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G = M4(2)×C21order 336 = 24·3·7

Direct product of C21 and M4(2)

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: M4(2)×C21, C4.C84, C83C42, C247C14, C5615C6, C16815C2, 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

C1C2 — M4(2)×C21
C1C2C4C28C84C168 — M4(2)×C21
C1C2 — M4(2)×C21
C1C84 — M4(2)×C21

Generators and relations for M4(2)×C21
 G = < a,b,c | a21=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

2C2
2C6
2C14
2C42

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)
(1 112 48 146 155 74 103 29)(2 113 49 147 156 75 104 30)(3 114 50 127 157 76 105 31)(4 115 51 128 158 77 85 32)(5 116 52 129 159 78 86 33)(6 117 53 130 160 79 87 34)(7 118 54 131 161 80 88 35)(8 119 55 132 162 81 89 36)(9 120 56 133 163 82 90 37)(10 121 57 134 164 83 91 38)(11 122 58 135 165 84 92 39)(12 123 59 136 166 64 93 40)(13 124 60 137 167 65 94 41)(14 125 61 138 168 66 95 42)(15 126 62 139 148 67 96 22)(16 106 63 140 149 68 97 23)(17 107 43 141 150 69 98 24)(18 108 44 142 151 70 99 25)(19 109 45 143 152 71 100 26)(20 110 46 144 153 72 101 27)(21 111 47 145 154 73 102 28)
(22 139)(23 140)(24 141)(25 142)(26 143)(27 144)(28 145)(29 146)(30 147)(31 127)(32 128)(33 129)(34 130)(35 131)(36 132)(37 133)(38 134)(39 135)(40 136)(41 137)(42 138)(64 123)(65 124)(66 125)(67 126)(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)

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), (1,112,48,146,155,74,103,29)(2,113,49,147,156,75,104,30)(3,114,50,127,157,76,105,31)(4,115,51,128,158,77,85,32)(5,116,52,129,159,78,86,33)(6,117,53,130,160,79,87,34)(7,118,54,131,161,80,88,35)(8,119,55,132,162,81,89,36)(9,120,56,133,163,82,90,37)(10,121,57,134,164,83,91,38)(11,122,58,135,165,84,92,39)(12,123,59,136,166,64,93,40)(13,124,60,137,167,65,94,41)(14,125,61,138,168,66,95,42)(15,126,62,139,148,67,96,22)(16,106,63,140,149,68,97,23)(17,107,43,141,150,69,98,24)(18,108,44,142,151,70,99,25)(19,109,45,143,152,71,100,26)(20,110,46,144,153,72,101,27)(21,111,47,145,154,73,102,28), (22,139)(23,140)(24,141)(25,142)(26,143)(27,144)(28,145)(29,146)(30,147)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(64,123)(65,124)(66,125)(67,126)(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)>;

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), (1,112,48,146,155,74,103,29)(2,113,49,147,156,75,104,30)(3,114,50,127,157,76,105,31)(4,115,51,128,158,77,85,32)(5,116,52,129,159,78,86,33)(6,117,53,130,160,79,87,34)(7,118,54,131,161,80,88,35)(8,119,55,132,162,81,89,36)(9,120,56,133,163,82,90,37)(10,121,57,134,164,83,91,38)(11,122,58,135,165,84,92,39)(12,123,59,136,166,64,93,40)(13,124,60,137,167,65,94,41)(14,125,61,138,168,66,95,42)(15,126,62,139,148,67,96,22)(16,106,63,140,149,68,97,23)(17,107,43,141,150,69,98,24)(18,108,44,142,151,70,99,25)(19,109,45,143,152,71,100,26)(20,110,46,144,153,72,101,27)(21,111,47,145,154,73,102,28), (22,139)(23,140)(24,141)(25,142)(26,143)(27,144)(28,145)(29,146)(30,147)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(64,123)(65,124)(66,125)(67,126)(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) );

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)], [(1,112,48,146,155,74,103,29),(2,113,49,147,156,75,104,30),(3,114,50,127,157,76,105,31),(4,115,51,128,158,77,85,32),(5,116,52,129,159,78,86,33),(6,117,53,130,160,79,87,34),(7,118,54,131,161,80,88,35),(8,119,55,132,162,81,89,36),(9,120,56,133,163,82,90,37),(10,121,57,134,164,83,91,38),(11,122,58,135,165,84,92,39),(12,123,59,136,166,64,93,40),(13,124,60,137,167,65,94,41),(14,125,61,138,168,66,95,42),(15,126,62,139,148,67,96,22),(16,106,63,140,149,68,97,23),(17,107,43,141,150,69,98,24),(18,108,44,142,151,70,99,25),(19,109,45,143,152,71,100,26),(20,110,46,144,153,72,101,27),(21,111,47,145,154,73,102,28)], [(22,139),(23,140),(24,141),(25,142),(26,143),(27,144),(28,145),(29,146),(30,147),(31,127),(32,128),(33,129),(34,130),(35,131),(36,132),(37,133),(38,134),(39,135),(40,136),(41,137),(42,138),(64,123),(65,124),(66,125),(67,126),(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)])

210 conjugacy classes

class 1 2A2B3A3B4A4B4C6A6B6C6D7A···7F8A8B8C8D12A12B12C12D12E12F14A···14F14G···14L21A···21L24A···24H28A···28L28M···28R42A···42L42M···42X56A···56X84A···84X84Y···84AJ168A···168AV
order1223344466667···7888812121212121214···1414···1421···2124···2428···2828···2842···4242···4256···5684···8484···84168···168
size1121111211221···122221111221···12···21···12···21···12···21···12···22···21···12···22···2

210 irreducible representations

dim111111111111111111112222
type+++
imageC1C2C2C3C4C4C6C6C7C12C12C14C14C21C28C28C42C42C84C84M4(2)C3×M4(2)C7×M4(2)M4(2)×C21
kernelM4(2)×C21C168C2×C84C7×M4(2)C84C2×C42C56C2×C28C3×M4(2)C28C2×C14C24C2×C12M4(2)C12C2×C6C8C2×C4C4C22C21C7C3C1
# reps1212224264412612121224122424241224

Matrix representation of M4(2)×C21 in GL3(𝔽337) generated by

20800
0520
0052
,
33600
0336335
02431
,
33600
010
0336336
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

Export

Subgroup lattice of M4(2)×C21 in TeX

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