metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D42.3C4, C28.34D6, C21⋊3M4(2), C12.34D14, C84.34C22, Dic21.3C4, C7⋊C8⋊5S3, C3⋊C8⋊5D7, C6.2(C4×D7), C7⋊1(C8⋊S3), C3⋊1(C8⋊D7), C14.2(C4×S3), C42.6(C2×C4), C4.27(S3×D7), (C4×D21).5C2, C2.3(D21⋊C4), (C7×C3⋊C8)⋊7C2, (C3×C7⋊C8)⋊7C2, SmallGroup(336,28)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D42.C4
G = < a,b,c | a42=b2=1, c4=a21, bab=a-1, cac-1=a13, cbc-1=a33b >
Smallest permutation representation of 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 62)(44 61)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(85 117)(86 116)(87 115)(88 114)(89 113)(90 112)(91 111)(92 110)(93 109)(94 108)(95 107)(96 106)(97 105)(98 104)(99 103)(100 102)(118 126)(119 125)(120 124)(121 123)(128 168)(129 167)(130 166)(131 165)(132 164)(133 163)(134 162)(135 161)(136 160)(137 159)(138 158)(139 157)(140 156)(141 155)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)
(1 138 53 91 22 159 74 112)(2 151 54 104 23 130 75 125)(3 164 55 117 24 143 76 96)(4 135 56 88 25 156 77 109)(5 148 57 101 26 127 78 122)(6 161 58 114 27 140 79 93)(7 132 59 85 28 153 80 106)(8 145 60 98 29 166 81 119)(9 158 61 111 30 137 82 90)(10 129 62 124 31 150 83 103)(11 142 63 95 32 163 84 116)(12 155 64 108 33 134 43 87)(13 168 65 121 34 147 44 100)(14 139 66 92 35 160 45 113)(15 152 67 105 36 131 46 126)(16 165 68 118 37 144 47 97)(17 136 69 89 38 157 48 110)(18 149 70 102 39 128 49 123)(19 162 71 115 40 141 50 94)(20 133 72 86 41 154 51 107)(21 146 73 99 42 167 52 120)
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)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(85,117)(86,116)(87,115)(88,114)(89,113)(90,112)(91,111)(92,110)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(99,103)(100,102)(118,126)(119,125)(120,124)(121,123)(128,168)(129,167)(130,166)(131,165)(132,164)(133,163)(134,162)(135,161)(136,160)(137,159)(138,158)(139,157)(140,156)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149), (1,138,53,91,22,159,74,112)(2,151,54,104,23,130,75,125)(3,164,55,117,24,143,76,96)(4,135,56,88,25,156,77,109)(5,148,57,101,26,127,78,122)(6,161,58,114,27,140,79,93)(7,132,59,85,28,153,80,106)(8,145,60,98,29,166,81,119)(9,158,61,111,30,137,82,90)(10,129,62,124,31,150,83,103)(11,142,63,95,32,163,84,116)(12,155,64,108,33,134,43,87)(13,168,65,121,34,147,44,100)(14,139,66,92,35,160,45,113)(15,152,67,105,36,131,46,126)(16,165,68,118,37,144,47,97)(17,136,69,89,38,157,48,110)(18,149,70,102,39,128,49,123)(19,162,71,115,40,141,50,94)(20,133,72,86,41,154,51,107)(21,146,73,99,42,167,52,120)>;
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)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(85,117)(86,116)(87,115)(88,114)(89,113)(90,112)(91,111)(92,110)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(99,103)(100,102)(118,126)(119,125)(120,124)(121,123)(128,168)(129,167)(130,166)(131,165)(132,164)(133,163)(134,162)(135,161)(136,160)(137,159)(138,158)(139,157)(140,156)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149), (1,138,53,91,22,159,74,112)(2,151,54,104,23,130,75,125)(3,164,55,117,24,143,76,96)(4,135,56,88,25,156,77,109)(5,148,57,101,26,127,78,122)(6,161,58,114,27,140,79,93)(7,132,59,85,28,153,80,106)(8,145,60,98,29,166,81,119)(9,158,61,111,30,137,82,90)(10,129,62,124,31,150,83,103)(11,142,63,95,32,163,84,116)(12,155,64,108,33,134,43,87)(13,168,65,121,34,147,44,100)(14,139,66,92,35,160,45,113)(15,152,67,105,36,131,46,126)(16,165,68,118,37,144,47,97)(17,136,69,89,38,157,48,110)(18,149,70,102,39,128,49,123)(19,162,71,115,40,141,50,94)(20,133,72,86,41,154,51,107)(21,146,73,99,42,167,52,120) );
