metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C5×D4)⋊14D4, (C5×Q8)⋊14D4, C5⋊7(D4⋊D4), D4⋊6(C5⋊D4), Q8⋊6(C5⋊D4), C20⋊7D4⋊26C2, C20.215(C2×D4), (C2×C20).452D4, C10.78C22≀C2, (C2×D4).204D10, D4⋊Dic5⋊41C2, Q8⋊Dic5⋊41C2, (C2×Q8).173D10, C10.112(C4○D8), C20.55D4⋊18C2, (C2×C20).484C23, (C22×C4).164D10, (C22×C10).115D4, C23.32(C5⋊D4), C2.23(D4⋊D10), C10.125(C8⋊C22), (C2×D20).137C22, (D4×C10).245C22, C2.12(C24⋊2D5), C4⋊Dic5.189C22, (Q8×C10).208C22, C2.30(D4.8D10), (C22×C20).210C22, (C2×C4○D4)⋊1D5, (C2×D4⋊D5)⋊24C2, (C10×C4○D4)⋊1C2, (C2×Q8⋊D5)⋊24C2, C4.62(C2×C5⋊D4), (C2×C10).567(C2×D4), (C2×C4).225(C5⋊D4), (C2×C4).568(C22×D5), C22.224(C2×C5⋊D4), (C2×C5⋊2C8).178C22, SmallGroup(320,865)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C5×D4)⋊14D4
G = < a,b,c,d,e | a5=b4=c2=d4=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, cbc=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >
Subgroups: 590 in 162 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C5⋊2C8, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×D5, C22×C10, C22×C10, D4⋊D4, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, D4⋊D5, Q8⋊D5, C2×D20, C2×C5⋊D4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C20.55D4, D4⋊Dic5, Q8⋊Dic5, C20⋊7D4, C2×D4⋊D5, C2×Q8⋊D5, C10×C4○D4, (C5×D4)⋊14D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C4○D8, C8⋊C22, C5⋊D4, C22×D5, D4⋊D4, C2×C5⋊D4, D4⋊D10, D4.8D10, C24⋊2D5, (C5×D4)⋊14D4
►On 160 points
Generators in S160
(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 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 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)(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)(121 131 126 136)(122 132 127 137)(123 133 128 138)(124 134 129 139)(125 135 130 140)(141 151 146 156)(142 152 147 157)(143 153 148 158)(144 154 149 159)(145 155 150 160)
(1 116)(2 117)(3 118)(4 119)(5 120)(6 111)(7 112)(8 113)(9 114)(10 115)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 96)(22 97)(23 98)(24 99)(25 100)(26 91)(27 92)(28 93)(29 94)(30 95)(31 81)(32 82)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 156)(42 157)(43 158)(44 159)(45 160)(46 151)(47 152)(48 153)(49 154)(50 155)(51 141)(52 142)(53 143)(54 144)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 136)(62 137)(63 138)(64 139)(65 140)(66 131)(67 132)(68 133)(69 134)(70 135)(71 121)(72 122)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 129)(80 130)
(1 51 26 76)(2 55 27 80)(3 54 28 79)(4 53 29 78)(5 52 30 77)(6 56 21 71)(7 60 22 75)(8 59 23 74)(9 58 24 73)(10 57 25 72)(11 41 36 66)(12 45 37 70)(13 44 38 69)(14 43 39 68)(15 42 40 67)(16 46 31 61)(17 50 32 65)(18 49 33 64)(19 48 34 63)(20 47 35 62)(81 121 106 146)(82 125 107 150)(83 124 108 149)(84 123 109 148)(85 122 110 147)(86 126 101 141)(87 130 102 145)(88 129 103 144)(89 128 104 143)(90 127 105 142)(91 136 116 151)(92 140 117 155)(93 139 118 154)(94 138 119 153)(95 137 120 152)(96 131 111 156)(97 135 112 160)(98 134 113 159)(99 133 114 158)(100 132 115 157)
(2 5)(3 4)(7 10)(8 9)(11 16)(12 20)(13 19)(14 18)(15 17)(22 25)(23 24)(27 30)(28 29)(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 76)(52 80)(53 79)(54 78)(55 77)(56 71)(57 75)(58 74)(59 73)(60 72)(81 91)(82 95)(83 94)(84 93)(85 92)(86 96)(87 100)(88 99)(89 98)(90 97)(101 111)(102 115)(103 114)(104 113)(105 112)(106 116)(107 120)(108 119)(109 118)(110 