direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C21, C4.C42, C84.7C2, C28.7C6, C12.3C14, C42.24C22, C2.2(C2×C42), C6.7(C2×C14), C14.15(C2×C6), SmallGroup(168,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C21
G = < a,b,c | a21=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C21
►Regular action 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)
(1 112 150 101)(2 113 151 102)(3 114 152 103)(4 115 153 104)(5 116 154 105)(6 117 155 85)(7 118 156 86)(8 119 157 87)(9 120 158 88)(10 121 159 89)(11 122 160 90)(12 123 161 91)(13 124 162 92)(14 125 163 93)(15 126 164 94)(16 106 165 95)(17 107 166 96)(18 108 167 97)(19 109 168 98)(20 110 148 99)(21 111 149 100)(22 129 84 46)(23 130 64 47)(24 131 65 48)(25 132 66 49)(26 133 67 50)(27 134 68 51)(28 135 69 52)(29 136 70 53)(30 137 71 54)(31 138 72 55)(32 139 73 56)(33 140 74 57)(34 141 75 58)(35 142 76 59)(36 143 77 60)(37 144 78 61)(38 145 79 62)(39 146 80 63)(40 147 81 43)(41 127 82 44)(42 128 83 45)
(1 41 150 82)(2 42 151 83)(3 22 152 84)(4 23 153 64)(5 24 154 65)(6 25 155 66)(7 26 156 67)(8 27 157 68)(9 28 158 69)(10 29 159 70)(11 30 160 71)(12 31 161 72)(13 32 162 73)(14 33 163 74)(15 34 164 75)(16 35 165 76)(17 36 166 77)(18 37 167 78)(19 38 168 79)(20 39 148 80)(21 40 149 81)(43 100 147 111)(44 101 127 112)(45 102 128 113)(46 103 129 114)(47 104 130 115)(48 105 131 116)(49 85 132 117)(50 86 133 118)(51 87 134 119)(52 88 135 120)(53 89 136 121)(54 90 137 122)(55 91 138 123)(56 92 139 124)(57 93 140 125)(58 94 141 126)(59 95 142 106)(60 96 143 107)(61 97 144 108)(62 98 145 109)(63 99 146 110)
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), (1,112,150,101)(2,113,151,102)(3,114,152,103)(4,115,153,104)(5,116,154,105)(6,117,155,85)(7,118,156,86)(8,119,157,87)(9,120,158,88)(10,121,159,89)(11,122,160,90)(12,123,161,91)(13,124,162,92)(14,125,163,93)(15,126,164,94)(16,106,165,95)(17,107,166,96)(18,108,167,97)(19,109,168,98)(20,110,148,99)(21,111,149,100)(22,129,84,46)(23,130,64,47)(24,131,65,48)(25,132,66,49)(26,133,67,50)(27,134,68,51)(28,135,69,52)(29,136,70,53)(30,137,71,54)(31,138,72,55)(32,139,73,56)(33,140,74,57)(34,141,75,58)(35,142,76,59)(36,143,77,60)(37,144,78,61)(38,145,79,62)(39,146,80,63)(40,147,81,43)(41,127,82,44)(42,128,83,45), (1,41,150,82)(2,42,151,83)(3,22,152,84)(4,23,153,64)(5,24,154,65)(6,25,155,66)(7,26,156,67)(8,27,157,68)(9,28,158,69)(10,29,159,70)(11,30,160,71)(12,31,161,72)(13,32,162,73)(14,33,163,74)(15,34,164,75)(16,35,165,76)(17,36,166,77)(18,37,167,78)(19,38,168,79)(20,39,148,80)(21,40,149,81)(43,100,147,111)(44,101,127,112)(45,102,128,113)(46,103,129,114)(47,104,130,115)(48,105,131,116)(49,85,132,117)(50,86,133,118)(51,87,134,119)(52,88,135,120)(53,89,136,121)(54,90,137,122)(55,91,138,123)(56,92,139,124)(57,93,140,125)(58,94,141,126)(59,95,142,106)(60,96,143,107)(61,97,144,108)(62,98,145,109)(63,99,146,110)>;
