direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×M6(2), C4.C112, C224⋊7C2, C32⋊3C14, C56.9C8, C8.3C56, C28.4C16, C16.2C28, C112.6C4, C22.C112, C112.30C22, (C2×C4).5C56, (C2×C28).14C8, C8.22(C2×C28), C4.13(C2×C56), (C2×C8).13C28, (C2×C14).1C16, (C2×C56).32C4, (C2×C16).8C14, C56.91(C2×C4), C2.3(C2×C112), C28.53(C2×C8), C16.7(C2×C14), (C2×C112).18C2, C14.13(C2×C16), SmallGroup(448,174)
Series: Derived ►Chief ►Lower central ►Upper central
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
(1 205 56 162 131 105 75)(2 206 57 163 132 106 76)(3 207 58 164 133 107 77)(4 208 59 165 134 108 78)(5 209 60 166 135 109 79)(6 210 61 167 136 110 80)(7 211 62 168 137 111 81)(8 212 63 169 138 112 82)(9 213 64 170 139 113 83)(10 214 33 171 140 114 84)(11 215 34 172 141 115 85)(12 216 35 173 142 116 86)(13 217 36 174 143 117 87)(14 218 37 175 144 118 88)(15 219 38 176 145 119 89)(16 220 39 177 146 120 90)(17 221 40 178 147 121 91)(18 222 41 179 148 122 92)(19 223 42 180 149 123 93)(20 224 43 181 150 124 94)(21 193 44 182 151 125 95)(22 194 45 183 152 126 96)(23 195 46 184 153 127 65)(24 196 47 185 154 128 66)(25 197 48 186 155 97 67)(26 198 49 187 156 98 68)(27 199 50 188 157 99 69)(28 200 51 189 158 100 70)(29 201 52 190 159 101 71)(30 202 53 191 160 102 72)(31 203 54 192 129 103 73)(32 204 55 161 130 104 74)
(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)(66 82)(68 84)(70 86)(72 88)(74 90)(76 92)(78 94)(80 96)(98 114)(100 116)(102 118)(104 120)(106 122)(108 124)(110 126)(112 128)(130 146)(132 148)(134 150)(136 152)(138 154)(140 156)(142 158)(144 160)(161 177)(163 179)(165 181)(167 183)(169 185)(171 187)(173 189)(175 191)(194 210)(196 212)(198 214)(200 216)(202 218)(204 220)(206 222)(208 224)
G:=sub<Sym(224)| (1,205,56,162,131,105,75)(2,206,57,163,132,106,76)(3,207,58,164,133,107,77)(4,208,59,165,134,108,78)(5,209,60,166,135,109,79)(6,210,61,167,136,110,80)(7,211,62,168,137,111,81)(8,212,63,169,138,112,82)(9,213,64,170,139,113,83)(10,214,33,171,140,114,84)(11,215,34,172,141,115,85)(12,216,35,173,142,116,86)(13,217,36,174,143,117,87)(14,218,37,175,144,118,88)(15,219,38,176,145,119,89)(16,220,39,177,146,120,90)(17,221,40,178,147,121,91)(18,222,41,179,148,122,92)(19,223,42,180,149,123,93)(20,224,43,181,150,124,94)(21,193,44,182,151,125,95)(22,194,45,183,152,126,96)(23,195,46,184,153,127,65)(24,196,47,185,154,128,66)(25,197,48,186,155,97,67)(26,198,49,187,156,98,68)(27,199,50,188,157,99,69)(28,200,51,189,158,100,70)(29,201,52,190,159,101,71)(30,202,53,191,160,102,72)(31,203,54,192,129,103,73)(32,204,55,161,130,104,74), (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)(66,82)(68,84)(70,86)(72,88)(74,90)(76,92)(78,94)(80,96)(98,114)(100,116)(102,118)(104,120)(106,122)(108,124)(110,126)(112,128)(130,146)(132,148)(134,150)(136,152)(138,154)(140,156)(142,158)(144,160)(161,177)(163,179)(165,181)(167,183)(169,185)(171,187)(173,189)(175,191)(194,210)(196,212)(198,214)(200,216)(202,218)(204,220)(206,222)(208,224)>;
G:=Group( (1,205,56,162,131,105,75)(2,206,57,163,132,106,76)(3,207,58,164,133,107,77)(4,208,59,165,134,108,78)(5,209,60,166,135,109,79)(6,210,61,167,136,110,80)(7,211,62,168,137,111,81)(8,212,63,169,138,112,82)(9,213,64,170,139,113,83)(10,214,33,171,140,114,84)(11,215,34,172,141,115,85)(12,216,35,173,142,116,86)(13,217,36,174,143,117,87)(14,218,37,175,144,118,88)(15,219,38,176,145,119,89)(16,220,39,177,146,120,90)(17,221,40,178,147,121,91)(18,222,41,179,148,122,92)(19,223,42,180,149,123,93)(20,224,43,181,150,124,94)(21,193,44,182,151,125,95)(22,194,45,183,152,126,96)(23,195,46,184,153,127,65)(24,196,47,185,154,128,66)(25,197,48,186,155,97,67)(26,198,49,187,156,98,68)(27,199,50,188,157,99,69)(28,200,51,189,158,100,70)(29,201,52,190,159,101,71)(30,202,53,191,160,102,72)(31,203,54,192,129,103,73)(32,204,55,161,130,104,74), (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)(66,82)(68,84)(70,86)(72,88)(74,90)(76,92)(78,94)(80,96)(98,114)(100,116)(102,118)(104,120)(106,122)(108,124)(110,126)(112,128)(130,146)(132,148)(134,150)(136,152)(138,154)(140,156)(142,158)(144,160)(161,177)(163,179)(165,181)(167,183)(169,185)(171,187)(173,189)(175,191)(194,210)(196,212)(198,214)(200,216)(202,218)(204,220)(206,222)(208,224) );
G=PermutationGroup([[(1,205,56,162,131,105,75),(2,206,57,163,132,106,76),(3,207,58,164,133,107,77),(4,208,59,165,134,108,78),(5,209,60,166,135,109,79),(6,210,61,167,136,110,80),(7,211,62,168,137,111,81),(8,212,63,169,138,112,82),(9,213,64,170,139,113,83),(10,214,33,171,140,114,84),(11,215,34,172,141,115,85),(12,216,35,173,142,116,86),(13,217,36,174,143,117,87),(14,218,37,175,144,118,88),(15,219,38,176,145,119,89),(16,220,39,177,146,120,90),(17,221,40,178,147,121,91),(18,222,41,179,148,122,92),(19,223,42,180,149,123,93),(20,224,43,181,150,124,94),(21,193,44,182,151,125,95),(22,194,45,183,152,126,96),(23,195,46,184,153,127,65),(24,196,47,185,154,128,66),(25,197,48,186,155,97,67),(26,198,49,187,156,98,68),(27,199,50,188,157,99,69),(28,200,51,189,158,100,70),(29,201,52,190,159,101,71),(30,202,53,191,160,102,72),(31,203,54,192,129,103,73),(32,204,55,161,130,104,74)], [(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),(66,82),(68,84),(70,86),(72,88),(74,90),(76,92),(78,94),(80,96),(98,114),(100,116),(102,118),(104,120),(106,122),(108,124),(110,126),(112,128),(130,146),(132,148),(134,150),(136,152),(138,154),(140,156),(142,158),(144,160),(161,177),(163,179),(165,181),(167,183),(169,185),(171,187),(173,189),(175,191),(194,210),(196,212),(198,214),(200,216),(202,218),(204,220),(206,222),(208,224)]])
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
Export