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## G = C4○D4×C21order 336 = 24·3·7

### Direct product of C21 and C4○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C21
 Chief series C1 — C2 — C14 — C42 — C2×C42 — D4×C21 — C4○D4×C21
 Lower central C1 — C2 — C4○D4×C21
 Upper central C1 — C84 — C4○D4×C21

Generators and relations for C4○D4×C21
G = < a,b,c,d | a21=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 92 in 80 conjugacy classes, 68 normal (20 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C6, C6 [×3], C7, C2×C4 [×3], D4 [×3], Q8, C12, C12 [×3], C2×C6 [×3], C14, C14 [×3], C4○D4, C21, C2×C12 [×3], C3×D4 [×3], C3×Q8, C28, C28 [×3], C2×C14 [×3], C42, C42 [×3], C3×C4○D4, C2×C28 [×3], C7×D4 [×3], C7×Q8, C84, C84 [×3], C2×C42 [×3], C7×C4○D4, C2×C84 [×3], D4×C21 [×3], Q8×C21, C4○D4×C21
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], C7, C23, C2×C6 [×7], C14 [×7], C4○D4, C21, C22×C6, C2×C14 [×7], C42 [×7], C3×C4○D4, C22×C14, C2×C42 [×7], C7×C4○D4, C22×C42, C4○D4×C21

Smallest permutation representation of C4○D4×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)
(1 158 141 33)(2 159 142 34)(3 160 143 35)(4 161 144 36)(5 162 145 37)(6 163 146 38)(7 164 147 39)(8 165 127 40)(9 166 128 41)(10 167 129 42)(11 168 130 22)(12 148 131 23)(13 149 132 24)(14 150 133 25)(15 151 134 26)(16 152 135 27)(17 153 136 28)(18 154 137 29)(19 155 138 30)(20 156 139 31)(21 157 140 32)(43 115 74 104)(44 116 75 105)(45 117 76 85)(46 118 77 86)(47 119 78 87)(48 120 79 88)(49 121 80 89)(50 122 81 90)(51 123 82 91)(52 124 83 92)(53 125 84 93)(54 126 64 94)(55 106 65 95)(56 107 66 96)(57 108 67 97)(58 109 68 98)(59 110 69 99)(60 111 70 100)(61 112 71 101)(62 113 72 102)(63 114 73 103)
(1 33 141 158)(2 34 142 159)(3 35 143 160)(4 36 144 161)(5 37 145 162)(6 38 146 163)(7 39 147 164)(8 40 127 165)(9 41 128 166)(10 42 129 167)(11 22 130 168)(12 23 131 148)(13 24 132 149)(14 25 133 150)(15 26 134 151)(16 27 135 152)(17 28 136 153)(18 29 137 154)(19 30 138 155)(20 31 139 156)(21 32 140 157)(43 115 74 104)(44 116 75 105)(45 117 76 85)(46 118 77 86)(47 119 78 87)(48 120 79 88)(49 121 80 89)(50 122 81 90)(51 123 82 91)(52 124 83 92)(53 125 84 93)(54 126 64 94)(55 106 65 95)(56 107 66 96)(57 108 67 97)(58 109 68 98)(59 110 69 99)(60 111 70 100)(61 112 71 101)(62 113 72 102)(63 114 73 103)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 88)(23 89)(24 90)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 97)(32 98)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 105)(40 85)(41 86)(42 87)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)(73 145)(74 146)(75 147)(76 127)(77 128)(78 129)(79 130)(80 131)(81 132)(82 133)(83 134)(84 135)(106 154)(107 155)(108 156)(109 157)(110 158)(111 159)(112 160)(113 161)(114 162)(115 163)(116 164)(117 165)(118 166)(119 167)(120 168)(121 148)(122 149)(123 150)(124 151)(125 152)(126 153)

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

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

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

210 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C ··· 6H 7A ··· 7F 12A 12B 12C 12D 12E ··· 12J 14A ··· 14F 14G ··· 14X 21A ··· 21L 28A ··· 28L 28M ··· 28AD 42A ··· 42L 42M ··· 42AV 84A ··· 84X 84Y ··· 84BH order 1 2 2 2 2 3 3 4 4 4 4 4 6 6 6 ··· 6 7 ··· 7 12 12 12 12 12 ··· 12 14 ··· 14 14 ··· 14 21 ··· 21 28 ··· 28 28 ··· 28 42 ··· 42 42 ··· 42 84 ··· 84 84 ··· 84 size 1 1 2 2 2 1 1 1 1 2 2 2 1 1 2 ··· 2 1 ··· 1 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

210 irreducible representations

 dim 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 C2 C3 C6 C6 C6 C7 C14 C14 C14 C21 C42 C42 C42 C4○D4 C3×C4○D4 C7×C4○D4 C4○D4×C21 kernel C4○D4×C21 C2×C84 D4×C21 Q8×C21 C7×C4○D4 C2×C28 C7×D4 C7×Q8 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C4○D4 C2×C4 D4 Q8 C21 C7 C3 C1 # reps 1 3 3 1 2 6 6 2 6 18 18 6 12 36 36 12 2 4 12 24

Matrix representation of C4○D4×C21 in GL3(𝔽337) generated by

 128 0 0 0 64 0 0 0 64
,
 1 0 0 0 189 0 0 0 189
,
 336 0 0 0 148 0 0 148 189
,
 336 0 0 0 189 296 0 189 148
G:=sub<GL(3,GF(337))| [128,0,0,0,64,0,0,0,64],[1,0,0,0,189,0,0,0,189],[336,0,0,0,148,148,0,0,189],[336,0,0,0,189,189,0,296,148] >;

C4○D4×C21 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_{21}
% in TeX

G:=Group("C4oD4xC21");
// GroupNames label

G:=SmallGroup(336,207);
// by ID

G=gap.SmallGroup(336,207);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-7,-2,2041,770]);
// Polycyclic

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

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