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), (1,112,150,101)(2,113,151,102)(3,114,152,103)(4,115,153,104)(5,116,154,105)(6,117,155,85)(7,118,156,86)(8,119,157,87)(9,120,158,88)(10,121,159,89)(11,122,160,90)(12,123,161,91)(13,124,162,92)(14,125,163,93)(15,126,164,94)(16,106,165,95)(17,107,166,96)(18,108,167,97)(19,109,168,98)(20,110,148,99)(21,111,149,100)(22,129,84,46)(23,130,64,47)(24,131,65,48)(25,132,66,49)(26,133,67,50)(27,134,68,51)(28,135,69,52)(29,136,70,53)(30,137,71,54)(31,138,72,55)(32,139,73,56)(33,140,74,57)(34,141,75,58)(35,142,76,59)(36,143,77,60)(37,144,78,61)(38,145,79,62)(39,146,80,63)(40,147,81,43)(41,127,82,44)(42,128,83,45), (1,41,150,82)(2,42,151,83)(3,22,152,84)(4,23,153,64)(5,24,154,65)(6,25,155,66)(7,26,156,67)(8,27,157,68)(9,28,158,69)(10,29,159,70)(11,30,160,71)(12,31,161,72)(13,32,162,73)(14,33,163,74)(15,34,164,75)(16,35,165,76)(17,36,166,77)(18,37,167,78)(19,38,168,79)(20,39,148,80)(21,40,149,81)(43,100,147,111)(44,101,127,112)(45,102,128,113)(46,103,129,114)(47,104,130,115)(48,105,131,116)(49,85,132,117)(50,86,133,118)(51,87,134,119)(52,88,135,120)(53,89,136,121)(54,90,137,122)(55,91,138,123)(56,92,139,124)(57,93,140,125)(58,94,141,126)(59,95,142,106)(60,96,143,107)(61,97,144,108)(62,98,145,109)(63,99,146,110) );
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)], [(1,112,150,101),(2,113,151,102),(3,114,152,103),(4,115,153,104),(5,116,154,105),(6,117,155,85),(7,118,156,86),(8,119,157,87),(9,120,158,88),(10,121,159,89),(11,122,160,90),(12,123,161,91),(13,124,162,92),(14,125,163,93),(15,126,164,94),(16,106,165,95),(17,107,166,96),(18,108,167,97),(19,109,168,98),(20,110,148,99),(21,111,149,100),(22,129,84,46),(23,130,64,47),(24,131,65,48),(25,132,66,49),(26,133,67,50),(27,134,68,51),(28,135,69,52),(29,136,70,53),(30,137,71,54),(31,138,72,55),(32,139,73,56),(33,140,74,57),(34,141,75,58),(35,142,76,59),(36,143,77,60),(37,144,78,61),(38,145,79,62),(39,146,80,63),(40,147,81,43),(41,127,82,44),(42,128,83,45)], [(1,41,150,82),(2,42,151,83),(3,22,152,84),(4,23,153,64),(5,24,154,65),(6,25,155,66),(7,26,156,67),(8,27,157,68),(9,28,158,69),(10,29,159,70),(11,30,160,71),(12,31,161,72),(13,32,162,73),(14,33,163,74),(15,34,164,75),(16,35,165,76),(17,36,166,77),(18,37,167,78),(19,38,168,79),(20,39,148,80),(21,40,149,81),(43,100,147,111),(44,101,127,112),(45,102,128,113),(46,103,129,114),(47,104,130,115),(48,105,131,116),(49,85,132,117),(50,86,133,118),(51,87,134,119),(52,88,135,120),(53,89,136,121),(54,90,137,122),(55,91,138,123),(56,92,139,124),(57,93,140,125),(58,94,141,126),(59,95,142,106),(60,96,143,107),(61,97,144,108),(62,98,145,109),(63,99,146,110)]])
Q8×C21 is a maximal subgroup of
Q8⋊2D21 C21⋊7Q16 Q8⋊3D21
105 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 7A | ··· | 7F | 12A | ··· | 12F | 14A | ··· | 14F | 21A | ··· | 21L | 28A | ··· | 28R | 42A | ··· | 42L | 84A | ··· | 84AJ |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 7 | ··· | 7 | 12 | ··· | 12 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | - | |||||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 | Q8 | C3×Q8 | C7×Q8 | Q8×C21 |
| kernel | Q8×C21 | C84 | C7×Q8 | C28 | C3×Q8 | C12 | Q8 | C4 | C21 | C7 | C3 | C1 |
| # reps | 1 | 3 | 2 | 6 | 6 | 18 | 12 | 36 | 1 | 2 | 6 | 12 |
Matrix representation of Q8×C21 ►in GL2(𝔽43) generated by
| 9 | 0 |
| 0 | 9 |
| 9 | 29 |
| 12 | 34 |
| 0 | 36 |
| 37 | 0 |
G:=sub<GL(2,GF(43))| [9,0,0,9],[9,12,29,34],[0,37,36,0] >;
Q8×C21 in GAP, Magma, Sage, TeX
Q_8\times C_{21} % in TeX
G:=Group("Q8xC21"); // GroupNames label
G:=SmallGroup(168,41);
// by ID
G=gap.SmallGroup(168,41);
# by ID
G:=PCGroup([5,-2,-2,-3,-7,-2,420,861,426]);
// Polycyclic
G:=Group<a,b,c|a^21=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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