(89,134),(90,135),(91,126),(92,127),(93,128),(94,129),(95,130),(96,121),(97,122),(98,123),(99,124),(100,125),(101,156),(102,157),(103,158),(104,159),(105,160),(106,151),(107,152),(108,153),(109,154),(110,155),(111,146),(112,147),(113,148),(114,149),(115,150),(116,141),(117,142),(118,143),(119,144),(120,145)], [(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)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(21,26),(22,30),(23,29),(24,28),(25,27),(31,36),(32,40),(33,39),(34,38),(35,37),(41,61),(42,65),(43,64),(44,63),(45,62),(46,66),(47,70),(48,69),(49,68),(50,67),(51,71),(52,75),(53,74),(54,73),(55,72),(56,76),(57,80),(58,79),(59,78),(60,77),(81,91),(82,95),(83,94),(84,93),(85,92),(86,96),(87,100),(88,99),(89,98),(90,97),(101,116),(102,120),(103,119),(104,118),(105,117),(106,111),(107,115),(108,114),(109,113),(110,112),(121,151),(122,155),(123,154),(124,153),(125,152),(126,156),(127,160),(128,159),(129,158),(130,157),(131,141),(132,145),(133,144),(134,143),(135,142),(136,146),(137,150),(138,149),(139,148),(140,147)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 40 | 2 | 2 | 4 | 8 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8⋊C22 | D4×D5 | D4⋊2D5 | D4.D10 | D4⋊D10 |
| kernel | C4⋊D4⋊D5 | C20.Q8 | D20⋊6C4 | D4⋊Dic5 | C2×C4.Dic5 | C20⋊7D4 | C2×D4⋊D5 | C5×C4⋊D4 | C5⋊2C8 | C2×C20 | C22×C10 | C4⋊D4 | C20 | C4⋊C4 | C22×C4 | C2×D4 | C2×C4 | C23 | C10 | C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C4⋊D4⋊D5 ►in GL8(𝔽41)
| 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 32 | 25 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 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 | 11 | 30 | 0 | 6 |
| 0 | 0 | 0 | 0 | 30 | 30 | 35 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 | 11 | 30 |
| 0 | 0 | 0 | 0 | 35 | 0 | 30 | 30 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 37 | 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 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 34 | 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 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0],[32,0,0,0,0,0,0,0,25,9,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,11,30,0,35,0,0,0,0,30,30,6,0,0,0,0,0,0,35,11,30,0,0,0,0,6,0,30,30],[40,37,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,34,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,4,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,34,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
C4⋊D4⋊D5 in GAP, Magma, Sage, TeX
C_4\rtimes D_4\rtimes D_5
% in TeX
G:=Group("C4:D4:D5"); // GroupNames label
G:=SmallGroup(320,666);
// by ID
G=gap.SmallGroup(320,666);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,555,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^5=e^2=1,b*a*b^-1=c*a*c=e*a*e=a^-1,a*d=d*a,c*b*c=b^-1,b*d=d*b,e*b*e=a*b^-1,c*d=d*c,e*c*e=a^-1*c,e*d*e=d^-1>;
// generators/relations