direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7×C4○D12, D12⋊5C14, C28.60D6, Dic6⋊5C14, C42.51C23, C84.72C22, (C2×C28)⋊7S3, (S3×C28)⋊9C2, (C4×S3)⋊4C14, (C2×C84)⋊12C2, (C2×C12)⋊4C14, C3⋊D4⋊3C14, (C7×D12)⋊11C2, C21⋊13(C4○D4), C4.16(S3×C14), D6.1(C2×C14), (C2×C14).20D6, C12.16(C2×C14), (C7×Dic6)⋊11C2, C22.2(S3×C14), C6.4(C22×C14), C14.41(C22×S3), (C2×C42).50C22, Dic3.2(C2×C14), (S3×C14).12C22, (C7×Dic3).14C22, C3⋊1(C7×C4○D4), (C2×C4)⋊3(S3×C7), C2.5(S3×C2×C14), (C7×C3⋊D4)⋊7C2, (C2×C6).11(C2×C14), SmallGroup(336,187)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4○D12
G = < a,b,c,d | a7=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >
Subgroups: 152 in 80 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C7, C2×C4, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C14, C14, C4○D4, C21, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C28, C28, C2×C14, C2×C14, S3×C7, C42, C42, C4○D12, C2×C28, C2×C28, C7×D4, C7×Q8, C7×Dic3, C84, S3×C14, C2×C42, C7×C4○D4, C7×Dic6, S3×C28, C7×D12, C7×C3⋊D4, C2×C84, C7×C4○D12
Quotients: C1, C2, C22, S3, C7, C23, D6, C14, C4○D4, C22×S3, C2×C14, S3×C7, C4○D12, C22×C14, S3×C14, C7×C4○D4, S3×C2×C14, C7×C4○D12
►On 168 points
Generators in S168
(1 157 15 117 75 121 137)(2 158 16 118 76 122 138)(3 159 17 119 77 123 139)(4 160 18 120 78 124 140)(5 161 19 109 79 125 141)(6 162 20 110 80 126 142)(7 163 21 111 81 127 143)(8 164 22 112 82 128 144)(9 165 23 113 83 129 133)(10 166 24 114 84 130 134)(11 167 13 115 73 131 135)(12 168 14 116 74 132 136)(25 40 105 92 147 67 56)(26 41 106 93 148 68 57)(27 42 107 94 149 69 58)(28 43 108 95 150 70 59)(29 44 97 96 151 71 60)(30 45 98 85 152 72 49)(31 46 99 86 153 61 50)(32 47 100 87 154 62 51)(33 48 101 88 155 63 52)(34 37 102 89 156 64 53)(35 38 103 90 145 65 54)(36 39 104 91 146 66 55)
(1 85 7 91)(2 86 8 92)(3 87 9 93)(4 88 10 94)(5 89 11 95)(6 90 12 96)(13 70 19 64)(14 71 20 65)(15 72 21 66)(16 61 22 67)(17 62 23 68)(18 63 24 69)(25 76 31 82)(26 77 32 83)(27 78 33 84)(28 79 34 73)(29 80 35 74)(30 81 36 75)(37 131 43 125)(38 132 44 126)(39 121 45 127)(40 122 46 128)(41 123 47 129)(42 124 48 130)(49 111 55 117)(50 112 56 118)(51 113 57 119)(52 114 58 120)(53 115 59 109)(54 116 60 110)(97 142 103 136)(98 143 104 137)(99 144 105 138)(100 133 106 139)(101 134 107 140)(102 135 108 141)(145 168 151 162)(146 157 152 163)(147 158 153 164)(148 159 154 165)(149 160 155 166)(150 161 156 167)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 19)(14 18)(15 17)(20 24)(21 23)(26 36)(27 35)(28 34)(29 33)(30 32)(37 43)(38 42)(39 41)(44 48)(45 47)(49 51)(52 60)(53 59)(54 58)(55 57)(62 72)(63 71)(64 70)(65 69)(66 68)(73 79)(74 78)(75 77)(80 84)(81 83)(85 87)(88 96)(89 95)(90 94)(91 93)(97 101)(98 100)(102 108)(103 107)(104 106)(109 115)(110 114)(111 113)(116 120)(117 119)(121 123)(124 132)(125 131)(126 130)(127 129)(133 143)(134 142)(135 141)(136 140)(137 139)(145 149)(146 148)(150 156)(151 155)(152 154)(157 159)(160 168)(161 167)(162 166)(163 165)