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),(43,62),(44,61),(45,60),(46,59),(47,58),(48,57),(49,56),(50,55),(51,54),(52,53),(63,84),(64,83),(65,82),(66,81),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(85,117),(86,116),(87,115),(88,114),(89,113),(90,112),(91,111),(92,110),(93,109),(94,108),(95,107),(96,106),(97,105),(98,104),(99,103),(100,102),(118,126),(119,125),(120,124),(121,123),(128,168),(129,167),(130,166),(131,165),(132,164),(133,163),(134,162),(135,161),(136,160),(137,159),(138,158),(139,157),(140,156),(141,155),(142,154),(143,153),(144,152),(145,151),(146,150),(147,149)], [(1,138,53,91,22,159,74,112),(2,151,54,104,23,130,75,125),(3,164,55,117,24,143,76,96),(4,135,56,88,25,156,77,109),(5,148,57,101,26,127,78,122),(6,161,58,114,27,140,79,93),(7,132,59,85,28,153,80,106),(8,145,60,98,29,166,81,119),(9,158,61,111,30,137,82,90),(10,129,62,124,31,150,83,103),(11,142,63,95,32,163,84,116),(12,155,64,108,33,134,43,87),(13,168,65,121,34,147,44,100),(14,139,66,92,35,160,45,113),(15,152,67,105,36,131,46,126),(16,165,68,118,37,144,47,97),(17,136,69,89,38,157,48,110),(18,149,70,102,39,128,49,123),(19,162,71,115,40,141,50,94),(20,133,72,86,41,154,51,107),(21,146,73,99,42,167,52,120)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6 | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 12A | 12B | 14A | 14B | 14C | 21A | 21B | 21C | 24A | 24B | 24C | 24D | 28A | ··· | 28F | 42A | 42B | 42C | 56A | ··· | 56L | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 12 | 12 | 14 | 14 | 14 | 21 | 21 | 21 | 24 | 24 | 24 | 24 | 28 | ··· | 28 | 42 | 42 | 42 | 56 | ··· | 56 | 84 | ··· | 84 |
| size | 1 | 1 | 42 | 2 | 1 | 1 | 42 | 2 | 2 | 2 | 2 | 6 | 6 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D6 | D7 | M4(2) | C4×S3 | D14 | C8⋊S3 | C4×D7 | C8⋊D7 | S3×D7 | D21⋊C4 | D42.C4 |
| kernel | D42.C4 | C7×C3⋊C8 | C3×C7⋊C8 | C4×D21 | Dic21 | D42 | C7⋊C8 | C28 | C3⋊C8 | C21 | C14 | C12 | C7 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 2 | 2 | 3 | 4 | 6 | 12 | 3 | 3 | 6 |
Matrix representation of D42.C4 ►in GL4(𝔽337) generated by
| 0 | 336 | 0 | 0 |
| 1 | 110 | 0 | 0 |
| 0 | 0 | 1 | 336 |
| 0 | 0 | 1 | 0 |
| 0 | 336 | 0 | 0 |
| 336 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 336 |
| 13 | 37 | 0 | 0 |
| 292 | 324 | 0 | 0 |
| 0 | 0 | 264 | 146 |
| 0 | 0 | 191 | 73 |
G:=sub<GL(4,GF(337))| [0,1,0,0,336,110,0,0,0,0,1,1,0,0,336,0],[0,336,0,0,336,0,0,0,0,0,1,1,0,0,0,336],[13,292,0,0,37,324,0,0,0,0,264,191,0,0,146,73] >;
D42.C4 in GAP, Magma, Sage, TeX
D_{42}.C_4 % in TeX
G:=Group("D42.C4"); // GroupNames label
G:=SmallGroup(336,28);
// by ID
G=gap.SmallGroup(336,28);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,24,121,31,50,490,10373]);
// Polycyclic
G:=Group<a,b,c|a^42=b^2=1,c^4=a^21,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^33*b>;
// generators/relations
Export