Copied to
clipboard

## G = C2×D21⋊C4order 336 = 24·3·7

### Direct product of C2 and D21⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C2×D21⋊C4
 Chief series C1 — C7 — C21 — C42 — C3×Dic7 — D21⋊C4 — C2×D21⋊C4
 Lower central C21 — C2×D21⋊C4
 Upper central C1 — C22

Generators and relations for C2×D21⋊C4
G = < a,b,c,d | a2=b21=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b8, dcd-1=b7c >

Subgroups: 604 in 108 conjugacy classes, 48 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C7, C2×C4, C23, Dic3, C12, D6, C2×C6, D7, C14, C14, C22×C4, C21, C4×S3, C2×Dic3, C2×C12, C22×S3, Dic7, C28, D14, C2×C14, D21, C42, C42, S3×C2×C4, C4×D7, C2×Dic7, C2×C28, C22×D7, C7×Dic3, C3×Dic7, D42, C2×C42, C2×C4×D7, D21⋊C4, C6×Dic7, Dic3×C14, C22×D21, C2×D21⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, D7, C22×C4, C4×S3, C22×S3, D14, S3×C2×C4, C4×D7, C22×D7, S3×D7, C2×C4×D7, D21⋊C4, C2×S3×D7, C2×D21⋊C4

Smallest permutation representation of C2×D21⋊C4
On 168 points
Generators in S168
(1 60)(2 61)(3 62)(4 63)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(85 128)(86 129)(87 130)(88 131)(89 132)(90 133)(91 134)(92 135)(93 136)(94 137)(95 138)(96 139)(97 140)(98 141)(99 142)(100 143)(101 144)(102 145)(103 146)(104 147)(105 127)(106 160)(107 161)(108 162)(109 163)(110 164)(111 165)(112 166)(113 167)(114 168)(115 148)(116 149)(117 150)(118 151)(119 152)(120 153)(121 154)(122 155)(123 156)(124 157)(125 158)(126 159)
(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 74)(2 73)(3 72)(4 71)(5 70)(6 69)(7 68)(8 67)(9 66)(10 65)(11 64)(12 84)(13 83)(14 82)(15 81)(16 80)(17 79)(18 78)(19 77)(20 76)(21 75)(22 49)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(29 63)(30 62)(31 61)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(85 161)(86 160)(87 159)(88 158)(89 157)(90 156)(91 155)(92 154)(93 153)(94 152)(95 151)(96 150)(97 149)(98 148)(99 168)(100 167)(101 166)(102 165)(103 164)(104 163)(105 162)(106 129)(107 128)(108 127)(109 147)(110 146)(111 145)(112 144)(113 143)(114 142)(115 141)(116 140)(117 139)(118 138)(119 137)(120 136)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)
(1 126 33 88)(2 113 34 96)(3 121 35 104)(4 108 36 91)(5 116 37 99)(6 124 38 86)(7 111 39 94)(8 119 40 102)(9 106 41 89)(10 114 42 97)(11 122 22 105)(12 109 23 92)(13 117 24 100)(14 125 25 87)(15 112 26 95)(16 120 27 103)(17 107 28 90)(18 115 29 98)(19 123 30 85)(20 110 31 93)(21 118 32 101)(43 149 79 142)(44 157 80 129)(45 165 81 137)(46 152 82 145)(47 160 83 132)(48 168 84 140)(49 155 64 127)(50 163 65 135)(51 150 66 143)(52 158 67 130)(53 166 68 138)(54 153 69 146)(55 161 70 133)(56 148 71 141)(57 156 72 128)(58 164 73 136)(59 151 74 144)(60 159 75 131)(61 167 76 139)(62 154 77 147)(63 162 78 134)

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D6 D6 D7 C4×S3 D14 D14 C4×D7 S3×D7 D21⋊C4 C2×S3×D7 kernel C2×D21⋊C4 D21⋊C4 C6×Dic7 Dic3×C14 C22×D21 D42 C2×Dic7 Dic7 C2×C14 C2×Dic3 C14 Dic3 C2×C6 C6 C22 C2 C2 # reps 1 4 1 1 1 8 1 2 1 3 4 6 3 12 3 6 3

Matrix representation of C2×D21⋊C4 in GL6(𝔽337)

 336 0 0 0 0 0 0 336 0 0 0 0 0 0 336 0 0 0 0 0 0 336 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 143 143 0 0 0 0 51 84 0 0 0 0 0 0 336 303 0 0 0 0 34 144 0 0 0 0 0 0 0 336 0 0 0 0 1 336
,
 1 0 0 0 0 0 75 336 0 0 0 0 0 0 1 34 0 0 0 0 0 336 0 0 0 0 0 0 1 336 0 0 0 0 0 336
,
 189 0 0 0 0 0 0 189 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 336 0 0 0 0 336 0

G:=sub<GL(6,GF(337))| [336,0,0,0,0,0,0,336,0,0,0,0,0,0,336,0,0,0,0,0,0,336,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[143,51,0,0,0,0,143,84,0,0,0,0,0,0,336,34,0,0,0,0,303,144,0,0,0,0,0,0,0,1,0,0,0,0,336,336],[1,75,0,0,0,0,0,336,0,0,0,0,0,0,1,0,0,0,0,0,34,336,0,0,0,0,0,0,1,0,0,0,0,0,336,336],[189,0,0,0,0,0,0,189,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,336,0,0,0,0,336,0] >;

C2×D21⋊C4 in GAP, Magma, Sage, TeX

C_2\times D_{21}\rtimes C_4
% in TeX

G:=Group("C2xD21:C4");
// GroupNames label

G:=SmallGroup(336,156);
// by ID

G=gap.SmallGroup(336,156);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,48,55,490,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^21=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^8,d*c*d^-1=b^7*c>;
// generators/relations

׿
×
𝔽