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## G = D4×C72order 392 = 23·72

### Direct product of C72 and D4

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

Aliases: D4×C72, C283C14, C1421C2, C2.1C142, C4⋊(C7×C14), (C7×C28)⋊5C2, C22⋊(C7×C14), (C2×C14)⋊1C14, C14.8(C2×C14), (C7×C14).16C22, SmallGroup(392,34)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C72
 Chief series C1 — C2 — C14 — C7×C14 — C142 — D4×C72
 Lower central C1 — C2 — D4×C72
 Upper central C1 — C7×C14 — D4×C72

Generators and relations for D4×C72
G = < a,b,c,d | a7=b7=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 100 in 80 conjugacy classes, 60 normal (8 characteristic)
C1, C2, C2, C4, C22, C7, D4, C14, C14, C28, C2×C14, C72, C7×D4, C7×C14, C7×C14, C7×C28, C142, D4×C72
Quotients: C1, C2, C22, C7, D4, C14, C2×C14, C72, C7×D4, C7×C14, C142, D4×C72

Smallest permutation representation of D4×C72
On 196 points
Generators in S196
(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)(169 170 171 172 173 174 175)(176 177 178 179 180 181 182)(183 184 185 186 187 188 189)(190 191 192 193 194 195 196)
(1 111 121 138 13 26 36)(2 112 122 139 14 27 37)(3 106 123 140 8 28 38)(4 107 124 134 9 22 39)(5 108 125 135 10 23 40)(6 109 126 136 11 24 41)(7 110 120 137 12 25 42)(15 32 49 183 150 167 177)(16 33 43 184 151 168 178)(17 34 44 185 152 162 179)(18 35 45 186 153 163 180)(19 29 46 187 154 164 181)(20 30 47 188 148 165 182)(21 31 48 189 149 166 176)(50 142 159 169 87 57 74)(51 143 160 170 88 58 75)(52 144 161 171 89 59 76)(53 145 155 172 90 60 77)(54 146 156 173 91 61 71)(55 147 157 174 85 62 72)(56 141 158 175 86 63 73)(64 81 98 102 119 129 194)(65 82 92 103 113 130 195)(66 83 93 104 114 131 196)(67 84 94 105 115 132 190)(68 78 95 99 116 133 191)(69 79 96 100 117 127 192)(70 80 97 101 118 128 193)
(1 96 172 176)(2 97 173 177)(3 98 174 178)(4 92 175 179)(5 93 169 180)(6 94 170 181)(7 95 171 182)(8 194 55 184)(9 195 56 185)(10 196 50 186)(11 190 51 187)(12 191 52 188)(13 192 53 189)(14 193 54 183)(15 112 101 91)(16 106 102 85)(17 107 103 86)(18 108 104 87)(19 109 105 88)(20 110 99 89)(21 111 100 90)(22 65 141 152)(23 66 142 153)(24 67 143 154)(25 68 144 148)(26 69 145 149)(27 70 146 150)(28 64 147 151)(29 126 115 58)(30 120 116 59)(31 121 117 60)(32 122 118 61)(33 123 119 62)(34 124 113 63)(35 125 114 57)(36 79 155 166)(37 80 156 167)(38 81 157 168)(39 82 158 162)(40 83 159 163)(41 84 160 164)(42 78 161 165)(43 140 129 72)(44 134 130 73)(45 135 131 74)(46 136 132 75)(47 137 133 76)(48 138 127 77)(49 139 128 71)
(1 176)(2 177)(3 178)(4 179)(5 180)(6 181)(7 182)(8 184)(9 185)(10 186)(11 187)(12 188)(13 189)(14 183)(15 112)(16 106)(17 107)(18 108)(19 109)(20 110)(21 111)(22 152)(23 153)(24 154)(25 148)(26 149)(27 150)(28 151)(29 126)(30 120)(31 121)(32 122)(33 123)(34 124)(35 125)(36 166)(37 167)(38 168)(39 162)(40 163)(41 164)(42 165)(43 140)(44 134)(45 135)(46 136)(47 137)(48 138)(49 139)(50 196)(51 190)(52 191)(53 192)(54 193)(55 194)(56 195)(57 114)(58 115)(59 116)(60 117)(61 118)(62 119)(63 113)(64 147)(65 141)(66 142)(67 143)(68 144)(69 145)(70 146)(71 128)(72 129)(73 130)(74 131)(75 132)(76 133)(77 127)(78 161)(79 155)(80 156)(81 157)(82 158)(83 159)(84 160)(85 102)(86 103)(87 104)(88 105)(89 99)(90 100)(91 101)(92 175)(93 169)(94 170)(95 171)(96 172)(97 173)(98 174)

G:=sub<Sym(196)| (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)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196), (1,111,121,138,13,26,36)(2,112,122,139,14,27,37)(3,106,123,140,8,28,38)(4,107,124,134,9,22,39)(5,108,125,135,10,23,40)(6,109,126,136,11,24,41)(7,110,120,137,12,25,42)(15,32,49,183,150,167,177)(16,33,43,184,151,168,178)(17,34,44,185,152,162,179)(18,35,45,186,153,163,180)(19,29,46,187,154,164,181)(20,30,47,188,148,165,182)(21,31,48,189,149,166,176)(50,142,159,169,87,57,74)(51,143,160,170,88,58,75)(52,144,161,171,89,59,76)(53,145,155,172,90,60,77)(54,146,156,173,91,61,71)(55,147,157,174,85,62,72)(56,141,158,175,86,63,73)(64,81,98,102,119,129,194)(65,82,92,103,113,130,195)(66,83,93,104,114,131,196)(67,84,94,105,115,132,190)(68,78,95,99,116,133,191)(69,79,96,100,117,127,192)(70,80,97,101,118,128,193), (1,96,172,176)(2,97,173,177)(3,98,174,178)(4,92,175,179)(5,93,169,180)(6,94,170,181)(7,95,171,182)(8,194,55,184)(9,195,56,185)(10,196,50,186)(11,190,51,187)(12,191,52,188)(13,192,53,189)(14,193,54,183)(15,112,101,91)(16,106,102,85)(17,107,103,86)(18,108,104,87)(19,109,105,88)(20,110,99,89)(21,111,100,90)(22,65,141,152)(23,66,142,153)(24,67,143,154)(25,68,144,148)(26,69,145,149)(27,70,146,150)(28,64,147,151)(29,126,115,58)(30,120,116,59)(31,121,117,60)(32,122,118,61)(33,123,119,62)(34,124,113,63)(35,125,114,57)(36,79,155,166)(37,80,156,167)(38,81,157,168)(39,82,158,162)(40,83,159,163)(41,84,160,164)(42,78,161,165)(43,140,129,72)(44,134,130,73)(45,135,131,74)(46,136,132,75)(47,137,133,76)(48,138,127,77)(49,139,128,71), (1,176)(2,177)(3,178)(4,179)(5,180)(6,181)(7,182)(8,184)(9,185)(10,186)(11,187)(12,188)(13,189)(14,183)(15,112)(16,106)(17,107)(18,108)(19,109)(20,110)(21,111)(22,152)(23,153)(24,154)(25,148)(26,149)(27,150)(28,151)(29,126)(30,120)(31,121)(32,122)(33,123)(34,124)(35,125)(36,166)(37,167)(38,168)(39,162)(40,163)(41,164)(42,165)(43,140)(44,134)(45,135)(46,136)(47,137)(48,138)(49,139)(50,196)(51,190)(52,191)(53,192)(54,193)(55,194)(56,195)(57,114)(58,115)(59,116)(60,117)(61,118)(62,119)(63,113)(64,147)(65,141)(66,142)(67,143)(68,144)(69,145)(70,146)(71,128)(72,129)(73,130)(74,131)(75,132)(76,133)(77,127)(78,161)(79,155)(80,156)(81,157)(82,158)(83,159)(84,160)(85,102)(86,103)(87,104)(88,105)(89,99)(90,100)(91,101)(92,175)(93,169)(94,170)(95,171)(96,172)(97,173)(98,174)>;

