direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C7×C4.Dic3, C84.7C4, C12.1C28, C28.59D6, C21⋊9M4(2), C28.4Dic3, C84.76C22, C3⋊C8⋊5C14, C4.(C7×Dic3), (C2×C42).7C4, (C2×C28).8S3, C6.6(C2×C28), (C2×C6).3C28, C3⋊2(C7×M4(2)), C4.15(S3×C14), (C2×C84).17C2, C42.38(C2×C4), (C2×C12).5C14, C12.15(C2×C14), C22.(C7×Dic3), (C2×C14).1Dic3, C2.3(Dic3×C14), C14.12(C2×Dic3), (C7×C3⋊C8)⋊12C2, (C2×C4).2(S3×C7), SmallGroup(336,80)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4.Dic3
G = < a,b,c,d | a7=b4=1, c6=b2, d2=b2c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >
Smallest permutation representation of C7×C4.Dic3
►On 168 points
Generators in S168
(1 76 64 49 40 31 19)(2 77 65 50 41 32 20)(3 78 66 51 42 33 21)(4 79 67 52 43 34 22)(5 80 68 53 44 35 23)(6 81 69 54 45 36 24)(7 82 70 55 46 25 13)(8 83 71 56 47 26 14)(9 84 72 57 48 27 15)(10 73 61 58 37 28 16)(11 74 62 59 38 29 17)(12 75 63 60 39 30 18)(85 162 155 143 128 119 98)(86 163 156 144 129 120 99)(87 164 145 133 130 109 100)(88 165 146 134 131 110 101)(89 166 147 135 132 111 102)(90 167 148 136 121 112 103)(91 168 149 137 122 113 104)(92 157 150 138 123 114 105)(93 158 151 139 124 115 106)(94 159 152 140 125 116 107)(95 160 153 141 126 117 108)(96 161 154 142 127 118 97)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 46 43 40)(38 47 44 41)(39 48 45 42)(49 58 55 52)(50 59 56 53)(51 60 57 54)(61 70 67 64)(62 71 68 65)(63 72 69 66)(73 82 79 76)(74 83 80 77)(75 84 81 78)(85 88 91 94)(86 89 92 95)(87 90 93 96)(97 100 103 106)(98 101 104 107)(99 102 105 108)(109 112 115 118)(110 113 116 119)(111 114 117 120)(121 124 127 130)(122 125 128 131)(123 126 129 132)(133 136 139 142)(134 137 140 143)(135 138 141 144)(145 148 151 154)(146 149 152 155)(147 150 153 156)(157 160 163 166)(158 161 164 167)(159 162 165 168)
(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 94 10 91 7 88 4 85)(2 87 11 96 8 93 5 90)(3 92 12 89 9 86 6 95)(13 101 22 98 19 107 16 104)(14 106 23 103 20 100 17 97)(15 99 24 108 21 105 18 102)(25 110 34 119 31 116 28 113)(26 115 35 112 32 109 29 118)(27 120 36 117 33 114 30 111)(37 122 46 131 43 128 40 125)(38 127 47 124 44 121 41 130)(39 132 48 129 45 126 42 123)(49 140 58 137 55 134 52 143)(50 133 59 142 56 139 53 136)(51 138 60 135 57 144 54 141)(61 149 70 146 67 155 64 152)(62 154 71 151 68 148 65 145)(63 147 72 156 69 153 66 150)(73 168 82 165 79 162 76 159)(74 161 83 158 80 167 77 164)(75 166 84 163 81 160 78 157)
G:=sub<Sym(168)| (1,76,64,49,40,31,19)(2,77,65,50,41,32,20)(3,78,66,51,42,33,21)(4,79,67,52,43,34,22)(5,80,68,53,44,35,23)(6,81,69,54,45,36,24)(7,82,70,55,46,25,13)(8,83,71,56,47,26,14)(9,84,72,57,48,27,15)(10,73,61,58,37,28,16)(11,74,62,59,38,29,17)(12,75,63,60,39,30,18)(85,162,155,143,128,119,98)(86,163,156,144,129,120,99)(87,164,145,133,130,109,100)(88,165,146,134,131,110,101)(89,166,147,135,132,111,102)(90,167,148,136,121,112,103)(91,168,149,137,122,113,104)(92,157,150,138,123,114,105)(93,158,151,139,124,115,106)(94,159,152,140,125,116,107)(95,160,153,141,126,117,108)(96,161,154,142,127,118,97), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,46,43,40)(38,47,