metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D42⋊1C4, C6.5D28, C2.2D84, C42.33D4, C14.5D12, C22.6D42, (C2×C28)⋊2S3, (C2×C84)⋊3C2, C7⋊2(D6⋊C4), (C2×C4)⋊1D21, (C2×C12)⋊2D7, C6.9(C4×D7), C3⋊2(D14⋊C4), C14.9(C4×S3), C2.5(C4×D21), C21⋊4(C22⋊C4), C42.18(C2×C4), (C2×C6).24D14, (C2×C14).24D6, (C2×Dic21)⋊1C2, C6.15(C7⋊D4), C2.2(C21⋊7D4), C14.15(C3⋊D4), (C2×C42).25C22, (C22×D21).1C2, SmallGroup(336,100)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.D84
G = < a,b,c | a2=b84=1, c2=a, ab=ba, ac=ca, cbc-1=ab-1 >
Subgroups: 512 in 68 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C7, C2×C4, 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, Dic21, C84, D42, D42, C2×C42, D14⋊C4, C2×Dic21, C2×C84, C22×D21, C2.D84
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, D7, C22⋊C4, C4×S3, D12, C3⋊D4, D14, D21, D6⋊C4, C4×D7, D28, C7⋊D4, D42, D14⋊C4, C4×D21, D84, C21⋊7D4, C2.D84
►On 168 points
Generators in S168
(1 123)(2 124)(3 125)(4 126)(5 127)(6 128)(7 129)(8 130)(9 131)(10 132)(11 133)(12 134)(13 135)(14 136)(15 137)(16 138)(17 139)(18 140)(19 141)(20 142)(21 143)(22 144)(23 145)(24 146)(25 147)(26 148)(27 149)(28 150)(29 151)(30 152)(31 153)(32 154)(33 155)(34 156)(35 157)(36 158)(37 159)(38 160)(39 161)(40 162)(41 163)(42 164)(43 165)(44 166)(45 167)(46 168)(47 85)(48 86)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 97)(60 98)(61 99)(62 100)(63 101)(64 102)(65 103)(66 104)(67 105)(68 106)(69 107)(70 108)(71 109)(72 110)(73 111)(74 112)(75 113)(76 114)(77 115)(78 116)(79 117)(80 118)(81 119)(82 120)(83 121)(84 122)
(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)
(1 122 123 84)(2 83 124 121)(3 120 125 82)(4 81 126 119)(5 118 127 80)(6 79 128 117)(7 116 129 78)(8 77 130 115)(9 114 131 76)(10 75 132 113)(11 112 133 74)(12 73 134 111)(13 110 135 72)(14 71 136 109)(15 108 137 70)(16 69 138 107)(17 106 139 68)(18 67 140 105)(19 104 141 66)(20 65 142 103)(21 102 143 64)(22 63 144 101)(23 100 145 62)(24 61 146 99)(25 98 147 60)(26 59 148 97)(27 96 149 58)(28 57 150 95)(29 94 151 56)(30 55 152 93)(31 92 153 54)(32 53 154 91)(33 90 155 52)(34 51 156 89)(35 88 157 50)(36 49 158 87)(37 86 159 48)(38 47 160 85)(39 168 161 46)(40 45 162 167)(41 166 163 44)(42 43 164 165)
G:=sub<Sym(168)| (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,129)(8,130)(9,131)(10,132)(11,133)(12,134)(13,135)(14,136)(15,137)(16,138)(17,139)(18,140)(19,141)(20,142)(21,143)(22,144)(23,145)(24,146)(25,147)(26,148)(27,149)(28,150)(29,151)(30,152)(31,153)(32,154)(33,155)(34,156)(35,157)(36,158)(37,159)(38,160)(39,161)(40,162)(41,163)(42,164)(43,165)(44,166)(45,167)(46,168)(47,85)(48,86)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,97)(60,98)(61,99)(62,100)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118)(81,119)(82,120)(83,121)(84,122), (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), (1,122,123,84)(2,83,124,121)(3,120,125,82)(4,81,126,119)(5,118,127,80)(6,79,128,117)(7,116,129,78)(8,77,130,115)(9,114,131,76)(10,75,132,113)(11,112,133,74)(12,73,134,111)(13,110,135,72)(14,71,136,109)(15,108,137,70)(16,69,138,107)(17,106,139,68)(18,67,140,105)(19,104,141,66)(20,65,142,103)(21,102,143,64)(22,63,144,101)(23,100,145,62)(24,61,146,99)(25,98,147,60)(26,59,148,97)(27,96,149,58)(28,57,150,95)(29,94,151,56)(30,55,152,93)(31,92,153,54)(32,53,154,91)(33,90,155,52)(34,51,156,89)(35,88,157,50)(36,49,158,87)(37,86,159,48)(38,47,160,85)(39,168,161,46)(40,45,162,167)(41,166,163,44)(42,43,164,165)>;
