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## G = C14×M5(2)  order 448 = 26·7

### Direct product of C14 and M5(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C14×M5(2)
 Chief series C1 — C2 — C4 — C8 — C56 — C112 — C7×M5(2) — C14×M5(2)
 Lower central C1 — C2 — C14×M5(2)
 Upper central C1 — C2×C56 — C14×M5(2)

Generators and relations for C14×M5(2)
G = < a,b,c | a14=b16=c2=1, ab=ba, ac=ca, cbc=b9 >

Subgroups: 98 in 90 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, C14, C14, C14, C16, C2×C8, C2×C8, C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C2×C16, M5(2), C22×C8, C56, C56, C2×C28, C2×C28, C22×C14, C2×M5(2), C112, C2×C56, C2×C56, C22×C28, C2×C112, C7×M5(2), C22×C56, C14×M5(2)
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, C23, C14, C2×C8, C22×C4, C28, C2×C14, M5(2), C22×C8, C56, C2×C28, C22×C14, C2×M5(2), C2×C56, C22×C28, C7×M5(2), C22×C56, C14×M5(2)

Smallest permutation representation of C14×M5(2)
On 224 points
Generators in S224
(1 127 102 62 138 183 165 160 44 92 205 30 213 65)(2 128 103 63 139 184 166 145 45 93 206 31 214 66)(3 113 104 64 140 185 167 146 46 94 207 32 215 67)(4 114 105 49 141 186 168 147 47 95 208 17 216 68)(5 115 106 50 142 187 169 148 48 96 193 18 217 69)(6 116 107 51 143 188 170 149 33 81 194 19 218 70)(7 117 108 52 144 189 171 150 34 82 195 20 219 71)(8 118 109 53 129 190 172 151 35 83 196 21 220 72)(9 119 110 54 130 191 173 152 36 84 197 22 221 73)(10 120 111 55 131 192 174 153 37 85 198 23 222 74)(11 121 112 56 132 177 175 154 38 86 199 24 223 75)(12 122 97 57 133 178 176 155 39 87 200 25 224 76)(13 123 98 58 134 179 161 156 40 88 201 26 209 77)(14 124 99 59 135 180 162 157 41 89 202 27 210 78)(15 125 100 60 136 181 163 158 42 90 203 28 211 79)(16 126 101 61 137 182 164 159 43 91 204 29 212 80)
(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)
(1 152)(2 145)(3 154)(4 147)(5 156)(6 149)(7 158)(8 151)(9 160)(10 153)(11 146)(12 155)(13 148)(14 157)(15 150)(16 159)(17 141)(18 134)(19 143)(20 136)(21 129)(22 138)(23 131)(24 140)(25 133)(26 142)(27 135)(28 144)(29 137)(30 130)(31 139)(32 132)(33 116)(34 125)(35 118)(36 127)(37 120)(38 113)(39 122)(40 115)(41 124)(42 117)(43 126)(44 119)(45 128)(46 121)(47 114)(48 123)(49 208)(50 201)(51 194)(52 203)(53 196)(54 205)(55 198)(56 207)(57 200)(58 193)(59 202)(60 195)(61 204)(62 197)(63 206)(64 199)(65 173)(66 166)(67 175)(68 168)(69 161)(70 170)(71 163)(72 172)(73 165)(74 174)(75 167)(76 176)(77 169)(78 162)(79 171)(80 164)(81 107)(82 100)(83 109)(84 102)(85 111)(86 104)(87 97)(88 106)(89 99)(90 108)(91 101)(92 110)(93 103)(94 112)(95 105)(96 98)(177 215)(178 224)(179 217)(180 210)(181 219)(182 212)(183 221)(184 214)(185 223)(186 216)(187 209)(188 218)(189 211)(190 220)(191 213)(192 222)

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

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

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

280 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 7A ··· 7F 8A ··· 8H 8I 8J 8K 8L 14A ··· 14R 14S ··· 14AD 16A ··· 16P 28A ··· 28X 28Y ··· 28AJ 56A ··· 56AV 56AW ··· 56BT 112A ··· 112CR order 1 2 2 2 2 2 4 4 4 4 4 4 7 ··· 7 8 ··· 8 8 8 8 8 14 ··· 14 14 ··· 14 16 ··· 16 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 112 ··· 112 size 1 1 1 1 2 2 1 1 1 1 2 2 1 ··· 1 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 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 M5(2) C7×M5(2) kernel C14×M5(2) C2×C112 C7×M5(2) C22×C56 C2×C56 C22×C28 C2×M5(2) C2×C28 C22×C14 C2×C16 M5(2) C22×C8 C2×C8 C22×C4 C2×C4 C23 C14 C2 # reps 1 2 4 1 6 2 6 12 4 12 24 6 36 12 72 24 8 48

Matrix representation of C14×M5(2) in GL3(𝔽113) generated by

 112 0 0 0 7 0 0 0 7
,
 112 0 0 0 90 111 0 86 23
,
 1 0 0 0 1 0 0 90 112
G:=sub<GL(3,GF(113))| [112,0,0,0,7,0,0,0,7],[112,0,0,0,90,86,0,111,23],[1,0,0,0,1,90,0,0,112] >;

C14×M5(2) in GAP, Magma, Sage, TeX

C_{14}\times M_5(2)
% in TeX

G:=Group("C14xM5(2)");
// GroupNames label

G:=SmallGroup(448,911);
// by ID

G=gap.SmallGroup(448,911);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,3165,102,124]);
// Polycyclic

G:=Group<a,b,c|a^14=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations

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