44,41)(39,48,45,42)(49,58,55,52)(50,59,56,53)(51,60,57,54)(61,70,67,64)(62,71,68,65)(63,72,69,66)(73,82,79,76)(74,83,80,77)(75,84,81,78)(85,88,91,94)(86,89,92,95)(87,90,93,96)(97,100,103,106)(98,101,104,107)(99,102,105,108)(109,112,115,118)(110,113,116,119)(111,114,117,120)(121,124,127,130)(122,125,128,131)(123,126,129,132)(133,136,139,142)(134,137,140,143)(135,138,141,144)(145,148,151,154)(146,149,152,155)(147,150,153,156)(157,160,163,166)(158,161,164,167)(159,162,165,168), (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), (1,94,10,91,7,88,4,85)(2,87,11,96,8,93,5,90)(3,92,12,89,9,86,6,95)(13,101,22,98,19,107,16,104)(14,106,23,103,20,100,17,97)(15,99,24,108,21,105,18,102)(25,110,34,119,31,116,28,113)(26,115,35,112,32,109,29,118)(27,120,36,117,33,114,30,111)(37,122,46,131,43,128,40,125)(38,127,47,124,44,121,41,130)(39,132,48,129,45,126,42,123)(49,140,58,137,55,134,52,143)(50,133,59,142,56,139,53,136)(51,138,60,135,57,144,54,141)(61,149,70,146,67,155,64,152)(62,154,71,151,68,148,65,145)(63,147,72,156,69,153,66,150)(73,168,82,165,79,162,76,159)(74,161,83,158,80,167,77,164)(75,166,84,163,81,160,78,157)>;
G:=Group( (1,76,64,49,40,31,19)(2,77,65,50,41,32,20)(3,78,66,51,42,33,21)(4,79,67,52,43,34,22)(5,80,68,53,44,35,23)(6,81,69,54,45,36,24)(7,82,70,55,46,25,13)(8,83,71,56,47,26,14)(9,84,72,57,48,27,15)(10,73,61,58,37,28,16)(11,74,62,59,38,29,17)(12,75,63,60,39,30,18)(85,162,155,143,128,119,98)(86,163,156,144,129,120,99)(87,164,145,133,130,109,100)(88,165,146,134,131,110,101)(89,166,147,135,132,111,102)(90,167,148,136,121,112,103)(91,168,149,137,122,113,104)(92,157,150,138,123,114,105)(93,158,151,139,124,115,106)(94,159,152,140,125,116,107)(95,160,153,141,126,117,108)(96,161,154,142,127,118,97), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,46,43,40)(38,47,44,41)(39,48,45,42)(49,58,55,52)(50,59,56,53)(51,60,57,54)(61,70,67,64)(62,71,68,65)(63,72,69,66)(73,82,79,76)(74,83,80,77)(75,84,81,78)(85,88,91,94)(86,89,92,95)(87,90,93,96)(97,100,103,106)(98,101,104,107)(99,102,105,108)(109,112,115,118)(110,113,116,119)(111,114,117,120)(121,124,127,130)(122,125,128,131)(123,126,129,132)(133,136,139,142)(134,137,140,143)(135,138,141,144)(145,148,151,154)(146,149,152,155)(147,150,153,156)(157,160,163,166)(158,161,164,167)(159,162,165,168), (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), (1,94,10,91,7,88,4,85)(2,87,11,96,8,93,5,90)(3,92,12,89,9,86,6,95)(13,101,22,98,19,107,16,104)(14,106,23,103,20,100,17,97)(15,99,24,108,21,105,18,102)(25,110,34,119,31,116,28,113)(26,115,35,112,32,109,29,118)(27,120,36,117,33,114,30,111)(37,122,46,131,43,128,40,125)(38,127,47,124,44,121,41,130)(39,132,48,129,45,126,42,123)(49,140,58,137,55,134,52,143)(50,133,59,142,56,139,53,136)(51,138,60,135,57,144,54,141)(61,149,70,146,67,155,64,152)(62,154,71,151,68,148,65,145)(63,147,72,156,69,153,66,150)(73,168,82,165,79,162,76,159)(74,161,83,158,80,167,77,164)(75,166,84,163,81,160,78,157) );