G:=Group( (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,129)(8,130)(9,131)(10,132)(11,133)(12,134)(13,135)(14,136)(15,137)(16,138)(17,139)(18,140)(19,141)(20,142)(21,143)(22,144)(23,145)(24,146)(25,147)(26,148)(27,149)(28,150)(29,151)(30,152)(31,153)(32,154)(33,155)(34,156)(35,157)(36,158)(37,159)(38,160)(39,161)(40,162)(41,163)(42,164)(43,165)(44,166)(45,167)(46,168)(47,85)(48,86)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,97)(60,98)(61,99)(62,100)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118)(81,119)(82,120)(83,121)(84,122), (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), (1,122,123,84)(2,83,124,121)(3,120,125,82)(4,81,126,119)(5,118,127,80)(6,79,128,117)(7,116,129,78)(8,77,130,115)(9,114,131,76)(10,75,132,113)(11,112,133,74)(12,73,134,111)(13,110,135,72)(14,71,136,109)(15,108,137,70)(16,69,138,107)(17,106,139,68)(18,67,140,105)(19,104,141,66)(20,65,142,103)(21,102,143,64)(22,63,144,101)(23,100,145,62)(24,61,146,99)(25,98,147,60)(26,59,148,97)(27,96,149,58)(28,57,150,95)(29,94,151,56)(30,55,152,93)(31,92,153,54)(32,53,154,91)(33,90,155,52)(34,51,156,89)(35,88,157,50)(36,49,158,87)(37,86,159,48)(38,47,160,85)(39,168,161,46)(40,45,162,167)(41,166,163,44)(42,43,164,165) );
G=PermutationGroup([[(1,123),(2,124),(3,125),(4,126),(5,127),(6,128),(7,129),(8,130),(9,131),(10,132),(11,133),(12,134),(13,135),(14,136),(15,137),(16,138),(17,139),(18,140),(19,141),(20,142),(21,143),(22,144),(23,145),(24,146),(25,147),(26,148),(27,149),(28,150),(29,151),(30,152),(31,153),(32,154),(33,155),(34,156),(35,157),(36,158),(37,159),(38,160),(39,161),(40,162),(41,163),(42,164),(43,165),(44,166),(45,167),(46,168),(47,85),(48,86),(49,87),(50,88),(51,89),(52,90),(53,91),(54,92),(55,93),(56,94),(57,95),(58,96),(59,97),(60,98),(61,99),(62,100),(63,101),(64,102),(65,103),(66,104),(67,105),(68,106),(69,107),(70,108),(71,109),(72,110),(73,111),(74,112),(75,113),(76,114),(77,115),(78,116),(79,117),(80,118),(81,119),(82,120),(83,121),(84,122)], [(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)], [(1,122,123,84),(2,83,124,121),(3,120,125,82),(4,81,126,119),(5,118,127,80),(6,79,128,117),(7,116,129,78),(8,77,130,115),(9,114,131,76),(10,75,132,113),(11,112,133,74),(12,73,134,111),(13,110,135,72),(14,71,136,109),(15,108,137,70),(16,69,138,107),(17,106,139,68),(18,67,140,105),(19,104,141,66),(20,65,142,103),(21,102,143,64),(22,63,144,101),(23,100,145,62),(24,61,146,99),(25,98,147,60),(26,59,148,97),(27,96,149,58),(28,57,150,95),(29,94,151,56),(30,55,152,93),(31,92,153,54),(32,53,154,91),(33,90,155,52),(34,51,156,89),(35,88,157,50),(36,49,158,87),(37,86,159,48),(38,47,160,85),(39,168,161,46),(40,45,162,167),(41,166,163,44),(42,43,164,165)]])
90 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 | ··· | 21F | 28A | ··· | 28L | 42A | ··· | 42R | 84A | ··· | 84X |
| 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 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 1 | 1 | 42 | 42 | 2 | 2 | 2 | 42 | 42 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D7 | C4×S3 | D12 | C3⋊D4 | D14 | D21 | C4×D7 | D28 | C7⋊D4 | D42 | C4×D21 | D84 | C21⋊7D4 |
| kernel | C2.D84 | C2×Dic21 | C2×C84 | C22×D21 | D42 | C2×C28 | C42 | C2×C14 | C2×C12 | C14 | C14 | C14 | C2×C6 | C2×C4 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 3 | 2 | 2 | 2 | 3 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 |
Matrix representation of C2.D84 ►in GL5(𝔽337)
| 336 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 0 | 336 |
| 148 | 0 | 0 | 0 | 0 |
| 0 | 177 | 16 | 0 | 0 |
| 0 | 321 | 307 | 0 | 0 |
| 0 | 0 | 0 | 203 | 286 |
| 0 | 0 | 0 | 89 | 54 |
| 189 | 0 | 0 | 0 | 0 |
| 0 | 177 | 16 | 0 | 0 |
| 0 | 64 | 160 | 0 | 0 |
| 0 | 0 | 0 | 203 | 286 |
| 0 | 0 | 0 | 101 | 134 |
G:=sub<GL(5,GF(337))| [336,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,336,0,0,0,0,0,336],[148,0,0,0,0,0,177,321,0,0,0,16,307,0,0,0,0,0,203,89,0,0,0,286,54],[189,0,0,0,0,0,177,64,0,0,0,16,160,0,0,0,0,0,203,101,0,0,0,286,134] >;
C2.D84 in GAP, Magma, Sage, TeX
C_2.D_{84} % in TeX
G:=Group("C2.D84"); // GroupNames label
G:=SmallGroup(336,100);
// by ID
G=gap.SmallGroup(336,100);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,121,31,964,10373]);
// Polycyclic
G:=Group<a,b,c|a^2=b^84=1,c^2=a,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations