metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D42⋊2C4, C42.15D4, C6.11D28, C14.11D12, C7⋊1(D6⋊C4), C6.3(C4×D7), C3⋊1(D14⋊C4), C14.3(C4×S3), (C2×C6).8D14, (C2×C14).8D6, C21⋊3(C22⋊C4), C42.10(C2×C4), (C2×Dic3)⋊2D7, (C2×Dic7)⋊2S3, (C6×Dic7)⋊2C2, C6.5(C7⋊D4), C22.7(S3×D7), C2.4(D21⋊C4), (Dic3×C14)⋊2C2, C2.2(C3⋊D28), C2.2(C7⋊D12), C14.5(C3⋊D4), (C2×C42).5C22, (C22×D21).2C2, SmallGroup(336,44)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D42⋊C4
G = < a,b,c | a42=b2=c4=1, bab=a-1, cac-1=a29, cbc-1=a7b >
Subgroups: 476 in 68 conjugacy classes, 28 normal (26 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C7, C2×C4, C23, Dic3, C12, D6, C2×C6, D7, C14, C22⋊C4, C21, C2×Dic3, C2×C12, C22×S3, Dic7, C28, D14, C2×C14, D21, C42, D6⋊C4, C2×Dic7, C2×C28, C22×D7, C7×Dic3, C3×Dic7, D42, D42, C2×C42, D14⋊C4, C6×Dic7, Dic3×C14, C22×D21, D42⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, D7, C22⋊C4, C4×S3, D12, C3⋊D4, D14, D6⋊C4, C4×D7, D28, C7⋊D4, S3×D7, D14⋊C4, D21⋊C4, C3⋊D28, C7⋊D12, D42⋊C4
►On 168 points
Generators in S168
(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)(127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(43 60)(44 59)(45 58)(46 57)(47 56)(48 55)(49 54)(50 53)(51 52)(61 84)(62 83)(63 82)(64 81)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(85 107)(86 106)(87 105)(88 104)(89 103)(90 102)(91 101)(92 100)(93 99)(94 98)(95 97)(108 126)(109 125)(110 124)(111 123)(112 122)(113 121)(114 120)(115 119)(116 118)(127 147)(128 146)(129 145)(130 144)(131 143)(132 142)(133 141)(134 140)(135 139)(136 138)(148 168)(149 167)(150 166)(151 165)(152 164)(153 163)(154 162)(155 161)(156 160)(157 159)
(1 127 52 107)(2 156 53 94)(3 143 54 123)(4 130 55 110)(5 159 56 97)(6 146 57 126)(7 133 58 113)(8 162 59 100)(9 149 60 87)(10 136 61 116)(11 165 62 103)(12 152 63 90)(13 139 64 119)(14 168 65 106)(15 155 66 93)(16 142 67 122)(17 129 68 109)(18 158 69 96)(19 145 70 125)(20 132 71 112)(21 161 72 99)(22 148 73 86)(23 135 74 115)(24 164 75 102)(25 151 76 89)(26 138 77 118)(27 167 78 105)(28 154 79 92)(29 141 80 121)(30 128 81 108)(31 157 82 95)(32 144 83 124)(33 131 84 111)(34 160 43 98)(35 147 44 85)(36 134 45 114)(37 163 46 101)(38 150 47 88)(39 137 48 117)(40 166 49 104)(41 153 50 91)(42 140 51 120)
G:=sub<Sym(168)| (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)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(85,107)(86,106)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(108,126)(109,125)(110,124)(111,123)(112,122)(113,121)(114,120)(115,119)(116,118)(127,147)(128,146)(129,145)(130,144)(131,143)(132,142)(133,141)(134,140)(135,139)(136,138)(148,168)(149,167)(150,166)(151,165)(152,164)(153,163)(154,162)(155,161)(156,160)(157,159), (1,127,52,107)(2,156,53,94)(3,143,54,123)(4,130,55,110)(5,159,56,97)(6,146,57,126)(7,133,58,113)(8,162,59,100)(9,149,60,87)(10,136,61,116)(11,165,62,103)(12,152,63,90)(13,139,64,119)(14,168,65,106)(15,155,66,93)(16,142,67,122)(17,129,68,109)(18,158,69,96)(19,145,70,125)(20,132,71,112)(21,161,72,99)(22,148,73,86)(23,135,74,115)(24,164,75,102)(25,151,76,89)(26,138,77,118)(27,167,78,105)(28,154,79,92)(29,141,80,121)(30,128,81,108)(31,157,82,95)(32,144,83,124)(33,131,84,111)(34,160,43,98)(35,147,44,85)(36,134,45,114)(37,163,46,101)(38,150,47,88)(39,137,48,117)(40,166,49,104)(41,153,50,91)(42,140,51,120)>;
