direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: M4(2)×C21, C4.C84, C8⋊3C42, C24⋊7C14, C56⋊15C6, C168⋊15C2, C22.C84, C84.10C4, C12.4C28, C28.10C12, C84.82C22, (C2×C42).1C4, (C2×C6).1C28, C2.3(C2×C84), (C2×C4).2C42, C4.6(C2×C42), (C2×C28).19C6, (C2×C12).8C14, (C2×C14).5C12, C42.43(C2×C4), C28.45(C2×C6), C6.12(C2×C28), (C2×C84).20C2, C12.22(C2×C14), C14.26(C2×C12), SmallGroup(336,110)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)×C21
G = < a,b,c | a21=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Smallest permutation representation of M4(2)×C21
►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 26 140 76 160 102 116 49)(2 27 141 77 161 103 117 50)(3 28 142 78 162 104 118 51)(4 29 143 79 163 105 119 52)(5 30 144 80 164 85 120 53)(6 31 145 81 165 86 121 54)(7 32 146 82 166 87 122 55)(8 33 147 83 167 88 123 56)(9 34 127 84 168 89 124 57)(10 35 128 64 148 90 125 58)(11 36 129 65 149 91 126 59)(12 37 130 66 150 92 106 60)(13 38 131 67 151 93 107 61)(14 39 132 68 152 94 108 62)(15 40 133 69 153 95 109 63)(16 41 134 70 154 96 110 43)(17 42 135 71 155 97 111 44)(18 22 136 72 156 98 112 45)(19 23 137 73 157 99 113 46)(20 24 138 74 158 100 114 47)(21 25 139 75 159 101 115 48)
(22 98)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 105)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 81)(55 82)(56 83)(57 84)(58 64)(59 65)(60 66)(61 67)(62 68)(63 69)
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), (1,26,140,76,160,102,116,49)(2,27,141,77,161,103,117,50)(3,28,142,78,162,104,118,51)(4,29,143,79,163,105,119,52)(5,30,144,80,164,85,120,53)(6,31,145,81,165,86,121,54)(7,32,146,82,166,87,122,55)(8,33,147,83,167,88,123,56)(9,34,127,84,168,89,124,57)(10,35,128,64,148,90,125,58)(11,36,129,65,149,91,126,59)(12,37,130,66,150,92,106,60)(13,38,131,67,151,93,107,61)(14,39,132,68,152,94,108,62)(15,40,133,69,153,95,109,63)(16,41,134,70,154,96,110,43)(17,42,135,71,155,97,111,44)(18,22,136,72,156,98,112,45)(19,23,137,73,157,99,113,46)(20,24,138,74,158,100,114,47)(21,25,139,75,159,101,115,48), (22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81)(55,82)(56,83)(57,84)(58,64)(59,65)(60,66)(61,67)(62,68)(63,69)>;
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), (1,26,140,76,160,102,116,49)(2,27,141,77,161,103,117,50)(3,28,142,78,162,104,118,51)(4,29,143,79,163,105,119,52)(5,30,144,80,164,85,120,53)(6,31,145,81,165,86,121,54)(7,32,146,82,166,87,122,55)(8,33,147,83,167,88,123,56)(9,34,127,84,168,89,124,57)(10,35,128,64,148,90,125,58)(11,36,129,65,149,91,126,59)(12,37,130,66,150,92,106,60)(13,38,131,67,151,93,107,61)(14,39,132,68,152,94,108,62)(15,40,133,69,153,95,109,63)(16,41,134,70,154,96,110,43)(17,42,135,71,155,97,111,44)(18,22,136,72,156,98,112,45)(19,23,137,73,157,99,113,46)(20,24,138,74,158,100,114,47)(21,25,139,75,159,101,115,48), (22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81)(55,82)(56,83)(57,84)(58,64)(59,65)(60,66)(61,67)(62,68)(63,69) );
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)], [(1,26,140,76,160,102,116,49),(2,27,141,77,161,103,117,50),(3,28,142,78,162,104,118,51),(4,29,143,79,163,105,119,52),(5,30,144,80,164,85,120,53),(6,31,145,81,165,86,121,54),(7,32,146,82,166,87,122,55),(8,33,147,83,167,88,123,56),(9,34,127,84,168,89,124,57),(10,35,128,64,148,90,125,58),(11,36,129,65,149,91,126,59),(12,37,130,66,150,92,106,60),(13,38,131,67,151,93,107,61),(14,39,132,68,152,94,108,62),(15,40,133,69,153,95,109,63),(16,41,134,70,154,96,110,43),(17,42,135,71,155,97,111,44),(18,22,136,72,156,98,112,45),(19,23,137,73,157,99,113,46),(20,24,138,74,158,100,114,47),(21,25,139,75,159,101,115,48)], [(22,98),(23,99),(24,100),(25,101),(26,102),(27,103),(28,104),(29,105),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,97),(43,70),(44,71),(45,72),(46,73),(47,74),(48,75),(49,76),(50,77),(51,78),(52,79),(53,80),(54,81),(55,82),(56,83),(57,84),(58,64),(59,65),(60,66),(61,67),(62,68),(63,69)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 14A | ··· | 14F | 14G | ··· | 14L | 21A | ··· | 21L | 24A | ··· | 24H | 28A | ··· | 28L | 28M | ··· | 28R | 42A | ··· | 42L | 42M | ··· | 42X | 56A | ··· | 56X | 84A | ··· | 84X | 84Y | ··· | 84AJ | 168A | ··· | 168AV |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 24 | ··· | 24 | 28 | ··· | 28 | 28 | ··· | 28 | 42 | ··· | 42 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
210 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C7 | C12 | C12 | C14 | C14 | C21 | C28 | C28 | C42 | C42 | C84 | C84 | M4(2) | C3×M4(2) | C7×M4(2) | M4(2)×C21 |
| kernel | M4(2)×C21 | C168 | C2×C84 | C7×M4(2) | C84 | C2×C42 | C56 | C2×C28 | C3×M4(2) | C28 | C2×C14 | C24 | C2×C12 | M4(2) | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C21 | C7 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 6 | 4 | 4 | 12 | 6 | 12 | 12 | 12 | 24 | 12 | 24 | 24 | 2 | 4 | 12 | 24 |
Matrix representation of M4(2)×C21 ►in GL3(𝔽337) generated by
| 208 | 0 | 0 |
| 0 | 52 | 0 |
| 0 | 0 | 52 |
| 336 | 0 | 0 |
| 0 | 336 | 335 |
| 0 | 243 | 1 |
| 336 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 336 | 336 |
G:=sub<GL(3,GF(337))| [208,0,0,0,52,0,0,0,52],[336,0,0,0,336,243,0,335,1],[336,0,0,0,1,336,0,0,336] >;
M4(2)×C21 in GAP, Magma, Sage, TeX
M_4(2)\times C_{21} % in TeX
G:=Group("M4(2)xC21"); // GroupNames label
G:=SmallGroup(336,110);
// by ID
G=gap.SmallGroup(336,110);
# by ID
G:=PCGroup([6,-2,-2,-3,-7,-2,-2,504,2041,88]);
// Polycyclic
G:=Group<a,b,c|a^21=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations
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