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## G = C7×D32order 448 = 26·7

### Direct product of C7 and D32

direct product, metacyclic, nilpotent (class 5), monomial, 2-elementary

Aliases: C7×D32, C321C14, C2243C2, D161C14, C28.39D8, C56.68D4, C14.15D16, C112.19C22, C8.5(C7×D4), C4.1(C7×D8), (C7×D16)⋊5C2, C2.3(C7×D16), C16.2(C2×C14), SmallGroup(448,175)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C16 — C7×D32
 Chief series C1 — C2 — C4 — C8 — C16 — C112 — C7×D16 — C7×D32
 Lower central C1 — C2 — C4 — C8 — C16 — C7×D32
 Upper central C1 — C14 — C28 — C56 — C112 — C7×D32

Generators and relations for C7×D32
G = < a,b,c | a7=b32=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C7×D32
On 224 points
Generators in S224
(1 168 72 222 137 58 101)(2 169 73 223 138 59 102)(3 170 74 224 139 60 103)(4 171 75 193 140 61 104)(5 172 76 194 141 62 105)(6 173 77 195 142 63 106)(7 174 78 196 143 64 107)(8 175 79 197 144 33 108)(9 176 80 198 145 34 109)(10 177 81 199 146 35 110)(11 178 82 200 147 36 111)(12 179 83 201 148 37 112)(13 180 84 202 149 38 113)(14 181 85 203 150 39 114)(15 182 86 204 151 40 115)(16 183 87 205 152 41 116)(17 184 88 206 153 42 117)(18 185 89 207 154 43 118)(19 186 90 208 155 44 119)(20 187 91 209 156 45 120)(21 188 92 210 157 46 121)(22 189 93 211 158 47 122)(23 190 94 212 159 48 123)(24 191 95 213 160 49 124)(25 192 96 214 129 50 125)(26 161 65 215 130 51 126)(27 162 66 216 131 52 127)(28 163 67 217 132 53 128)(29 164 68 218 133 54 97)(30 165 69 219 134 55 98)(31 166 70 220 135 56 99)(32 167 71 221 136 57 100)
(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)(193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(52 64)(53 63)(54 62)(55 61)(56 60)(57 59)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(80 96)(81 95)(82 94)(83 93)(84 92)(85 91)(86 90)(87 89)(97 105)(98 104)(99 103)(100 102)(106 128)(107 127)(108 126)(109 125)(110 124)(111 123)(112 122)(113 121)(114 120)(115 119)(116 118)(129 145)(130 144)(131 143)(132 142)(133 141)(134 140)(135 139)(136 138)(146 160)(147 159)(148 158)(149 157)(150 156)(151 155)(152 154)(161 175)(162 174)(163 173)(164 172)(165 171)(166 170)(167 169)(176 192)(177 191)(178 190)(179 189)(180 188)(181 187)(182 186)(183 185)(193 219)(194 218)(195 217)(196 216)(197 215)(198 214)(199 213)(200 212)(201 211)(202 210)(203 209)(204 208)(205 207)(220 224)(221 223)

G:=sub<Sym(224)| (1,168,72,222,137,58,101)(2,169,73,223,138,59,102)(3,170,74,224,139,60,103)(4,171,75,193,140,61,104)(5,172,76,194,141,62,105)(6,173,77,195,142,63,106)(7,174,78,196,143,64,107)(8,175,79,197,144,33,108)(9,176,80,198,145,34,109)(10,177,81,199,146,35,110)(11,178,82,200,147,36,111)(12,179,83,201,148,37,112)(13,180,84,202,149,38,113)(14,181,85,203,150,39,114)(15,182,86,204,151,40,115)(16,183,87,205,152,41,116)(17,184,88,206,153,42,117)(18,185,89,207,154,43,118)(19,186,90,208,155,44,119)(20,187,91,209,156,45,120)(21,188,92,210,157,46,121)(22,189,93,211,158,47,122)(23,190,94,212,159,48,123)(24,191,95,213,160,49,124)(25,192,96,214,129,50,125)(26,161,65,215,130,51,126)(27,162,66,216,131,52,127)(28,163,67,217,132,53,128)(29,164,68,218,133,54,97)(30,165,69,219,134,55,98)(31,166,70,220,135,56,99)(32,167,71,221,136,57,100), (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)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(80,96)(81,95)(82,94)(83,93)(84,92)(85,91)(86,90)(87,89)(97,105)(98,104)(99,103)(100,102)(106,128)(107,127)(108,126)(109,125)(110,124)(111,123)(112,122)(113,121)(114,120)(115,119)(116,118)(129,145)(130,144)(131,143)(132,142)(133,141)(134,140)(135,139)(136,138)(146,160)(147,159)(148,158)(149,157)(150,156)(151,155)(152,154)(161,175)(162,174)(163,173)(164,172)(165,171)(166,170)(167,169)(176,192)(177,191)(178,190)(179,189)(180,188)(181,187)(182,186)(183,185)(193,219)(194,218)(195,217)(196,216)(197,215)(198,214)(199,213)(200,212)(201,211)(202,210)(203,209)(204,208)(205,207)(220,224)(221,223)>;

