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## G = C7×Q32⋊C2order 448 = 26·7

### Direct product of C7 and Q32⋊C2

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C7×Q32⋊C2
 Chief series C1 — C2 — C4 — C8 — C56 — C7×D8 — C7×SD32 — C7×Q32⋊C2
 Lower central C1 — C2 — C4 — C8 — C7×Q32⋊C2
 Upper central C1 — C14 — C2×C28 — C2×C56 — C7×Q32⋊C2

Generators and relations for C7×Q32⋊C2
G = < a,b,c,d | a7=b16=d2=1, c2=b8, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b9, cd=dc >

Subgroups: 162 in 82 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C14, C14, C16, C2×C8, D8, SD16, Q16, Q16, Q16, C2×Q8, C4○D4, C28, C28, C2×C14, C2×C14, M5(2), SD32, Q32, C2×Q16, C4○D8, C56, C2×C28, C2×C28, C7×D4, C7×Q8, Q32⋊C2, C112, C2×C56, C7×D8, C7×SD16, C7×Q16, C7×Q16, C7×Q16, Q8×C14, C7×C4○D4, C7×M5(2), C7×SD32, C7×Q32, C14×Q16, C7×C4○D8, C7×Q32⋊C2
Quotients: C1, C2, C22, C7, D4, C23, C14, D8, C2×D4, C2×C14, C2×D8, C7×D4, C22×C14, Q32⋊C2, C7×D8, D4×C14, C14×D8, C7×Q32⋊C2

Smallest permutation representation of C7×Q32⋊C2
On 224 points
Generators in S224
(1 53 155 18 112 123 167)(2 54 156 19 97 124 168)(3 55 157 20 98 125 169)(4 56 158 21 99 126 170)(5 57 159 22 100 127 171)(6 58 160 23 101 128 172)(7 59 145 24 102 113 173)(8 60 146 25 103 114 174)(9 61 147 26 104 115 175)(10 62 148 27 105 116 176)(11 63 149 28 106 117 161)(12 64 150 29 107 118 162)(13 49 151 30 108 119 163)(14 50 152 31 109 120 164)(15 51 153 32 110 121 165)(16 52 154 17 111 122 166)(33 68 189 144 215 89 205)(34 69 190 129 216 90 206)(35 70 191 130 217 91 207)(36 71 192 131 218 92 208)(37 72 177 132 219 93 193)(38 73 178 133 220 94 194)(39 74 179 134 221 95 195)(40 75 180 135 222 96 196)(41 76 181 136 223 81 197)(42 77 182 137 224 82 198)(43 78 183 138 209 83 199)(44 79 184 139 210 84 200)(45 80 185 140 211 85 201)(46 65 186 141 212 86 202)(47 66 187 142 213 87 203)(48 67 188 143 214 88 204)
(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 74 9 66)(2 73 10 65)(3 72 11 80)(4 71 12 79)(5 70 13 78)(6 69 14 77)(7 68 15 76)(8 67 16 75)(17 222 25 214)(18 221 26 213)(19 220 27 212)(20 219 28 211)(21 218 29 210)(22 217 30 209)(23 216 31 224)(24 215 32 223)(33 165 41 173)(34 164 42 172)(35 163 43 171)(36 162 44 170)(37 161 45 169)(38 176 46 168)(39 175 47 167)(40 174 48 166)(49 183 57 191)(50 182 58 190)(51 181 59 189)(52 180 60 188)(53 179 61 187)(54 178 62 186)(55 177 63 185)(56 192 64 184)(81 102 89 110)(82 101 90 109)(83 100 91 108)(84 99 92 107)(85 98 93 106)(86 97 94 105)(87 112 95 104)(88 111 96 103)(113 205 121 197)(114 204 122 196)(115 203 123 195)(116 202 124 194)(117 201 125 193)(118 200 126 208)(119 199 127 207)(120 198 128 206)(129 152 137 160)(130 151 138 159)(131 150 139 158)(132 149 140 157)(133 148 141 156)(134 147 142 155)(135 146 143 154)(136 145 144 153)
(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)(50 58)(52 60)(54 62)(56 64)(65 73)(67 75)(69 77)(71 79)(82 90)(84 92)(86 94)(88 96)(97 105)(99 107)(101 109)(103 111)(114 122)(116 124)(118 126)(120 128)(129 137)(131 139)(133 141)(135 143)(146 154)(148 156)(150 158)(152 160)(162 170)(164 172)(166 174)(168 176)(178 186)(180 188)(182 190)(184 192)(194 202)(196 204)(198 206)(200 208)(210 218)(212 220)(214 222)(216 224)

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

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

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

112 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 7A ··· 7F 8A 8B 8C 14A ··· 14F 14G ··· 14L 14M ··· 14R 16A 16B 16C 16D 28A ··· 28L 28M ··· 28AD 56A ··· 56L 56M ··· 56R 112A ··· 112X order 1 2 2 2 4 4 4 4 4 7 ··· 7 8 8 8 14 ··· 14 14 ··· 14 14 ··· 14 16 16 16 16 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 112 ··· 112 size 1 1 2 8 2 2 8 8 8 1 ··· 1 2 2 4 1 ··· 1 2 ··· 2 8 ··· 8 4 4 4 4 2 ··· 2 8 ··· 8 2 ··· 2 4 ··· 4 4 ··· 4

112 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 D4 D4 D8 D8 C7×D4 C7×D4 C7×D8 C7×D8 Q32⋊C2 C7×Q32⋊C2 kernel C7×Q32⋊C2 C7×M5(2) C7×SD32 C7×Q32 C14×Q16 C7×C4○D8 Q32⋊C2 M5(2) SD32 Q32 C2×Q16 C4○D8 C56 C2×C28 C28 C2×C14 C8 C2×C4 C4 C22 C7 C1 # reps 1 1 2 2 1 1 6 6 12 12 6 6 1 1 2 2 6 6 12 12 2 12

Matrix representation of C7×Q32⋊C2 in GL4(𝔽113) generated by

 106 0 0 0 0 106 0 0 0 0 106 0 0 0 0 106
,
 47 0 59 73 27 0 106 66 13 12 86 93 22 104 86 93
,
 47 73 0 0 27 66 0 0 34 93 104 101 25 93 101 9
,
 1 0 0 0 0 1 0 0 1 0 112 0 1 0 0 112
G:=sub<GL(4,GF(113))| [106,0,0,0,0,106,0,0,0,0,106,0,0,0,0,106],[47,27,13,22,0,0,12,104,59,106,86,86,73,66,93,93],[47,27,34,25,73,66,93,93,0,0,104,101,0,0,101,9],[1,0,1,1,0,1,0,0,0,0,112,0,0,0,0,112] >;

C7×Q32⋊C2 in GAP, Magma, Sage, TeX

C_7\times Q_{32}\rtimes C_2
% in TeX

G:=Group("C7xQ32:C2");
// GroupNames label

G:=SmallGroup(448,918);
// by ID

G=gap.SmallGroup(448,918);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,1576,4790,5884,2951,242,14117,7068,124]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^16=d^2=1,c^2=b^8,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^9,c*d=d*c>;
// generators/relations

׿
×
𝔽