117)(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)| (1,2,3,4,5)(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), (1,16,6,11)(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)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75)(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)(121,131,126,136)(122,132,127,137)(123,133,128,138)(124,134,129,139)(125,135,130,140)(141,151,146,156)(142,152,147,157)(143,153,148,158)(144,154,149,159)(145,155,150,160), (1,116)(2,117)(3,118)(4,119)(5,120)(6,111)(7,112)(8,113)(9,114)(10,115)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,96)(22,97)(23,98)(24,99)(25,100)(26,91)(27,92)(28,93)(29,94)(30,95)(31,81)(32,82)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,156)(42,157)(43,158)(44,159)(45,160)(46,151)(47,152)(48,153)(49,154)(50,155)(51,141)(52,142)(53,143)(54,144)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(61,136)(62,137)(63,138)(64,139)(65,140)(66,131)(67,132)(68,133)(69,134)(70,135)(71,121)(72,122)(73,123)(74,124)(75,125)(76,126)(77,127)(78,128)(79,129)(80,130), (1,51,26,76)(2,55,27,80)(3,54,28,79)(4,53,29,78)(5,52,30,77)(6,56,21,71)(7,60,22,75)(8,59,23,74)(9,58,24,73)(10,57,25,72)(11,41,36,66)(12,45,37,70)(13,44,38,69)(14,43,39,68)(15,42,40,67)(16,46,31,61)(17,50,32,65)(18,49,33,64)(19,48,34,63)(20,47,35,62)(81,121,106,146)(82,125,107,150)(83,124,108,149)(84,123,109,148)(85,122,110,147)(86,126,101,141)(87,130,102,145)(88,129,103,144)(89,128,104,143)(90,127,105,142)(91,136,116,151)(92,140,117,155)(93,139,118,154)(94,138,119,153)(95,137,120,152)(96,131,111,156)(97,135,112,160)(98,134,113,159)(99,133,114,158)(100,132,115,157), (2,5)(3,4)(7,10)(8,9)(11,16)(12,20)(13,19)(14,18)(15,17)(22,25)(23,24)(27,30)(28,29)(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)(51,76)(52,80)(53,79)(54,78)(55,77)(56,71)(57,75)(58,74)(59,73)(60,72)(81,91)(82,95)(83,94)(84,93)(85,92)(86,96)(87,100)(88,99)(89,98)(90,97)(101,111)(102,115)(103,114)(104,113)(105,112)(106,116)(107,120)(108,119)(109,118)(110,117)(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( (1,2,3,4,5)(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), (1,16,6,11)(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)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75)(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)(121,131,126,136)(122,132,127,137)(123,133,128,138)(124,134,129,139)(125,135,130,140)(141,151,146,156)(142,152,147,157)(143,153,148,158)(144,154,149,159)(145,155,150,160), (1,116)(2,117)(3,118)(4,119)(5,120)(6,111)(7,112)(8,113)(9,114)(10,115)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,96)(22,97)(23,98)(24,99)(25,100)(26,91)(27,92)(28,93)(29,94)(30,95)(31,81)(32,82)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,156)(42,157)(43,158)(44,159)(45,160)(46,151)(47,152)(48,153)(49,154)(50,155)(51,141)(52,142)(53,143)(54,144)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(61,136)(62,137)(63,138)(64,139)(65,140)(66,131)(67,132)(68,133)(69,134)(70,135)(71,121)(72,122)(73,123)(74,124)(75,125)(76,126)(77,127)(78,128)(79,129)(80,130), (1,51,26,76)(2,55,27,80)(3,54,28,79)(4,53,29,78)(5,52,30,77)(6,56,21,71)(7,60,22,75)(8,59,23,74)(9,58,24,73)(10,57,25,72)(11,41,36,66)(12,45,37,70)(13,44,38,69)(14,43,39,68)(15,42,40,67)(16,46,31,61)(17,50,32,65)(18,49,33,64)(19,48,34,63)(20,47,35,62)(81,121,106,146)(82,125,107,150)(83,124,108,149)(84,123,109,148)(85,122,110,147)(86,126,101,141)(87,130,102,145)(88,129,103,144)(89,128,104,143)(90,127,105,142)(91,136,116,151)(92,140,117,155)(93,139,118,154)(94,138,119,153)(95,137,120,152)(96,131,111,156)(97,135,112,160)(98,134,113,159)(99,133,114,158)(100,132,115,157), (2,5)(3,4)(7,10)(8,9)(11,16)(12,20)(13,19)(14,18)(15,17)(22,25)(23,24)(27,30)(28,29)(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)(51,76)(52,80)(53,79)(54,78)(55,77)(56,71)(57,75)(58,74)(59,73)(60,72)(81,91)(82,95)(83,94)(84,93)(85,92)(86,96)(87,100)(88,99)(89,98)(90,97)(101,111)(102,115)(103,114)(104,113)(105,112)(106,116)(107,120)(108,119)(109,118)(110,117)(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([[(1,2,3,4,5),(