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)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(85,107)(86,106)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(108,126)(109,125)(110,124)(111,123)(112,122)(113,121)(114,120)(115,119)(116,118)(127,147)(128,146)(129,145)(130,144)(131,143)(132,142)(133,141)(134,140)(135,139)(136,138)(148,168)(149,167)(150,166)(151,165)(152,164)(153,163)(154,162)(155,161)(156,160)(157,159), (1,127,52,107)(2,156,53,94)(3,143,54,123)(4,130,55,110)(5,159,56,97)(6,146,57,126)(7,133,58,113)(8,162,59,100)(9,149,60,87)(10,136,61,116)(11,165,62,103)(12,152,63,90)(13,139,64,119)(14,168,65,106)(15,155,66,93)(16,142,67,122)(17,129,68,109)(18,158,69,96)(19,145,70,125)(20,132,71,112)(21,161,72,99)(22,148,73,86)(23,135,74,115)(24,164,75,102)(25,151,76,89)(26,138,77,118)(27,167,78,105)(28,154,79,92)(29,141,80,121)(30,128,81,108)(31,157,82,95)(32,144,83,124)(33,131,84,111)(34,160,43,98)(35,147,44,85)(36,134,45,114)(37,163,46,101)(38,150,47,88)(39,137,48,117)(40,166,49,104)(41,153,50,91)(42,140,51,120) );
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),(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,42),(2,41),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,30),(14,29),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23),(21,22),(43,60),(44,59),(45,58),(46,57),(47,56),(48,55),(49,54),(50,53),(51,52),(61,84),(62,83),(63,82),(64,81),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73),(85,107),(86,106),(87,105),(88,104),(89,103),(90,102),(91,101),(92,100),(93,99),(94,98),(95,97),(108,126),(109,125),(110,124),(111,123),(112,122),(113,121),(114,120),(115,119),(116,118),(127,147),(128,146),(129,145),(130,144),(131,143),(132,142),(133,141),(134,140),(135,139),(136,138),(148,168),(149,167),(150,166),(151,165),(152,164),(153,163),(154,162),(155,161),(156,160),(157,159)], [(1,127,52,107),(2,156,53,94),(3,143,54,123),(4,130,55,110),(5,159,56,97),(6,146,57,126),(7,133,58,113),(8,162,59,100),(9,149,60,87),(10,136,61,116),(11,165,62,103),(12,152,63,90),(13,139,64,119),(14,168,65,106),(15,155,66,93),(16,142,67,122),(17,129,68,109),(18,158,69,96),(19,145,70,125),(20,132,71,112),(21,161,72,99),(22,148,73,86),(23,135,74,115),(24,164,75,102),(25,151,76,89),(26,138,77,118),(27,167,78,105),(28,154,79,92),(29,141,80,121),(30,128,81,108),(31,157,82,95),(32,144,83,124),(33,131,84,111),(34,160,43,98),(35,147,44,85),(36,134,45,114),(37,163,46,101),(38,150,47,88),(39,137,48,117),(40,166,49,104),(41,153,50,91),(42,140,51,120)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 7A | 7B | 7C | 12A | 12B | 12C | 12D | 14A | ··· | 14I | 21A | 21B | 21C | 28A | ··· | 28L | 42A | ··· | 42I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 21 | 21 | 21 | 28 | ··· | 28 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 42 | 42 | 2 | 6 | 6 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D7 | C4×S3 | D12 | C3⋊D4 | D14 | C4×D7 | D28 | C7⋊D4 | S3×D7 | D21⋊C4 | C3⋊D28 | C7⋊D12 |
| kernel | D42⋊C4 | C6×Dic7 | Dic3×C14 | C22×D21 | D42 | C2×Dic7 | C42 | C2×C14 | C2×Dic3 | C14 | C14 | C14 | C2×C6 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 3 | 2 | 2 | 2 | 3 | 6 | 6 | 6 | 3 | 3 | 3 | 3 |
Matrix representation of D42⋊C4 ►in GL4(𝔽337) generated by
| 303 | 336 | 0 | 0 |
| 179 | 144 | 0 | 0 |
| 0 | 0 | 2 | 141 |
| 0 | 0 | 43 | 336 |
| 144 | 1 | 0 | 0 |
| 159 | 193 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 43 | 336 |
| 215 | 63 | 0 | 0 |
| 181 | 122 | 0 | 0 |
| 0 | 0 | 257 | 284 |
| 0 | 0 | 178 | 80 |
G:=sub<GL(4,GF(337))| [303,179,0,0,336,144,0,0,0,0,2,43,0,0,141,336],[144,159,0,0,1,193,0,0,0,0,1,43,0,0,0,336],[215,181,0,0,63,122,0,0,0,0,257,178,0,0,284,80] >;
D42⋊C4 in GAP, Magma, Sage, TeX
D_{42}\rtimes C_4 % in TeX
G:=Group("D42:C4"); // GroupNames label
G:=SmallGroup(336,44);
// by ID
G=gap.SmallGroup(336,44);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,24,121,79,490,10373]);
// Polycyclic
G:=Group<a,b,c|a^42=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^29,c*b*c^-1=a^7*b>;
// generators/relations