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## G = C7×M6(2)  order 448 = 26·7

### Direct product of C7 and M6(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×M6(2)
 Chief series C1 — C2 — C4 — C8 — C16 — C112 — C224 — C7×M6(2)
 Lower central C1 — C2 — C7×M6(2)
 Upper central C1 — C112 — C7×M6(2)

Generators and relations for C7×M6(2)
G = < a,b,c | a7=b32=c2=1, ab=ba, ac=ca, cbc=b17 >

Smallest permutation representation of C7×M6(2)
On 224 points
Generators in S224
(1 84 54 212 172 154 101)(2 85 55 213 173 155 102)(3 86 56 214 174 156 103)(4 87 57 215 175 157 104)(5 88 58 216 176 158 105)(6 89 59 217 177 159 106)(7 90 60 218 178 160 107)(8 91 61 219 179 129 108)(9 92 62 220 180 130 109)(10 93 63 221 181 131 110)(11 94 64 222 182 132 111)(12 95 33 223 183 133 112)(13 96 34 224 184 134 113)(14 65 35 193 185 135 114)(15 66 36 194 186 136 115)(16 67 37 195 187 137 116)(17 68 38 196 188 138 117)(18 69 39 197 189 139 118)(19 70 40 198 190 140 119)(20 71 41 199 191 141 120)(21 72 42 200 192 142 121)(22 73 43 201 161 143 122)(23 74 44 202 162 144 123)(24 75 45 203 163 145 124)(25 76 46 204 164 146 125)(26 77 47 205 165 147 126)(27 78 48 206 166 148 127)(28 79 49 207 167 149 128)(29 80 50 208 168 150 97)(30 81 51 209 169 151 98)(31 82 52 210 170 152 99)(32 83 53 211 171 153 100)
(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 18)(4 20)(6 22)(8 24)(10 26)(12 28)(14 30)(16 32)(33 49)(35 51)(37 53)(39 55)(41 57)(43 59)(45 61)(47 63)(65 81)(67 83)(69 85)(71 87)(73 89)(75 91)(77 93)(79 95)(98 114)(100 116)(102 118)(104 120)(106 122)(108 124)(110 126)(112 128)(129 145)(131 147)(133 149)(135 151)(137 153)(139 155)(141 157)(143 159)(161 177)(163 179)(165 181)(167 183)(169 185)(171 187)(173 189)(175 191)(193 209)(195 211)(197 213)(199 215)(201 217)(203 219)(205 221)(207 223)

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

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

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

280 conjugacy classes

 class 1 2A 2B 4A 4B 4C 7A ··· 7F 8A 8B 8C 8D 8E 8F 14A ··· 14F 14G ··· 14L 16A ··· 16H 16I 16J 16K 16L 28A ··· 28L 28M ··· 28R 32A ··· 32P 56A ··· 56X 56Y ··· 56AJ 112A ··· 112AV 112AW ··· 112BT 224A ··· 224CR order 1 2 2 4 4 4 7 ··· 7 8 8 8 8 8 8 14 ··· 14 14 ··· 14 16 ··· 16 16 16 16 16 28 ··· 28 28 ··· 28 32 ··· 32 56 ··· 56 56 ··· 56 112 ··· 112 112 ··· 112 224 ··· 224 size 1 1 2 1 1 2 1 ··· 1 1 1 1 1 2 2 1 ··· 1 2 ··· 2 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 1 1 2 2 type + + + image C1 C2 C2 C4 C4 C7 C8 C8 C14 C14 C16 C16 C28 C28 C56 C56 C112 C112 M6(2) C7×M6(2) kernel C7×M6(2) C224 C2×C112 C112 C2×C56 M6(2) C56 C2×C28 C32 C2×C16 C28 C2×C14 C16 C2×C8 C8 C2×C4 C4 C22 C7 C1 # reps 1 2 1 2 2 6 4 4 12 6 8 8 12 12 24 24 48 48 8 48

Matrix representation of C7×M6(2) in GL2(𝔽449) generated by

 176 0 0 176
,
 39 447 197 410
,
 1 0 39 448
G:=sub<GL(2,GF(449))| [176,0,0,176],[39,197,447,410],[1,39,0,448] >;

C7×M6(2) in GAP, Magma, Sage, TeX

C_7\times M_6(2)
% in TeX

G:=Group("C7xM6(2)");
// GroupNames label

G:=SmallGroup(448,174);
// by ID

G=gap.SmallGroup(448,174);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,196,3165,80,102,124]);
// 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^17>;
// generators/relations

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