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## G = C7×D4○C16order 448 = 26·7

### Direct product of C7 and D4○C16

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×D4○C16
 Chief series C1 — C2 — C4 — C8 — C56 — C112 — C2×C112 — C7×D4○C16
 Lower central C1 — C2 — C7×D4○C16
 Upper central C1 — C112 — C7×D4○C16

Generators and relations for C7×D4○C16
G = < a,b,c,d | a7=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, cd=dc >

Subgroups: 90 in 84 conjugacy classes, 78 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, Q8, C14, C14, C16, C16, C2×C8, M4(2), C4○D4, C28, C28, C2×C14, C2×C16, M5(2), C8○D4, C56, C56, C2×C28, C7×D4, C7×Q8, D4○C16, C112, C112, C2×C56, C7×M4(2), C7×C4○D4, C2×C112, C7×M5(2), C7×C8○D4, C7×D4○C16
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, C23, C14, C2×C8, C22×C4, C28, C2×C14, C22×C8, C56, C2×C28, C22×C14, D4○C16, C2×C56, C22×C28, C22×C56, C7×D4○C16

Smallest permutation representation of C7×D4○C16
On 224 points
Generators in S224
(1 76 201 124 150 103 96)(2 77 202 125 151 104 81)(3 78 203 126 152 105 82)(4 79 204 127 153 106 83)(5 80 205 128 154 107 84)(6 65 206 113 155 108 85)(7 66 207 114 156 109 86)(8 67 208 115 157 110 87)(9 68 193 116 158 111 88)(10 69 194 117 159 112 89)(11 70 195 118 160 97 90)(12 71 196 119 145 98 91)(13 72 197 120 146 99 92)(14 73 198 121 147 100 93)(15 74 199 122 148 101 94)(16 75 200 123 149 102 95)(17 134 221 169 191 39 51)(18 135 222 170 192 40 52)(19 136 223 171 177 41 53)(20 137 224 172 178 42 54)(21 138 209 173 179 43 55)(22 139 210 174 180 44 56)(23 140 211 175 181 45 57)(24 141 212 176 182 46 58)(25 142 213 161 183 47 59)(26 143 214 162 184 48 60)(27 144 215 163 185 33 61)(28 129 216 164 186 34 62)(29 130 217 165 187 35 63)(30 131 218 166 188 36 64)(31 132 219 167 189 37 49)(32 133 220 168 190 38 50)
(1 210 9 218)(2 211 10 219)(3 212 11 220)(4 213 12 221)(5 214 13 222)(6 215 14 223)(7 216 15 224)(8 217 16 209)(17 106 25 98)(18 107 26 99)(19 108 27 100)(20 109 28 101)(21 110 29 102)(22 111 30 103)(23 112 31 104)(24 97 32 105)(33 121 41 113)(34 122 42 114)(35 123 43 115)(36 124 44 116)(37 125 45 117)(38 126 46 118)(39 127 47 119)(40 128 48 120)(49 151 57 159)(50 152 58 160)(51 153 59 145)(52 154 60 146)(53 155 61 147)(54 156 62 148)(55 157 63 149)(56 158 64 150)(65 163 73 171)(66 164 74 172)(67 165 75 173)(68 166 76 174)(69 167 77 175)(70 168 78 176)(71 169 79 161)(72 170 80 162)(81 140 89 132)(82 141 90 133)(83 142 91 134)(84 143 92 135)(85 144 93 136)(86 129 94 137)(87 130 95 138)(88 131 96 139)(177 206 185 198)(178 207 186 199)(179 208 187 200)(180 193 188 201)(181 194 189 202)(182 195 190 203)(183 196 191 204)(184 197 192 205)
(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)(129 137)(130 138)(131 139)(132 140)(133 141)(134 142)(135 143)(136 144)(161 169)(162 170)(163 171)(164 172)(165 173)(166 174)(167 175)(168 176)(177 185)(178 186)(179 187)(180 188)(181 189)(182 190)(183 191)(184 192)(209 217)(210 218)(211 219)(212 220)(213 221)(214 222)(215 223)(216 224)
(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)

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

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

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

280 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 7A ··· 7F 8A 8B 8C 8D 8E ··· 8J 14A ··· 14F 14G ··· 14X 16A ··· 16H 16I ··· 16T 28A ··· 28L 28M ··· 28AD 56A ··· 56X 56Y ··· 56BH 112A ··· 112AV 112AW ··· 112DP order 1 2 2 2 2 4 4 4 4 4 7 ··· 7 8 8 8 8 8 ··· 8 14 ··· 14 14 ··· 14 16 ··· 16 16 ··· 16 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 112 ··· 112 112 ··· 112 size 1 1 2 2 2 1 1 2 2 2 1 ··· 1 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

280 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C4 C4 C7 C8 C8 C14 C14 C14 C28 C28 C56 C56 D4○C16 C7×D4○C16 kernel C7×D4○C16 C2×C112 C7×M5(2) C7×C8○D4 C7×M4(2) C7×C4○D4 D4○C16 C7×D4 C7×Q8 C2×C16 M5(2) C8○D4 M4(2) C4○D4 D4 Q8 C7 C1 # reps 1 3 3 1 6 2 6 12 4 18 18 6 36 12 72 24 8 48

Matrix representation of C7×D4○C16 in GL2(𝔽113) generated by

 28 0 0 28
,
 108 2 100 5
,
 1 0 5 112
,
 78 0 0 78
G:=sub<GL(2,GF(113))| [28,0,0,28],[108,100,2,5],[1,5,0,112],[78,0,0,78] >;

C7×D4○C16 in GAP, Magma, Sage, TeX

C_7\times D_4\circ C_{16}
% in TeX

G:=Group("C7xD4oC16");
// GroupNames label

G:=SmallGroup(448,912);
// by ID

G=gap.SmallGroup(448,912);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,2403,102,124]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^4=c^2=1,d^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations

׿
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