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)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196), (1,111,121,138,13,26,36)(2,112,122,139,14,27,37)(3,106,123,140,8,28,38)(4,107,124,134,9,22,39)(5,108,125,135,10,23,40)(6,109,126,136,11,24,41)(7,110,120,137,12,25,42)(15,32,49,183,150,167,177)(16,33,43,184,151,168,178)(17,34,44,185,152,162,179)(18,35,45,186,153,163,180)(19,29,46,187,154,164,181)(20,30,47,188,148,165,182)(21,31,48,189,149,166,176)(50,142,159,169,87,57,74)(51,143,160,170,88,58,75)(52,144,161,171,89,59,76)(53,145,155,172,90,60,77)(54,146,156,173,91,61,71)(55,147,157,174,85,62,72)(56,141,158,175,86,63,73)(64,81,98,102,119,129,194)(65,82,92,103,113,130,195)(66,83,93,104,114,131,196)(67,84,94,105,115,132,190)(68,78,95,99,116,133,191)(69,79,96,100,117,127,192)(70,80,97,101,118,128,193), (1,96,172,176)(2,97,173,177)(3,98,174,178)(4,92,175,179)(5,93,169,180)(6,94,170,181)(7,95,171,182)(8,194,55,184)(9,195,56,185)(10,196,50,186)(11,190,51,187)(12,191,52,188)(13,192,53,189)(14,193,54,183)(15,112,101,91)(16,106,102,85)(17,107,103,86)(18,108,104,87)(19,109,105,88)(20,110,99,89)(21,111,100,90)(22,65,141,152)(23,66,142,153)(24,67,143,154)(25,68,144,148)(26,69,145,149)(27,70,146,150)(28,64,147,151)(29,126,115,58)(30,120,116,59)(31,121,117,60)(32,122,118,61)(33,123,119,62)(34,124,113,63)(35,125,114,57)(36,79,155,166)(37,80,156,167)(38,81,157,168)(39,82,158,162)(40,83,159,163)(41,84,160,164)(42,78,161,165)(43,140,129,72)(44,134,130,73)(45,135,131,74)(46,136,132,75)(47,137,133,76)(48,138,127,77)(49,139,128,71), (1,176)(2,177)(3,178)(4,179)(5,180)(6,181)(7,182)(8,184)(9,185)(10,186)(11,187)(12,188)(13,189)(14,183)(15,112)(16,106)(17,107)(18,108)(19,109)(20,110)(21,111)(22,152)(23,153)(24,154)(25,148)(26,149)(27,150)(28,151)(29,126)(30,120)(31,121)(32,122)(33,123)(34,124)(35,125)(36,166)(37,167)(38,168)(39,162)(40,163)(41,164)(42,165)(43,140)(44,134)(45,135)(46,136)(47,137)(48,138)(49,139)(50,196)(51,190)(52,191)(53,192)(54,193)(55,194)(56,195)(57,114)(58,115)(59,116)(60,117)(61,118)(62,119)(63,113)(64,147)(65,141)(66,142)(67,143)(68,144)(69,145)(70,146)(71,128)(72,129)(73,130)(74,131)(75,132)(76,133)(77,127)(78,161)(79,155)(80,156)(81,157)(82,158)(83,159)(84,160)(85,102)(86,103)(87,104)(88,105)(89,99)(90,100)(91,101)(92,175)(93,169)(94,170)(95,171)(96,172)(97,173)(98,174) );