G:=Group( (1,168,72,222,137,58,101)(2,169,73,223,138,59,102)(3,170,74,224,139,60,103)(4,171,75,193,140,61,104)(5,172,76,194,141,62,105)(6,173,77,195,142,63,106)(7,174,78,196,143,64,107)(8,175,79,197,144,33,108)(9,176,80,198,145,34,109)(10,177,81,199,146,35,110)(11,178,82,200,147,36,111)(12,179,83,201,148,37,112)(13,180,84,202,149,38,113)(14,181,85,203,150,39,114)(15,182,86,204,151,40,115)(16,183,87,205,152,41,116)(17,184,88,206,153,42,117)(18,185,89,207,154,43,118)(19,186,90,208,155,44,119)(20,187,91,209,156,45,120)(21,188,92,210,157,46,121)(22,189,93,211,158,47,122)(23,190,94,212,159,48,123)(24,191,95,213,160,49,124)(25,192,96,214,129,50,125)(26,161,65,215,130,51,126)(27,162,66,216,131,52,127)(28,163,67,217,132,53,128)(29,164,68,218,133,54,97)(30,165,69,219,134,55,98)(31,166,70,220,135,56,99)(32,167,71,221,136,57,100), (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)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(80,96)(81,95)(82,94)(83,93)(84,92)(85,91)(86,90)(87,89)(97,105)(98,104)(99,103)(100,102)(106,128)(107,127)(108,126)(109,125)(110,124)(111,123)(112,122)(113,121)(114,120)(115,119)(116,118)(129,145)(130,144)(131,143)(132,142)(133,141)(134,140)(135,139)(136,138)(146,160)(147,159)(148,158)(149,157)(150,156)(151,155)(152,154)(161,175)(162,174)(163,173)(164,172)(165,171)(166,170)(167,169)(176,192)(177,191)(178,190)(179,189)(180,188)(181,187)(182,186)(183,185)(193,219)(194,218)(195,217)(196,216)(197,215)(198,214)(199,213)(200,212)(201,211)(202,210)(203,209)(204,208)(205,207)(220,224)(221,223) );

G=PermutationGroup([(1,168,72,222,137,58,101),(2,169,73,223,138,59,102),(3,170,74,224,139,60,103),(4,171,75,193,140,61,104),(5,172,76,194,141,62,105),(6,173,77,195,142,63,106),(7,174,78,196,143,64,107),(8,175,79,197,144,33,108),(9,176,80,198,145,34,109),(10,177,81,199,146,35,110),(11,178,82,200,147,36,111),(12,179,83,201,148,37,112),(13,180,84,202,149,38,113),(14,181,85,203,150,39,114),(15,182,86,204,151,40,115),(16,183,87,205,152,41,116),(17,184,88,206,153,42,117),(18,185,89,207,154,43,118),(19,186,90,208,155,44,119),(20,187,91,209,156,45,120),(21,188,92,210,157,46,121),(22,189,93,211,158,47,122),(23,190,94,212,159,48,123),(24,191,95,213,160,49,124),(25,192,96,214,129,50,125),(26,161,65,215,130,51,126),(27,162,66,216,131,52,127),(28,163,67,217,132,53,128),(29,164,68,218,133,54,97),(30,165,69,219,134,55,98),(31,166,70,220,135,56,99),(32,167,71,221,136,57,100)], [(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),(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(52,64),(53,63),(54,62),(55,61),(56,60),(57,59),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(80,96),(81,95),(82,94),(83,93),(84,92),(85,91),(86,90),(87,89),(97,105),(98,104),(99,103),(100,102),(106,128),(107,127),(108,126),(109,125),(110,124),(111,123),(112,122),(113,121),(114,120),(115,119),(116,118),(129,145),(130,144),(131,143),(132,142),(133,141),(134,140),(135,139),(136,138),(146,160),(147,159),(148,158),(149,157),(150,156),(151,155),(152,154),(161,175),(162,174),(163,173),(164,172),(165,171),(166,170),(167,169),(176,192),(177,191),(178,190),(179,189),(180,188),(181,187),(182,186),(183,185),(193,219),(194,218),(195,217),(196,216),(197,215),(198,214),(199,213),(200,212),(201,211),(202,210),(203,209),(204,208),(205,207),(220,224),(221,223)])

133 conjugacy classes

 class 1 2A 2B 2C 4 7A ··· 7F 8A 8B 14A ··· 14F 14G ··· 14R 16A 16B 16C 16D 28A ··· 28F 32A ··· 32H 56A ··· 56L 112A ··· 112X 224A ··· 224AV order 1 2 2 2 4 7 ··· 7 8 8 14 ··· 14 14 ··· 14 16 16 16 16 28 ··· 28 32 ··· 32 56 ··· 56 112 ··· 112 224 ··· 224 size 1 1 16 16 2 1 ··· 1 2 2 1 ··· 1 16 ··· 16 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

133 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C7 C14 C14 D4 D8 D16 C7×D4 D32 C7×D8 C7×D16 C7×D32 kernel C7×D32 C224 C7×D16 D32 C32 D16 C56 C28 C14 C8 C7 C4 C2 C1 # reps 1 1 2 6 6 12 1 2 4 6 8 12 24 48

Matrix representation of C7×D32 in GL2(𝔽449) generated by

 324 0 0 324
,
 409 113 336 409
,
 1 0 0 448
G:=sub<GL(2,GF(449))| [324,0,0,324],[409,336,113,409],[1,0,0,448] >;

C7×D32 in GAP, Magma, Sage, TeX

C_7\times D_{32}
% in TeX

G:=Group("C7xD32");
// GroupNames label

G:=SmallGroup(448,175);
// by ID

G=gap.SmallGroup(448,175);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,421,2355,1186,192,5884,2951,242,14117,7068,124]);
// Polycyclic

G:=Group<a,b,c|a^7=b^32=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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