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)], [(1,16,6,11),(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),(41,56,46,51),(42,57,47,52),(43,58,48,53),(44,59,49,54),(45,60,50,55),(61,76,66,71),(62,77,67,72),(63,78,68,73),(64,79,69,74),(65,80,70,75),(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),(121,131,126,136),(122,132,127,137),(123,133,128,138),(124,134,129,139),(125,135,130,140),(141,151,146,156),(142,152,147,157),(143,153,148,158),(144,154,149,159),(145,155,150,160)], [(1,116),(2,117),(3,118),(4,119),(5,120),(6,111),(7,112),(8,113),(9,114),(10,115),(11,101),(12,102),(13,103),(14,104),(15,105),(16,106),(17,107),(18,108),(19,109),(20,110),(21,96),(22,97),(23,98),(24,99),(25,100),(26,91),(27,92),(28,93),(29,94),(30,95),(31,81),(32,82),(33,83),(34,84),(35,85),(36,86),(37,87),(38,88),(39,89),(40,90),(41,156),(42,157),(43,158),(44,159),(45,160),(46,151),(47,152),(48,153),(49,154),(50,155),(51,141),(52,142),(53,143),(54,144),(55,145),(56,146),(57,147),(58,148),(59,149),(60,150),(61,136),(62,137),(63,138),(64,139),(65,140),(66,131),(67,132),(68,133),(69,134),(70,135),(71,121),(72,122),(73,123),(74,124),(75,125),(76,126),(77,127),(78,128),(79,129),(80,130)], [(1,51,26,76),(2,55,27,80),(3,54,28,79),(4,53,29,78),(5,52,30,77),(6,56,21,71),(7,60,22,75),(8,59,23,74),(9,58,24,73),(10,57,25,72),(11,41,36,66),(12,45,37,70),(13,44,38,69),(14,43,39,68),(15,42,40,67),(16,46,31,61),(17,50,32,65),(18,49,33,64),(19,48,34,63),(20,47,35,62),(81,121,106,146),(82,125,107,150),(83,124,108,149),(84,123,109,148),(85,122,110,147),(86,126,101,141),(87,130,102,145),(88,129,103,144),(89,128,104,143),(90,127,105,142),(91,136,116,151),(92,140,117,155),(93,139,118,154),(94,138,119,153),(95,137,120,152),(96,131,111,156),(97,135,112,160),(98,134,113,159),(99,133,114,158),(100,132,115,157)], [(2,5),(3,4),(7,10),(8,9),(11,16),(12,20),(13,19),(14,18),(15,17),(22,25),(23,24),(27,30),(28,29),(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),(51,76),(52,80),(53,79),(54,78),(55,77),(56,71),(57,75),(58,74),(59,73),(60,72),(81,91),(82,95),(83,94),(84,93),(85,92),(86,96),(87,100),(88,99),(89,98),(90,97),(101,111),(102,115),(103,114),(104,113),(105,112),(106,116),(107,120),(108,119),(109,118),(110,117),(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)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 40 | 2 | 2 | 2 | 2 | 4 | 4 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C5⋊D4 | C5⋊D4 | C5⋊D4 | C8⋊C22 | D4⋊D10 | D4.8D10 |
| kernel | (C5×D4)⋊14D4 | C20.55D4 | D4⋊Dic5 | Q8⋊Dic5 | C20⋊7D4 | C2×D4⋊D5 | C2×Q8⋊D5 | C10×C4○D4 | C2×C20 | C5×D4 | C5×Q8 | C22×C10 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C10 | C2×C4 | D4 | Q8 | C23 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 1 | 4 | 4 |
Matrix representation of (C5×D4)⋊14D4 ►in GL4(𝔽41) generated by
| 6 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 4 |
| 0 | 0 | 20 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 17 | 34 |
| 0 | 0 | 6 | 24 |
| 18 | 35 | 0 | 0 |
| 20 | 23 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 16 | 32 |
| 1 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 |
| 0 | 0 | 1 | 4 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [6,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,20,0,0,4,40],[1,0,0,0,0,1,0,0,0,0,17,6,0,0,34,24],[18,20,0,0,35,23,0,0,0,0,9,16,0,0,0,32],[1,6,0,0,0,40,0,0,0,0,1,0,0,0,4,40] >;
(C5×D4)⋊14D4 in GAP, Magma, Sage, TeX
(C_5\times D_4)\rtimes_{14}D_4 % in TeX
G:=Group("(C5xD4):14D4"); // GroupNames label
G:=SmallGroup(320,865);
// by ID
G=gap.SmallGroup(320,865);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,184,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a^-1,c*b*c=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations