metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C56.6C8, C7⋊2M6(2), C112.4C4, C28.1C16, C16.22D14, C16.2Dic7, C112.27C22, C7⋊C32⋊5C2, C4.(C7⋊C16), C8.3(C7⋊C8), (C2×C28).9C8, C22.(C7⋊C16), (C2×C16).7D7, C28.43(C2×C8), C14.9(C2×C16), (C2×C56).25C4, C56.76(C2×C4), (C2×C14).3C16, (C2×C112).13C2, C8.22(C2×Dic7), (C2×C8).14Dic7, C4.15(C2×C7⋊C8), C2.4(C2×C7⋊C16), (C2×C4).5(C7⋊C8), SmallGroup(448,56)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7⋊M6(2)
G = < a,b,c | a7=b32=c2=1, bab-1=a-1, ac=ca, cbc=b17 >
Smallest permutation representation of C7⋊M6(2)
►On 224 points
Generators in S224
(1 210 110 34 139 187 90)(2 91 188 140 35 111 211)(3 212 112 36 141 189 92)(4 93 190 142 37 113 213)(5 214 114 38 143 191 94)(6 95 192 144 39 115 215)(7 216 116 40 145 161 96)(8 65 162 146 41 117 217)(9 218 118 42 147 163 66)(10 67 164 148 43 119 219)(11 220 120 44 149 165 68)(12 69 166 150 45 121 221)(13 222 122 46 151 167 70)(14 71 168 152 47 123 223)(15 224 124 48 153 169 72)(16 73 170 154 49 125 193)(17 194 126 50 155 171 74)(18 75 172 156 51 127 195)(19 196 128 52 157 173 76)(20 77 174 158 53 97 197)(21 198 98 54 159 175 78)(22 79 176 160 55 99 199)(23 200 100 56 129 177 80)(24 81 178 130 57 101 201)(25 202 102 58 131 179 82)(26 83 180 132 59 103 203)(27 204 104 60 133 181 84)(28 85 182 134 61 105 205)(29 206 106 62 135 183 86)(30 87 184 136 63 107 207)(31 208 108 64 137 185 88)(32 89 186 138 33 109 209)
(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)(97 113)(99 115)(101 117)(103 119)(105 121)(107 123)(109 125)(111 127)(130 146)(132 148)(134 150)(136 152)(138 154)(140 156)(142 158)(144 160)(162 178)(164 180)(166 182)(168 184)(170 186)(172 188)(174 190)(176 192)(193 209)(195 211)(197 213)(199 215)(201 217)(203 219)(205 221)(207 223)
G:=sub<Sym(224)| (1,210,110,34,139,187,90)(2,91,188,140,35,111,211)(3,212,112,36,141,189,92)(4,93,190,142,37,113,213)(5,214,114,38,143,191,94)(6,95,192,144,39,115,215)(7,216,116,40,145,161,96)(8,65,162,146,41,117,217)(9,218,118,42,147,163,66)(10,67,164,148,43,119,219)(11,220,120,44,149,165,68)(12,69,166,150,45,121,221)(13,222,122,46,151,167,70)(14,71,168,152,47,123,223)(15,224,124,48,153,169,72)(16,73,170,154,49,125,193)(17,194,126,50,155,171,74)(18,75,172,156,51,127,195)(19,196,128,52,157,173,76)(20,77,174,158,53,97,197)(21,198,98,54,159,175,78)(22,79,176,160,55,99,199)(23,200,100,56,129,177,80)(24,81,178,130,57,101,201)(25,202,102,58,131,179,82)(26,83,180,132,59,103,203)(27,204,104,60,133,181,84)(28,85,182,134,61,105,205)(29,206,106,62,135,183,86)(30,87,184,136,63,107,207)(31,208,108,64,137,185,88)(32,89,186,138,33,109,209), (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)(97,113)(99,115)(101,117)(103,119)(105,121)(107,123)(109,125)(111,127)(130,146)(132,148)(134,150)(136,152)(138,154)(140,156)(142,158)(144,160)(162,178)(164,180)(166,182)(168,184)(170,186)(172,188)(174,190)(176,192)(193,209)(195,211)(197,213)(199,215)(201,217)(203,219)(205,221)(207,223)>;
G:=Group( (1,210,110,34,139,187,90)(2,91,188,140,35,111,211)(3,212,112,36,141,189,92)(4,93,190,142,37,113,213)(5,214,114,38,143,191,94)(6,95,192,144,39,115,215)(7,216,116,40,145,161,96)(8,65,162,146,41,117,217)(9,218,118,42,147,163,66)(10,67,164,148,43,119,219)(11,220,120,44,149,165,68)(12,69,166,150,45,121,221)(13,222,122,46,151,167,70)(14,71,168,152,47,123,223)(15,224,124,48,153,169,72)(16,73,170,154,49,125,193)(17,194,126,50,155,171,74)(18,75,172,156,51,127,195)(19,196,128,52,157,173,76)(20,77,174,158,53,97,197)(21,198,98,54,159,175,78)(22,79,176,160,55,99,199)(23,200,100,56,129,177,80)(24,81,178,130,57,101,201)(25,202,102,58,131,179,82)(26,83,180,132,59,103,203)(27,204,104,60,133,181,84)(28,85,182,134,61,105,205)(29,206,106,62,135,183,86)(30,87,184,136,63,107,207)(31,208,108,64,137,185,88)(32,89,186,138,33,109,209), (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)(97,113)(99,115)(101,117)(103,119)(105,121)(107,123)(109,125)(111,127)(130,146)(132,148)(134,150)(136,152)(138,154)(140,156)(142,158)(144,160)(162,178)(164,180)(166,182)(168,184)(170,186)(172,188)(174,190)(176,192)(193,209)(195,211)(197,213)(199,215)(201,217)(203,219)(205,221)(207,223) );
G=PermutationGroup([[(1,210,110,34,139,187,90),(2,91,188,140,35,111,211),(3,212,112,36,141,189,92),(4,93,190,142,37,113,213),(5,214,114,38,143,191,94),(6,95,192,144,39,115,215),(7,216,116,40,145,161,96),(8,65,162,146,41,117,217),(9,218,118,42,147,163,66),(10,67,164,148,43,119,219),(11,220,120,44,149,165,68),(12,69,166,150,45,121,221),(13,222,122,46,151,167,70),(14,71,168,152,47,123,223),(15,224,124,48,153,169,72),(16,73,170,154,49,125,193),(17,194,126,50,155,171,74),(18,75,172,156,51,127,195),(19,196,128,52,157,173,76),(20,77,174,158,53,97,197),(21,198,98,54,159,175,78),(22,79,176,160,55,99,199),(23,200,100,56,129,177,80),(24,81,178,130,57,101,201),(25,202,102,58,131,179,82),(26,83,180,132,59,103,203),(27,204,104,60,133,181,84),(28,85,182,134,61,105,205),(29,206,106,62,135,183,86),(30,87,184,136,63,107,207),(31,208,108,64,137,185,88),(32,89,186,138,33,109,209)], [(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),(97,113),(99,115),(101,117),(103,119),(105,121),(107,123),(109,125),(111,127),(130,146),(132,148),(134,150),(136,152),(138,154),(140,156),(142,158),(144,160),(162,178),(164,180),(166,182),(168,184),(170,186),(172,188),(174,190),(176,192),(193,209),(195,211),(197,213),(199,215),(201,217),(203,219),(205,221),(207,223)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 14A | ··· | 14I | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 28A | ··· | 28L | 32A | ··· | 32P | 56A | ··· | 56X | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 32 | ··· | 32 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | D7 | Dic7 | D14 | Dic7 | C7⋊C8 | C7⋊C8 | M6(2) | C7⋊C16 | C7⋊C16 | C7⋊M6(2) |
| kernel | C7⋊M6(2) | C7⋊C32 | C2×C112 | C112 | C2×C56 | C56 | C2×C28 | C28 | C2×C14 | C2×C16 | C16 | C16 | C2×C8 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 3 | 3 | 3 | 3 | 6 | 6 | 8 | 12 | 12 | 48 |
Matrix representation of C7⋊M6(2) ►in GL2(𝔽449) generated by
| 324 | 0 |
| 0 | 176 |
| 0 | 1 |
| 35 | 0 |
| 1 | 0 |
| 0 | 448 |
G:=sub<GL(2,GF(449))| [324,0,0,176],[0,35,1,0],[1,0,0,448] >;
C7⋊M6(2) in GAP, Magma, Sage, TeX
C_7\rtimes M_6(2)
% in TeX
G:=Group("C7:M6(2)"); // GroupNames label
G:=SmallGroup(448,56);
// by ID
G=gap.SmallGroup(448,56);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,477,58,80,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^7=b^32=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=b^17>;
// generators/relations
Export