G:=sub<Sym(168)| (1,157,15,117,75,121,137)(2,158,16,118,76,122,138)(3,159,17,119,77,123,139)(4,160,18,120,78,124,140)(5,161,19,109,79,125,141)(6,162,20,110,80,126,142)(7,163,21,111,81,127,143)(8,164,22,112,82,128,144)(9,165,23,113,83,129,133)(10,166,24,114,84,130,134)(11,167,13,115,73,131,135)(12,168,14,116,74,132,136)(25,40,105,92,147,67,56)(26,41,106,93,148,68,57)(27,42,107,94,149,69,58)(28,43,108,95,150,70,59)(29,44,97,96,151,71,60)(30,45,98,85,152,72,49)(31,46,99,86,153,61,50)(32,47,100,87,154,62,51)(33,48,101,88,155,63,52)(34,37,102,89,156,64,53)(35,38,103,90,145,65,54)(36,39,104,91,146,66,55), (1,85,7,91)(2,86,8,92)(3,87,9,93)(4,88,10,94)(5,89,11,95)(6,90,12,96)(13,70,19,64)(14,71,20,65)(15,72,21,66)(16,61,22,67)(17,62,23,68)(18,63,24,69)(25,76,31,82)(26,77,32,83)(27,78,33,84)(28,79,34,73)(29,80,35,74)(30,81,36,75)(37,131,43,125)(38,132,44,126)(39,121,45,127)(40,122,46,128)(41,123,47,129)(42,124,48,130)(49,111,55,117)(50,112,56,118)(51,113,57,119)(52,114,58,120)(53,115,59,109)(54,116,60,110)(97,142,103,136)(98,143,104,137)(99,144,105,138)(100,133,106,139)(101,134,107,140)(102,135,108,141)(145,168,151,162)(146,157,152,163)(147,158,153,164)(148,159,154,165)(149,160,155,166)(150,161,156,167), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,19)(14,18)(15,17)(20,24)(21,23)(26,36)(27,35)(28,34)(29,33)(30,32)(37,43)(38,42)(39,41)(44,48)(45,47)(49,51)(52,60)(53,59)(54,58)(55,57)(62,72)(63,71)(64,70)(65,69)(66,68)(73,79)(74,78)(75,77)(80,84)(81,83)(85,87)(88,96)(89,95)(90,94)(91,93)(97,101)(98,100)(102,108)(103,107)(104,106)(109,115)(110,114)(111,113)(116,120)(117,119)(121,123)(124,132)(125,131)(126,130)(127,129)(133,143)(134,142)(135,141)(136,140)(137,139)(145,149)(146,148)(150,156)(151,155)(152,154)(157,159)(160,168)(161,167)(162,166)(163,165)>;
G:=Group( (1,157,15,117,75,121,137)(2,158,16,118,76,122,138)(3,159,17,119,77,123,139)(4,160,18,120,78,124,140)(5,161,19,109,79,125,141)(6,162,20,110,80,126,142)(7,163,21,111,81,127,143)(8,164,22,112,82,128,144)(9,165,23,113,83,129,133)(10,166,24,114,84,130,134)(11,167,13,115,73,131,135)(12,168,14,116,74,132,136)(25,40,105,92,147,67,56)(26,41,106,93,148,68,57)(27,42,107,94,149,69,58)(28,43,108,95,150,70,59)(29,44,97,96,151,71,60)(30,45,98,85,152,72,49)(31,46,99,86,153,61,50)(32,47,100,87,154,62,51)(33,48,101,88,155,63,52)(34,37,102,89,156,64,53)(35,38,103,90,145,65,54)(36,39,104,91,146,66,55), (1,85,7,91)(2,86,8,92)(3,87,9,93)(4,88,10,94)(5,89,11,95)(6,90,12,96)(13,70,19,64)(14,71,20,65)(15,72,21,66)(16,61,22,67)(17,62,23,68)(18,63,24,69)(25,76,31,82)(26,77,32,83)(27,78,33,84)(28,79,34,73)(29,80,35,74)(30,81,36,75)(37,131,43,125)(38,132,44,126)(39,121,45,127)(40,122,46,128)(41,123,47,129)(42,124,48,130)(49,111,55,117)(50,112,56,118)(51,113,57,119)(52,114,58,120)(53,115,59,109)(54,116,60,110)(97,142,103,136)(98,143,104,137)(99,144,105,138)(100,133,106,139)(101,134,107,140)(102,135,108,141)(145,168,151,162)(146,157,152,163)(147,158,153,164)(148,159,154,165)(149,160,155,166)(150,161,156,167), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,19)(14,18)(15,17)(20,24)(21,23)(26,36)(27,35)(28,34)(29,33)(30,32)(37,43)(38,42)(39,41)(44,48)(45,47)(49,51)(52,60)(53,59)(54,58)(55,57)(62,72)(63,71)(64,70)(65,69)(66,68)(73,79)(74,78)(75,77)(80,84)(81,83)(85,87)(88,96)(89,95)(90,94)(91,93)(97,101)(98,100)(102,108)(103,107)(104,106)(109,115)(110,114)(111,113)(116,120)(117,119)(121,123)(124,132)(125,131)(126,130)(127,129)(133,143)(134,142)(135,141)(136,140)(137,139)(145,149)(146,148)(150,156)(151,155)(152,154)(157,159)(160,168)(161,167)(162,166)(163,165) );