G=PermutationGroup([[(1,76,64,49,40,31,19),(2,77,65,50,41,32,20),(3,78,66,51,42,33,21),(4,79,67,52,43,34,22),(5,80,68,53,44,35,23),(6,81,69,54,45,36,24),(7,82,70,55,46,25,13),(8,83,71,56,47,26,14),(9,84,72,57,48,27,15),(10,73,61,58,37,28,16),(11,74,62,59,38,29,17),(12,75,63,60,39,30,18),(85,162,155,143,128,119,98),(86,163,156,144,129,120,99),(87,164,145,133,130,109,100),(88,165,146,134,131,110,101),(89,166,147,135,132,111,102),(90,167,148,136,121,112,103),(91,168,149,137,122,113,104),(92,157,150,138,123,114,105),(93,158,151,139,124,115,106),(94,159,152,140,125,116,107),(95,160,153,141,126,117,108),(96,161,154,142,127,118,97)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,22,19,16),(14,23,20,17),(15,24,21,18),(25,34,31,28),(26,35,32,29),(27,36,33,30),(37,46,43,40),(38,47,44,41),(39,48,45,42),(49,58,55,52),(50,59,56,53),(51,60,57,54),(61,70,67,64),(62,71,68,65),(63,72,69,66),(73,82,79,76),(74,83,80,77),(75,84,81,78),(85,88,91,94),(86,89,92,95),(87,90,93,96),(97,100,103,106),(98,101,104,107),(99,102,105,108),(109,112,115,118),(110,113,116,119),(111,114,117,120),(121,124,127,130),(122,125,128,131),(123,126,129,132),(133,136,139,142),(134,137,140,143),(135,138,141,144),(145,148,151,154),(146,149,152,155),(147,150,153,156),(157,160,163,166),(158,161,164,167),(159,162,165,168)], [(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)], [(1,94,10,91,7,88,4,85),(2,87,11,96,8,93,5,90),(3,92,12,89,9,86,6,95),(13,101,22,98,19,107,16,104),(14,106,23,103,20,100,17,97),(15,99,24,108,21,105,18,102),(25,110,34,119,31,116,28,113),(26,115,35,112,32,109,29,118),(27,120,36,117,33,114,30,111),(37,122,46,131,43,128,40,125),(38,127,47,124,44,121,41,130),(39,132,48,129,45,126,42,123),(49,140,58,137,55,134,52,143),(50,133,59,142,56,139,53,136),(51,138,60,135,57,144,54,141),(61,149,70,146,67,155,64,152),(62,154,71,151,68,148,65,145),(63,147,72,156,69,153,66,150),(73,168,82,165,79,162,76,159),(74,161,83,158,80,167,77,164),(75,166,84,163,81,160,78,157)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 14A | ··· | 14F | 14G | ··· | 14L | 21A | ··· | 21F | 28A | ··· | 28L | 28M | ··· | 28R | 42A | ··· | 42R | 56A | ··· | 56X | 84A | ··· | 84X |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C7 | C14 | C14 | C28 | C28 | S3 | Dic3 | D6 | Dic3 | M4(2) | S3×C7 | C4.Dic3 | C7×Dic3 | S3×C14 | C7×Dic3 | C7×M4(2) | C7×C4.Dic3 |
| kernel | C7×C4.Dic3 | C7×C3⋊C8 | C2×C84 | C84 | C2×C42 | C4.Dic3 | C3⋊C8 | C2×C12 | C12 | C2×C6 | C2×C28 | C28 | C28 | C2×C14 | C21 | C2×C4 | C7 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 6 | 12 | 6 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 6 | 4 | 6 | 6 | 6 | 12 | 24 |
Matrix representation of C7×C4.Dic3 ►in GL2(𝔽337) generated by
| 79 | 0 |
| 0 | 79 |
| 189 | 0 |
| 0 | 148 |
| 220 | 0 |
| 0 | 265 |
| 0 | 1 |
| 189 | 0 |
G:=sub<GL(2,GF(337))| [79,0,0,79],[189,0,0,148],[220,0,0,265],[0,189,1,0] >;
C7×C4.Dic3 in GAP, Magma, Sage, TeX
C_7\times C_4.{\rm Dic}_3 % in TeX
G:=Group("C7xC4.Dic3"); // GroupNames label
G:=SmallGroup(336,80);
// by ID
G=gap.SmallGroup(336,80);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-3,168,697,69,8069]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=1,c^6=b^2,d^2=b^2*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations
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