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## G = C14×C8.C22order 448 = 26·7

### Direct product of C14 and C8.C22

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C14×C8.C22
 Chief series C1 — C2 — C4 — C28 — C7×D4 — C7×SD16 — C7×C8.C22 — C14×C8.C22
 Lower central C1 — C2 — C4 — C14×C8.C22
 Upper central C1 — C2×C14 — C22×C28 — C14×C8.C22

Generators and relations for C14×C8.C22
G = < a,b,c,d | a14=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >

Subgroups: 370 in 258 conjugacy classes, 162 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C14, C14, C14, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C28, C28, C28, C2×C14, C2×C14, C2×C14, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, C2×C8.C22, C2×C56, C7×M4(2), C7×SD16, C7×Q16, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, Q8×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C14×M4(2), C14×SD16, C14×Q16, C7×C8.C22, Q8×C2×C14, C14×C4○D4, C14×C8.C22
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C24, C2×C14, C8.C22, C22×D4, C7×D4, C22×C14, C2×C8.C22, D4×C14, C23×C14, C7×C8.C22, D4×C2×C14, C14×C8.C22

Smallest permutation representation of C14×C8.C22
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)
(1 199 38 181 109 124 55 157)(2 200 39 182 110 125 56 158)(3 201 40 169 111 126 43 159)(4 202 41 170 112 113 44 160)(5 203 42 171 99 114 45 161)(6 204 29 172 100 115 46 162)(7 205 30 173 101 116 47 163)(8 206 31 174 102 117 48 164)(9 207 32 175 103 118 49 165)(10 208 33 176 104 119 50 166)(11 209 34 177 105 120 51 167)(12 210 35 178 106 121 52 168)(13 197 36 179 107 122 53 155)(14 198 37 180 108 123 54 156)(15 67 194 82 145 86 139 215)(16 68 195 83 146 87 140 216)(17 69 196 84 147 88 127 217)(18 70 183 71 148 89 128 218)(19 57 184 72 149 90 129 219)(20 58 185 73 150 91 130 220)(21 59 186 74 151 92 131 221)(22 60 187 75 152 93 132 222)(23 61 188 76 153 94 133 223)(24 62 189 77 154 95 134 224)(25 63 190 78 141 96 135 211)(26 64 191 79 142 97 136 212)(27 65 192 80 143 98 137 213)(28 66 193 81 144 85 138 214)
(1 220)(2 221)(3 222)(4 223)(5 224)(6 211)(7 212)(8 213)(9 214)(10 215)(11 216)(12 217)(13 218)(14 219)(15 176)(16 177)(17 178)(18 179)(19 180)(20 181)(21 182)(22 169)(23 170)(24 171)(25 172)(26 173)(27 174)(28 175)(29 96)(30 97)(31 98)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 60)(44 61)(45 62)(46 63)(47 64)(48 65)(49 66)(50 67)(51 68)(52 69)(53 70)(54 57)(55 58)(56 59)(71 107)(72 108)(73 109)(74 110)(75 111)(76 112)(77 99)(78 100)(79 101)(80 102)(81 103)(82 104)(83 105)(84 106)(113 133)(114 134)(115 135)(116 136)(117 137)(118 138)(119 139)(120 140)(121 127)(122 128)(123 129)(124 130)(125 131)(126 132)(141 162)(142 163)(143 164)(144 165)(145 166)(146 167)(147 168)(148 155)(149 156)(150 157)(151 158)(152 159)(153 160)(154 161)(183 197)(184 198)(185 199)(186 200)(187 201)(188 202)(189 203)(190 204)(191 205)(192 206)(193 207)(194 208)(195 209)(196 210)
(57 90)(58 91)(59 92)(60 93)(61 94)(62 95)(63 96)(64 97)(65 98)(66 85)(67 86)(68 87)(69 88)(70 89)(71 218)(72 219)(73 220)(74 221)(75 222)(76 223)(77 224)(78 211)(79 212)(80 213)(81 214)(82 215)(83 216)(84 217)(113 202)(114 203)(115 204)(116 205)(117 206)(118 207)(119 208)(120 209)(121 210)(122 197)(123 198)(124 199)(125 200)(126 201)(155 179)(156 180)(157 181)(158 182)(159 169)(160 170)(161 171)(162 172)(163 173)(164 174)(165 175)(166 176)(167 177)(168 178)

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4J 7A ··· 7F 8A 8B 8C 8D 14A ··· 14R 14S ··· 14AD 14AE ··· 14AP 28A ··· 28X 28Y ··· 28BH 56A ··· 56X order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 7 ··· 7 8 8 8 8 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 4 4 2 2 2 2 4 ··· 4 1 ··· 1 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 C14 D4 D4 C7×D4 C7×D4 C8.C22 C7×C8.C22 kernel C14×C8.C22 C14×M4(2) C14×SD16 C14×Q16 C7×C8.C22 Q8×C2×C14 C14×C4○D4 C2×C8.C22 C2×M4(2) C2×SD16 C2×Q16 C8.C22 C22×Q8 C2×C4○D4 C2×C28 C22×C14 C2×C4 C23 C14 C2 # reps 1 1 2 2 8 1 1 6 6 12 12 48 6 6 3 1 18 6 2 12

Matrix representation of C14×C8.C22 in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 64
,
 18 72 0 0 0 0 41 95 0 0 0 0 0 0 110 36 37 21 0 0 96 91 94 58 0 0 95 66 0 79 0 0 55 18 0 25
,
 41 95 0 0 0 0 18 72 0 0 0 0 0 0 0 2 1 15 0 0 0 49 0 86 0 0 1 0 0 111 0 0 0 1 0 64
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 109 0 0 0 1 0 15 0 0 0 0 112 0 0 0 0 0 0 112

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[18,41,0,0,0,0,72,95,0,0,0,0,0,0,110,96,95,55,0,0,36,91,66,18,0,0,37,94,0,0,0,0,21,58,79,25],[41,18,0,0,0,0,95,72,0,0,0,0,0,0,0,0,1,0,0,0,2,49,0,1,0,0,1,0,0,0,0,0,15,86,111,64],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,109,15,0,112] >;

C14×C8.C22 in GAP, Magma, Sage, TeX

C_{14}\times C_8.C_2^2
% in TeX

G:=Group("C14xC8.C2^2");
// GroupNames label

G:=SmallGroup(448,1357);
// by ID

G=gap.SmallGroup(448,1357);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,1576,4790,14117,7068,124]);
// Polycyclic

G:=Group<a,b,c,d|a^14=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations

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