metabelian, supersoluble, monomial
Aliases: C7⋊1D28, C28⋊1D7, C72⋊5D4, C14.14D14, C4⋊(C7⋊D7), (C7×C28)⋊1C2, (C7×C14).13C22, (C2×C7⋊D7)⋊2C2, C2.4(C2×C7⋊D7), SmallGroup(392,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊D7
G = < a,b,c | a28=b7=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Subgroups: 772 in 80 conjugacy classes, 33 normal (7 characteristic)
C1, C2, C2, C4, C22, C7, D4, D7, C14, C28, D14, C72, D28, C7⋊D7, C7×C14, C7×C28, C2×C7⋊D7, C28⋊D7
Quotients: C1, C2, C22, D4, D7, D14, D28, C7⋊D7, C2×C7⋊D7, C28⋊D7
►On 196 points
Generators in S196
(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)
(1 187 90 120 31 141 78)(2 188 91 121 32 142 79)(3 189 92 122 33 143 80)(4 190 93 123 34 144 81)(5 191 94 124 35 145 82)(6 192 95 125 36 146 83)(7 193 96 126 37 147 84)(8 194 97 127 38 148 57)(9 195 98 128 39 149 58)(10 196 99 129 40 150 59)(11 169 100 130 41 151 60)(12 170 101 131 42 152 61)(13 171 102 132 43 153 62)(14 172 103 133 44 154 63)(15 173 104 134 45 155 64)(16 174 105 135 46 156 65)(17 175 106 136 47 157 66)(18 176 107 137 48 158 67)(19 177 108 138 49 159 68)(20 178 109 139 50 160 69)(21 179 110 140 51 161 70)(22 180 111 113 52 162 71)(23 181 112 114 53 163 72)(24 182 85 115 54 164 73)(25 183 86 116 55 165 74)(26 184 87 117 56 166 75)(27 185 88 118 29 167 76)(28 186 89 119 30 168 77)
(1 57)(2 84)(3 83)(4 82)(5 81)(6 80)(7 79)(8 78)(9 77)(10 76)(11 75)(12 74)(13 73)(14 72)(15 71)(16 70)(17 69)(18 68)(19 67)(20 66)(21 65)(22 64)(23 63)(24 62)(25 61)(26 60)(27 59)(28 58)(29 99)(30 98)(31 97)(32 96)(33 95)(34 94)(35 93)(36 92)(37 91)(38 90)(39 89)(40 88)(41 87)(42 86)(43 85)(44 112)(45 111)(46 110)(47 109)(48 108)(49 107)(50 106)(51 105)(52 104)(53 103)(54 102)(55 101)(56 100)(113 134)(114 133)(115 132)(116 131)(117 130)(118 129)(119 128)(120 127)(121 126)(122 125)(123 124)(135 140)(136 139)(137 138)(141 194)(142 193)(143 192)(144 191)(145 190)(146 189)(147 188)(148 187)(149 186)(150 185)(151 184)(152 183)(153 182)(154 181)(155 180)(156 179)(157 178)(158 177)(159 176)(160 175)(161 174)(162 173)(163 172)(164 171)(165 170)(166 169)(167 196)(168 195)
G:=sub<Sym(196)| (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), (1,187,90,120,31,141,78)(2,188,91,121,32,142,79)(3,189,92,122,33,143,80)(4,190,93,123,34,144,81)(5,191,94,124,35,145,82)(6,192,95,125,36,146,83)(7,193,96,126,37,147,84)(8,194,97,127,38,148,57)(9,195,98,128,39,149,58)(10,196,99,129,40,150,59)(11,169,100,130,41,151,60)(12,170,101,131,42,152,61)(13,171,102,132,43,153,62)(14,172,103,133,44,154,63)(15,173,104,134,45,155,64)(16,174,105,135,46,156,65)(17,175,106,136,47,157,66)(18,176,107,137,48,158,67)(19,177,108,138,49,159,68)(20,178,109,139,50,160,69)(21,179,110,140,51,161,70)(22,180,111,113,52,162,71)(23,181,112,114,53,163,72)(24,182,85,115,54,164,73)(25,183,86,116,55,165,74)(26,184,87,117,56,166,75)(27,185,88,118,29,167,76)(28,186,89,119,30,168,77), (1,57)(2,84)(3,83)(4,82)(5,81)(6,80)(7,79)(8,78)(9,77)(10,76)(11,75)(12,74)(13,73)(14,72)(15,71)(16,70)(17,69)(18,68)(19,67)(20,66)(21,65)(22,64)(23,63)(24,62)(25,61)(26,60)(27,59)(28,58)(29,99)(30,98)(31,97)(32,96)(33,95)(34,94)(35,93)(36,92)(37,91)(38,90)(39,89)(40,88)(41,87)(42,86)(43,85)(44,112)(45,111)(46,110)(47,109)(48,108)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,100)(113,134)(114,133)(115,132)(116,131)(117,130)(118,129)(119,128)(120,127)(121,126)(122,125)(123,124)(135,140)(136,139)(137,138)(141,194)(142,193)(143,192)(144,191)(145,190)(146,189)(147,188)(148,187)(149,186)(150,185)(151,184)(152,183)(153,182)(154,181)(155,180)(156,179)(157,178)(158,177)(159,176)(160,175)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,196)(168,195)>;
