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## G = Q8×D21order 336 = 24·3·7

### Direct product of Q8 and D21

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — Q8×D21
 Chief series C1 — C7 — C21 — C42 — D42 — C4×D21 — Q8×D21
 Lower central C21 — C42 — Q8×D21
 Upper central C1 — C2 — Q8

Generators and relations for Q8×D21
G = < a,b,c,d | a4=c21=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 440 in 76 conjugacy classes, 37 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C7, C2×C4, Q8, Q8, Dic3, C12, D6, D7, C14, C2×Q8, C21, Dic6, C4×S3, C3×Q8, Dic7, C28, D14, D21, C42, S3×Q8, Dic14, C4×D7, C7×Q8, Dic21, C84, D42, Q8×D7, Dic42, C4×D21, Q8×C21, Q8×D21
Quotients: C1, C2, C22, S3, Q8, C23, D6, D7, C2×Q8, C22×S3, D14, D21, S3×Q8, C22×D7, D42, Q8×D7, C22×D21, Q8×D21

Smallest permutation representation of Q8×D21
On 168 points
Generators in S168
(1 71 37 45)(2 72 38 46)(3 73 39 47)(4 74 40 48)(5 75 41 49)(6 76 42 50)(7 77 22 51)(8 78 23 52)(9 79 24 53)(10 80 25 54)(11 81 26 55)(12 82 27 56)(13 83 28 57)(14 84 29 58)(15 64 30 59)(16 65 31 60)(17 66 32 61)(18 67 33 62)(19 68 34 63)(20 69 35 43)(21 70 36 44)(85 131 113 157)(86 132 114 158)(87 133 115 159)(88 134 116 160)(89 135 117 161)(90 136 118 162)(91 137 119 163)(92 138 120 164)(93 139 121 165)(94 140 122 166)(95 141 123 167)(96 142 124 168)(97 143 125 148)(98 144 126 149)(99 145 106 150)(100 146 107 151)(101 147 108 152)(102 127 109 153)(103 128 110 154)(104 129 111 155)(105 130 112 156)
(1 117 37 89)(2 118 38 90)(3 119 39 91)(4 120 40 92)(5 121 41 93)(6 122 42 94)(7 123 22 95)(8 124 23 96)(9 125 24 97)(10 126 25 98)(11 106 26 99)(12 107 27 100)(13 108 28 101)(14 109 29 102)(15 110 30 103)(16 111 31 104)(17 112 32 105)(18 113 33 85)(19 114 34 86)(20 115 35 87)(21 116 36 88)(43 159 69 133)(44 160 70 134)(45 161 71 135)(46 162 72 136)(47 163 73 137)(48 164 74 138)(49 165 75 139)(50 166 76 140)(51 167 77 141)(52 168 78 142)(53 148 79 143)(54 149 80 144)(55 150 81 145)(56 151 82 146)(57 152 83 147)(58 153 84 127)(59 154 64 128)(60 155 65 129)(61 156 66 130)(62 157 67 131)(63 158 68 132)
(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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(43 72)(44 71)(45 70)(46 69)(47 68)(48 67)(49 66)(50 65)(51 64)(52 84)(53 83)(54 82)(55 81)(56 80)(57 79)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(85 120)(86 119)(87 118)(88 117)(89 116)(90 115)(91 114)(92 113)(93 112)(94 111)(95 110)(96 109)(97 108)(98 107)(99 106)(100 126)(101 125)(102 124)(103 123)(104 122)(105 121)(127 168)(128 167)(129 166)(130 165)(131 164)(132 163)(133 162)(134 161)(135 160)(136 159)(137 158)(138 157)(139 156)(140 155)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6 7A 7B 7C 12A 12B 12C 14A 14B 14C 21A ··· 21F 28A ··· 28I 42A ··· 42F 84A ··· 84R order 1 2 2 2 3 4 4 4 4 4 4 6 7 7 7 12 12 12 14 14 14 21 ··· 21 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 21 21 2 2 2 2 42 42 42 2 2 2 2 4 4 4 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + - + + + + + - - - image C1 C2 C2 C2 S3 Q8 D6 D7 D14 D21 D42 S3×Q8 Q8×D7 Q8×D21 kernel Q8×D21 Dic42 C4×D21 Q8×C21 C7×Q8 D21 C28 C3×Q8 C12 Q8 C4 C7 C3 C1 # reps 1 3 3 1 1 2 3 3 9 6 18 1 3 6

Matrix representation of Q8×D21 in GL4(𝔽337) generated by

 336 0 0 0 0 336 0 0 0 0 1 96 0 0 7 336
,
 1 0 0 0 0 1 0 0 0 0 41 46 0 0 66 296
,
 101 59 0 0 278 89 0 0 0 0 1 0 0 0 0 1
,
 237 107 0 0 253 100 0 0 0 0 336 0 0 0 0 336
G:=sub<GL(4,GF(337))| [336,0,0,0,0,336,0,0,0,0,1,7,0,0,96,336],[1,0,0,0,0,1,0,0,0,0,41,66,0,0,46,296],[101,278,0,0,59,89,0,0,0,0,1,0,0,0,0,1],[237,253,0,0,107,100,0,0,0,0,336,0,0,0,0,336] >;

Q8×D21 in GAP, Magma, Sage, TeX

Q_8\times D_{21}
% in TeX

G:=Group("Q8xD21");
// GroupNames label

G:=SmallGroup(336,200);
// by ID

G=gap.SmallGroup(336,200);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,55,116,50,964,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^21=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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