G=PermutationGroup([[(1,157,15,117,75,121,137),(2,158,16,118,76,122,138),(3,159,17,119,77,123,139),(4,160,18,120,78,124,140),(5,161,19,109,79,125,141),(6,162,20,110,80,126,142),(7,163,21,111,81,127,143),(8,164,22,112,82,128,144),(9,165,23,113,83,129,133),(10,166,24,114,84,130,134),(11,167,13,115,73,131,135),(12,168,14,116,74,132,136),(25,40,105,92,147,67,56),(26,41,106,93,148,68,57),(27,42,107,94,149,69,58),(28,43,108,95,150,70,59),(29,44,97,96,151,71,60),(30,45,98,85,152,72,49),(31,46,99,86,153,61,50),(32,47,100,87,154,62,51),(33,48,101,88,155,63,52),(34,37,102,89,156,64,53),(35,38,103,90,145,65,54),(36,39,104,91,146,66,55)], [(1,85,7,91),(2,86,8,92),(3,87,9,93),(4,88,10,94),(5,89,11,95),(6,90,12,96),(13,70,19,64),(14,71,20,65),(15,72,21,66),(16,61,22,67),(17,62,23,68),(18,63,24,69),(25,76,31,82),(26,77,32,83),(27,78,33,84),(28,79,34,73),(29,80,35,74),(30,81,36,75),(37,131,43,125),(38,132,44,126),(39,121,45,127),(40,122,46,128),(41,123,47,129),(42,124,48,130),(49,111,55,117),(50,112,56,118),(51,113,57,119),(52,114,58,120),(53,115,59,109),(54,116,60,110),(97,142,103,136),(98,143,104,137),(99,144,105,138),(100,133,106,139),(101,134,107,140),(102,135,108,141),(145,168,151,162),(146,157,152,163),(147,158,153,164),(148,159,154,165),(149,160,155,166),(150,161,156,167)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,19),(14,18),(15,17),(20,24),(21,23),(26,36),(27,35),(28,34),(29,33),(30,32),(37,43),(38,42),(39,41),(44,48),(45,47),(49,51),(52,60),(53,59),(54,58),(55,57),(62,72),(63,71),(64,70),(65,69),(66,68),(73,79),(74,78),(75,77),(80,84),(81,83),(85,87),(88,96),(89,95),(90,94),(91,93),(97,101),(98,100),(102,108),(103,107),(104,106),(109,115),(110,114),(111,113),(116,120),(117,119),(121,123),(124,132),(125,131),(126,130),(127,129),(133,143),(134,142),(135,141),(136,140),(137,139),(145,149),(146,148),(150,156),(151,155),(152,154),(157,159),(160,168),(161,167),(162,166),(163,165)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 7A | ··· | 7F | 12A | 12B | 12C | 12D | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14X | 21A | ··· | 21F | 28A | ··· | 28L | 28M | ··· | 28R | 28S | ··· | 28AD | 42A | ··· | 42R | 84A | ··· | 84X |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 7 | ··· | 7 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 2 | 6 | 6 | 2 | 1 | 1 | 2 | 6 | 6 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | S3 | D6 | D6 | C4○D4 | S3×C7 | C4○D12 | S3×C14 | S3×C14 | C7×C4○D4 | C7×C4○D12 |
| kernel | C7×C4○D12 | C7×Dic6 | S3×C28 | C7×D12 | C7×C3⋊D4 | C2×C84 | C4○D12 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | C2×C28 | C28 | C2×C14 | C21 | C2×C4 | C7 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 6 | 6 | 12 | 6 | 12 | 6 | 1 | 2 | 1 | 2 | 6 | 4 | 12 | 6 | 12 | 24 |
Matrix representation of C7×C4○D12 ►in GL2(𝔽337) generated by
| 64 | 0 |
| 0 | 64 |
| 148 | 0 |
| 0 | 148 |
| 15 | 15 |
| 322 | 30 |
| 0 | 336 |
| 336 | 0 |
G:=sub<GL(2,GF(337))| [64,0,0,64],[148,0,0,148],[15,322,15,30],[0,336,336,0] >;
C7×C4○D12 in GAP, Magma, Sage, TeX
C_7\times C_4\circ D_{12} % in TeX
G:=Group("C7xC4oD12"); // GroupNames label
G:=SmallGroup(336,187);
// by ID
G=gap.SmallGroup(336,187);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-3,343,1082,8069]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=d^2=1,c^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^5>;
// generators/relations