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), (1,187,90,120,31,141,78)(2,188,91,121,32,142,79)(3,189,92,122,33,143,80)(4,190,93,123,34,144,81)(5,191,94,124,35,145,82)(6,192,95,125,36,146,83)(7,193,96,126,37,147,84)(8,194,97,127,38,148,57)(9,195,98,128,39,149,58)(10,196,99,129,40,150,59)(11,169,100,130,41,151,60)(12,170,101,131,42,152,61)(13,171,102,132,43,153,62)(14,172,103,133,44,154,63)(15,173,104,134,45,155,64)(16,174,105,135,46,156,65)(17,175,106,136,47,157,66)(18,176,107,137,48,158,67)(19,177,108,138,49,159,68)(20,178,109,139,50,160,69)(21,179,110,140,51,161,70)(22,180,111,113,52,162,71)(23,181,112,114,53,163,72)(24,182,85,115,54,164,73)(25,183,86,116,55,165,74)(26,184,87,117,56,166,75)(27,185,88,118,29,167,76)(28,186,89,119,30,168,77), (1,57)(2,84)(3,83)(4,82)(5,81)(6,80)(7,79)(8,78)(9,77)(10,76)(11,75)(12,74)(13,73)(14,72)(15,71)(16,70)(17,69)(18,68)(19,67)(20,66)(21,65)(22,64)(23,63)(24,62)(25,61)(26,60)(27,59)(28,58)(29,99)(30,98)(31,97)(32,96)(33,95)(34,94)(35,93)(36,92)(37,91)(38,90)(39,89)(40,88)(41,87)(42,86)(43,85)(44,112)(45,111)(46,110)(47,109)(48,108)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,100)(113,134)(114,133)(115,132)(116,131)(117,130)(118,129)(119,128)(120,127)(121,126)(122,125)(123,124)(135,140)(136,139)(137,138)(141,194)(142,193)(143,192)(144,191)(145,190)(146,189)(147,188)(148,187)(149,186)(150,185)(151,184)(152,183)(153,182)(154,181)(155,180)(156,179)(157,178)(158,177)(159,176)(160,175)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,196)(168,195) );
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)], [(1,187,90,120,31,141,78),(2,188,91,121,32,142,79),(3,189,92,122,33,143,80),(4,190,93,123,34,144,81),(5,191,94,124,35,145,82),(6,192,95,125,36,146,83),(7,193,96,126,37,147,84),(8,194,97,127,38,148,57),(9,195,98,128,39,149,58),(10,196,99,129,40,150,59),(11,169,100,130,41,151,60),(12,170,101,131,42,152,61),(13,171,102,132,43,153,62),(14,172,103,133,44,154,63),(15,173,104,134,45,155,64),(16,174,105,135,46,156,65),(17,175,106,136,47,157,66),(18,176,107,137,48,158,67),(19,177,108,138,49,159,68),(20,178,109,139,50,160,69),(21,179,110,140,51,161,70),(22,180,111,113,52,162,71),(23,181,112,114,53,163,72),(24,182,85,115,54,164,73),(25,183,86,116,55,165,74),(26,184,87,117,56,166,75),(27,185,88,118,29,167,76),(28,186,89,119,30,168,77)], [(1,57),(2,84),(3,83),(4,82),(5,81),(6,80),(7,79),(8,78),(9,77),(10,76),(11,75),(12,74),(13,73),(14,72),(15,71),(16,70),(17,69),(18,68),(19,67),(20,66),(21,65),(22,64),(23,63),(24,62),(25,61),(26,60),(27,59),(28,58),(29,99),(30,98),(31,97),(32,96),(33,95),(34,94),(35,93),(36,92),(37,91),(38,90),(39,89),(40,88),(41,87),(42,86),(43,85),(44,112),(45,111),(46,110),(47,109),(48,108),(49,107),(50,106),(51,105),(52,104),(53,103),(54,102),(55,101),(56,100),(113,134),(114,133),(115,132),(116,131),(117,130),(118,129),(119,128),(120,127),(121,126),(122,125),(123,124),(135,140),(136,139),(137,138),(141,194),(142,193),(143,192),(144,191),(145,190),(146,189),(147,188),(148,187),(149,186),(150,185),(151,184),(152,183),(153,182),(154,181),(155,180),(156,179),(157,178),(158,177),(159,176),(160,175),(161,174),(162,173),(163,172),(164,171),(165,170),(166,169),(167,196),(168,195)]])
101 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 7A | ··· | 7X | 14A | ··· | 14X | 28A | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 98 | 98 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
101 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D7 | D14 | D28 |
| kernel | C28⋊D7 | C7×C28 | C2×C7⋊D7 | C72 | C28 | C14 | C7 |
| # reps | 1 | 1 | 2 | 1 | 24 | 24 | 48 |
Matrix representation of C28⋊D7 ►in GL4(𝔽29) generated by
| 10 | 10 | 0 | 0 |
| 19 | 22 | 0 | 0 |
| 0 | 0 | 23 | 24 |
| 0 | 0 | 7 | 25 |
| 0 | 1 | 0 | 0 |
| 28 | 7 | 0 | 0 |
| 0 | 0 | 22 | 1 |
| 0 | 0 | 16 | 10 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 20 | 2 |
| 0 | 0 | 18 | 9 |
G:=sub<GL(4,GF(29))| [10,19,0,0,10,22,0,0,0,0,23,7,0,0,24,25],[0,28,0,0,1,7,0,0,0,0,22,16,0,0,1,10],[0,1,0,0,1,0,0,0,0,0,20,18,0,0,2,9] >;
C28⋊D7 in GAP, Magma, Sage, TeX
C_{28}\rtimes D_7 % in TeX
G:=Group("C28:D7"); // GroupNames label
G:=SmallGroup(392,30);
// by ID
G=gap.SmallGroup(392,30);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,-7,61,26,963,8404]);
// Polycyclic
G:=Group<a,b,c|a^28=b^7=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations