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G = C7×SD64order 448 = 26·7

Direct product of C7 and SD64

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

Aliases: C7×SD64, C2244C2, C322C14, D16.C14, Q321C14, C56.69D4, C28.40D8, C14.16D16, C112.20C22, C4.2(C7×D8), C8.6(C7×D4), (C7×Q32)⋊5C2, C2.4(C7×D16), C16.3(C2×C14), (C7×D16).2C2, SmallGroup(448,176)

Series: Derived Chief Lower central Upper central

C1C16 — C7×SD64
C1C2C4C8C16C112C7×Q32 — C7×SD64
C1C2C4C8C16 — C7×SD64
C1C14C28C56C112 — C7×SD64

Generators and relations for C7×SD64
 G = < a,b,c | a7=b32=c2=1, ab=ba, ac=ca, cbc=b15 >

16C2
8C4
8C22
16C14
4D4
4Q8
8C28
8C2×C14
2D8
2Q16
4C7×Q8
4C7×D4
2C7×Q16
2C7×D8

Smallest permutation representation of C7×SD64
On 224 points
Generators in S224
(1 51 113 160 205 84 184)(2 52 114 129 206 85 185)(3 53 115 130 207 86 186)(4 54 116 131 208 87 187)(5 55 117 132 209 88 188)(6 56 118 133 210 89 189)(7 57 119 134 211 90 190)(8 58 120 135 212 91 191)(9 59 121 136 213 92 192)(10 60 122 137 214 93 161)(11 61 123 138 215 94 162)(12 62 124 139 216 95 163)(13 63 125 140 217 96 164)(14 64 126 141 218 65 165)(15 33 127 142 219 66 166)(16 34 128 143 220 67 167)(17 35 97 144 221 68 168)(18 36 98 145 222 69 169)(19 37 99 146 223 70 170)(20 38 100 147 224 71 171)(21 39 101 148 193 72 172)(22 40 102 149 194 73 173)(23 41 103 150 195 74 174)(24 42 104 151 196 75 175)(25 43 105 152 197 76 176)(26 44 106 153 198 77 177)(27 45 107 154 199 78 178)(28 46 108 155 200 79 179)(29 47 109 156 201 80 180)(30 48 110 157 202 81 181)(31 49 111 158 203 82 182)(32 50 112 159 204 83 183)
(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 16)(3 31)(4 14)(5 29)(6 12)(7 27)(8 10)(9 25)(11 23)(13 21)(15 19)(18 32)(20 30)(22 28)(24 26)(33 37)(34 52)(36 50)(38 48)(39 63)(40 46)(41 61)(42 44)(43 59)(45 57)(47 55)(49 53)(54 64)(56 62)(58 60)(65 87)(66 70)(67 85)(69 83)(71 81)(72 96)(73 79)(74 94)(75 77)(76 92)(78 90)(80 88)(82 86)(89 95)(91 93)(98 112)(99 127)(100 110)(101 125)(102 108)(103 123)(104 106)(105 121)(107 119)(109 117)(111 115)(114 128)(116 126)(118 124)(120 122)(129 143)(130 158)(131 141)(132 156)(133 139)(134 154)(135 137)(136 152)(138 150)(140 148)(142 146)(145 159)(147 157)(149 155)(151 153)(161 191)(162 174)(163 189)(164 172)(165 187)(166 170)(167 185)(169 183)(171 181)(173 179)(175 177)(176 192)(178 190)(180 188)(182 186)(193 217)(194 200)(195 215)(196 198)(197 213)(199 211)(201 209)(202 224)(203 207)(204 222)(206 220)(208 218)(210 216)(212 214)(219 223)

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

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

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

133 conjugacy classes

class 1 2A2B4A4B7A···7F8A8B14A···14F14G···14L16A16B16C16D28A···28F28G···28L32A···32H56A···56L112A···112X224A···224AV
order122447···78814···1414···141616161628···2828···2832···3256···56112···112224···224
size11162161···1221···116···1622222···216···162···22···22···22···2

133 irreducible representations

dim1111111122222222
type+++++++
imageC1C2C2C2C7C14C14C14D4D8D16C7×D4SD64C7×D8C7×D16C7×SD64
kernelC7×SD64C224C7×D16C7×Q32SD64C32D16Q32C56C28C14C8C7C4C2C1
# reps1111666612468122448

Matrix representation of C7×SD64 in GL2(𝔽449) generated by

250
025
,
271242
207271
,
10
0448
G:=sub<GL(2,GF(449))| [25,0,0,25],[271,207,242,271],[1,0,0,448] >;

C7×SD64 in GAP, Magma, Sage, TeX

C_7\times {\rm SD}_{64}
% in TeX

G:=Group("C7xSD64");
// GroupNames label

G:=SmallGroup(448,176);
// by ID

G=gap.SmallGroup(448,176);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,1568,421,2355,1186,192,5884,2951,242,14117,7068,124]);
// 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^15>;
// generators/relations

Export

Subgroup lattice of C7×SD64 in TeX

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