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),(169,170,171,172,173,174,175),(176,177,178,179,180,181,182),(183,184,185,186,187,188,189),(190,191,192,193,194,195,196)], [(1,111,121,138,13,26,36),(2,112,122,139,14,27,37),(3,106,123,140,8,28,38),(4,107,124,134,9,22,39),(5,108,125,135,10,23,40),(6,109,126,136,11,24,41),(7,110,120,137,12,25,42),(15,32,49,183,150,167,177),(16,33,43,184,151,168,178),(17,34,44,185,152,162,179),(18,35,45,186,153,163,180),(19,29,46,187,154,164,181),(20,30,47,188,148,165,182),(21,31,48,189,149,166,176),(50,142,159,169,87,57,74),(51,143,160,170,88,58,75),(52,144,161,171,89,59,76),(53,145,155,172,90,60,77),(54,146,156,173,91,61,71),(55,147,157,174,85,62,72),(56,141,158,175,86,63,73),(64,81,98,102,119,129,194),(65,82,92,103,113,130,195),(66,83,93,104,114,131,196),(67,84,94,105,115,132,190),(68,78,95,99,116,133,191),(69,79,96,100,117,127,192),(70,80,97,101,118,128,193)], [(1,96,172,176),(2,97,173,177),(3,98,174,178),(4,92,175,179),(5,93,169,180),(6,94,170,181),(7,95,171,182),(8,194,55,184),(9,195,56,185),(10,196,50,186),(11,190,51,187),(12,191,52,188),(13,192,53,189),(14,193,54,183),(15,112,101,91),(16,106,102,85),(17,107,103,86),(18,108,104,87),(19,109,105,88),(20,110,99,89),(21,111,100,90),(22,65,141,152),(23,66,142,153),(24,67,143,154),(25,68,144,148),(26,69,145,149),(27,70,146,150),(28,64,147,151),(29,126,115,58),(30,120,116,59),(31,121,117,60),(32,122,118,61),(33,123,119,62),(34,124,113,63),(35,125,114,57),(36,79,155,166),(37,80,156,167),(38,81,157,168),(39,82,158,162),(40,83,159,163),(41,84,160,164),(42,78,161,165),(43,140,129,72),(44,134,130,73),(45,135,131,74),(46,136,132,75),(47,137,133,76),(48,138,127,77),(49,139,128,71)], [(1,176),(2,177),(3,178),(4,179),(5,180),(6,181),(7,182),(8,184),(9,185),(10,186),(11,187),(12,188),(13,189),(14,183),(15,112),(16,106),(17,107),(18,108),(19,109),(20,110),(21,111),(22,152),(23,153),(24,154),(25,148),(26,149),(27,150),(28,151),(29,126),(30,120),(31,121),(32,122),(33,123),(34,124),(35,125),(36,166),(37,167),(38,168),(39,162),(40,163),(41,164),(42,165),(43,140),(44,134),(45,135),(46,136),(47,137),(48,138),(49,139),(50,196),(51,190),(52,191),(53,192),(54,193),(55,194),(56,195),(57,114),(58,115),(59,116),(60,117),(61,118),(62,119),(63,113),(64,147),(65,141),(66,142),(67,143),(68,144),(69,145),(70,146),(71,128),(72,129),(73,130),(74,131),(75,132),(76,133),(77,127),(78,161),(79,155),(80,156),(81,157),(82,158),(83,159),(84,160),(85,102),(86,103),(87,104),(88,105),(89,99),(90,100),(91,101),(92,175),(93,169),(94,170),(95,171),(96,172),(97,173),(98,174)]])

245 conjugacy classes

 class 1 2A 2B 2C 4 7A ··· 7AV 14A ··· 14AV 14AW ··· 14EN 28A ··· 28AV order 1 2 2 2 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

245 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C7 C14 C14 D4 C7×D4 kernel D4×C72 C7×C28 C142 C7×D4 C28 C2×C14 C72 C7 # reps 1 1 2 48 48 96 1 48

Matrix representation of D4×C72 in GL3(𝔽29) generated by

 20 0 0 0 16 0 0 0 16
,
 25 0 0 0 16 0 0 0 16
,
 28 0 0 0 0 1 0 28 0
,
 1 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(29))| [20,0,0,0,16,0,0,0,16],[25,0,0,0,16,0,0,0,16],[28,0,0,0,0,28,0,1,0],[1,0,0,0,0,1,0,1,0] >;

D4×C72 in GAP, Magma, Sage, TeX

D_4\times C_7^2
% in TeX

G:=Group("D4xC7^2");
// GroupNames label

G:=SmallGroup(392,34);
// by ID

G=gap.SmallGroup(392,34);
# by ID

G:=PCGroup([5,-2,-2,-7,-7,-2,1981]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^7=c^4=d^2=1,a*b=b*a,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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