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G = D7×C32order 448 = 26·7

Direct product of C32 and D7

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D7×C32, C2245C2, D14.2C16, C16.19D14, Dic7.2C16, C112.24C22, C7⋊C326C2, C71(C2×C32), C7⋊C8.4C8, C7⋊C16.3C4, (C4×D7).4C8, (C8×D7).5C4, C8.36(C4×D7), C2.1(D7×C16), C4.16(C8×D7), C28.21(C2×C8), C56.53(C2×C4), C14.1(C2×C16), (D7×C16).3C2, SmallGroup(448,3)

Series: Derived Chief Lower central Upper central

C1C7 — D7×C32
C1C7C14C28C56C112D7×C16 — D7×C32
C7 — D7×C32
C1C32

Generators and relations for D7×C32
 G = < a,b,c | a32=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C2
7C22
7C4
7C2×C4
7C8
7C2×C8
7C16
7C2×C16
7C32
7C2×C32

Smallest permutation representation of D7×C32
On 224 points
Generators in S224
(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)
(1 128 47 222 155 71 183)(2 97 48 223 156 72 184)(3 98 49 224 157 73 185)(4 99 50 193 158 74 186)(5 100 51 194 159 75 187)(6 101 52 195 160 76 188)(7 102 53 196 129 77 189)(8 103 54 197 130 78 190)(9 104 55 198 131 79 191)(10 105 56 199 132 80 192)(11 106 57 200 133 81 161)(12 107 58 201 134 82 162)(13 108 59 202 135 83 163)(14 109 60 203 136 84 164)(15 110 61 204 137 85 165)(16 111 62 205 138 86 166)(17 112 63 206 139 87 167)(18 113 64 207 140 88 168)(19 114 33 208 141 89 169)(20 115 34 209 142 90 170)(21 116 35 210 143 91 171)(22 117 36 211 144 92 172)(23 118 37 212 145 93 173)(24 119 38 213 146 94 174)(25 120 39 214 147 95 175)(26 121 40 215 148 96 176)(27 122 41 216 149 65 177)(28 123 42 217 150 66 178)(29 124 43 218 151 67 179)(30 125 44 219 152 68 180)(31 126 45 220 153 69 181)(32 127 46 221 154 70 182)
(1 167)(2 168)(3 169)(4 170)(5 171)(6 172)(7 173)(8 174)(9 175)(10 176)(11 177)(12 178)(13 179)(14 180)(15 181)(16 182)(17 183)(18 184)(19 185)(20 186)(21 187)(22 188)(23 189)(24 190)(25 191)(26 192)(27 161)(28 162)(29 163)(30 164)(31 165)(32 166)(33 157)(34 158)(35 159)(36 160)(37 129)(38 130)(39 131)(40 132)(41 133)(42 134)(43 135)(44 136)(45 137)(46 138)(47 139)(48 140)(49 141)(50 142)(51 143)(52 144)(53 145)(54 146)(55 147)(56 148)(57 149)(58 150)(59 151)(60 152)(61 153)(62 154)(63 155)(64 156)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 113)(73 114)(74 115)(75 116)(76 117)(77 118)(78 119)(79 120)(80 121)(81 122)(82 123)(83 124)(84 125)(85 126)(86 127)(87 128)(88 97)(89 98)(90 99)(91 100)(92 101)(93 102)(94 103)(95 104)(96 105)(193 209)(194 210)(195 211)(196 212)(197 213)(198 214)(199 215)(200 216)(201 217)(202 218)(203 219)(204 220)(205 221)(206 222)(207 223)(208 224)

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

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

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

160 conjugacy classes

class 1 2A2B2C4A4B4C4D7A7B7C8A8B8C8D8E8F8G8H14A14B14C16A···16H16I···16P28A···28F32A···32P32Q···32AF56A···56L112A···112X224A···224AV
order122244447778888888814141416···1616···1628···2832···3232···3256···56112···112224···224
size11771177222111177772221···17···72···21···17···72···22···22···2

160 irreducible representations

dim11111111111222222
type++++++
imageC1C2C2C2C4C4C8C8C16C16C32D7D14C4×D7C8×D7D7×C16D7×C32
kernelD7×C32C7⋊C32C224D7×C16C7⋊C16C8×D7C7⋊C8C4×D7Dic7D14D7C32C16C8C4C2C1
# reps111122448832336122448

Matrix representation of D7×C32 in GL3(𝔽449) generated by

12800
03490
00349
,
100
03982
0448405
,
100
09543
0354354
G:=sub<GL(3,GF(449))| [128,0,0,0,349,0,0,0,349],[1,0,0,0,398,448,0,2,405],[1,0,0,0,95,354,0,43,354] >;

D7×C32 in GAP, Magma, Sage, TeX

D_7\times C_{32}
% in TeX

G:=Group("D7xC32");
// GroupNames label

G:=SmallGroup(448,3);
// by ID

G=gap.SmallGroup(448,3);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,36,58,80,102,18822]);
// Polycyclic

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

Export

Subgroup lattice of D7×C32 in TeX

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