metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.5C42, C42.186D14, C4.Dic7⋊9C4, C4.5(C4×Dic7), C4⋊C4.10Dic7, (C2×C14).5C42, C14.35(C8○D4), C7⋊3(C8○2M4(2)), C14.21(C2×C42), C22⋊C4.6Dic7, C22.5(C4×Dic7), (C4×C28).231C22, C28.136(C22×C4), (C2×C28).845C23, C42⋊C2.14D7, C42.D7⋊21C2, (C22×C4).336D14, C2.1(Q8.Dic7), C23.14(C2×Dic7), (C22×C28).147C22, C22.21(C22×Dic7), (C2×C7⋊C8)⋊7C4, (C4×C7⋊C8)⋊25C2, (C7×C4⋊C4).7C4, C7⋊C8.23(C2×C4), C4.110(C2×C4×D7), (C2×C4).80(C4×D7), (C22×C7⋊C8).6C2, C2.10(C2×C4×Dic7), (C2×C28).88(C2×C4), (C7×C22⋊C4).7C4, (C2×C7⋊C8).332C22, (C2×C4).42(C2×Dic7), (C22×C14).56(C2×C4), (C7×C42⋊C2).6C2, (C2×C4).787(C22×D7), (C2×C4.Dic7).17C2, (C2×C14).182(C22×C4), SmallGroup(448,531)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.5C42
G = < a,b,c | a28=c4=1, b4=a14, bab-1=a13, cac-1=a15, bc=cb >
Subgroups: 308 in 130 conjugacy classes, 87 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C28, C28, C28, C2×C14, C2×C14, C2×C14, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C7⋊C8, C2×C28, C2×C28, C22×C14, C8○2M4(2), C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C4×C7⋊C8, C42.D7, C22×C7⋊C8, C2×C4.Dic7, C7×C42⋊C2, C28.5C42
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C42, C22×C4, Dic7, D14, C2×C42, C8○D4, C4×D7, C2×Dic7, C22×D7, C8○2M4(2), C4×Dic7, C2×C4×D7, C22×Dic7, C2×C4×Dic7, Q8.Dic7, C28.5C42
►On 224 points
Generators in S224
(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)
(1 101 152 189 15 87 166 175)(2 86 153 174 16 100 167 188)(3 99 154 187 17 85 168 173)(4 112 155 172 18 98 141 186)(5 97 156 185 19 111 142 171)(6 110 157 170 20 96 143 184)(7 95 158 183 21 109 144 169)(8 108 159 196 22 94 145 182)(9 93 160 181 23 107 146 195)(10 106 161 194 24 92 147 180)(11 91 162 179 25 105 148 193)(12 104 163 192 26 90 149 178)(13 89 164 177 27 103 150 191)(14 102 165 190 28 88 151 176)(29 137 73 200 43 123 59 214)(30 122 74 213 44 136 60 199)(31 135 75 198 45 121 61 212)(32 120 76 211 46 134 62 197)(33 133 77 224 47 119 63 210)(34 118 78 209 48 132 64 223)(35 131 79 222 49 117 65 208)(36 116 80 207 50 130 66 221)(37 129 81 220 51 115 67 206)(38 114 82 205 52 128 68 219)(39 127 83 218 53 113 69 204)(40 140 84 203 54 126 70 217)(41 125 57 216 55 139 71 202)(42 138 58 201 56 124 72 215)
(1 212 159 128)(2 199 160 115)(3 214 161 130)(4 201 162 117)(5 216 163 132)(6 203 164 119)(7 218 165 134)(8 205 166 121)(9 220 167 136)(10 207 168 123)(11 222 141 138)(12 209 142 125)(13 224 143 140)(14 211 144 127)(15 198 145 114)(16 213 146 129)(17 200 147 116)(18 215 148 131)(19 202 149 118)(20 217 150 133)(21 204 151 120)(22 219 152 135)(23 206 153 122)(24 221 154 137)(25 208 155 124)(26 223 156 139)(27 210 157 126)(28 197 158 113)(29 194 66 99)(30 181 67 86)(31 196 68 101)(32 183 69 88)(33 170 70 103)(34 185 71 90)(35 172 72 105)(36 187 73 92)(37 174 74 107)(38 189 75 94)(39 176 76 109)(40 191 77 96)(41 178 78 111)(42 193 79 98)(43 180 80 85)(44 195 81 100)(45 182 82 87)(46 169 83 102)(47 184 84 89)(48 171 57 104)(49 186 58 91)(50 173 59 106)(51 188 60 93)(52 175 61 108)(53 190 62 95)(54 177 63 110)(55 192 64 97)(56 179 65 112)
G:=sub<Sym(224)| (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), (1,101,152,189,15,87,166,175)(2,86,153,174,16,100,167,188)(3,99,154,187,17,85,168,173)(4,112,155,172,18,98,141,186)(5,97,156,185,19,111,142,171)(6,110,157,170,20,96,143,184)(7,95,158,183,21,109,144,169)(8,108,159,196,22,94,145,182)(9,93,160,181,23,107,146,195)(10,106,161,194,24,92,147,180)(11,91,162,179,25,105,148,193)(12,104,163,192,26,90,149,178)(13,89,164,177,27,103,150,191)(14,102,165,190,28,88,151,176)(29,137,73,200,43,123,59,214)(30,122,74,213,44,136,60,199)(31,135,75,198,45,121,61,212)(32,120,76,211,46,134,62,197)(33,133,77,224,47,119,63,210)(34,118,78,209,48,132,64,223)(35,131,79,222,49,117,65,208)(36,116,80,207,50,130,66,221)(37,129,81,220,51,115,67,206)(38,114,82,205,52,128,68,219)(39,127,83,218,53,113,69,204)(40,140,84,203,54,126,70,217)(41,125,57,216,55,139,71,202)(42,138,58,201,56,124,72,215), (1,212,159,128)(2,199,160,115)(3,214,161,130)(4,201,162,117)(5,216,163,132)(6,203,164,119)(7,218,165,134)(8,205,166,121)(9,220,167,136)(10,207,168,123)(11,222,141,138)(12,209,142,125)(13,224,143,140)(14,211,144,127)(15,198,145,114)(16,213,146,129)(17,200,147,116)(18,215,148,131)(19,202,149,118)(20,217,150,133)(21,204,151,120)(22,219,152,135)(23,206,153,122)(24,221,154,137)(25,208,155,124)(26,223,156,139)(27,210,157,126)(28,197,158,113)(29,194,66,99)(30,181,67,86)(31,196,68,101)(32,183,69,88)(33,170,70,103)(34,185,71,90)(35,172,72,105)(36,187,73,92)(37,174,74,107)(38,189,75,94)(39,176,76,109)(40,191,77,96)(41,178,78,111)(42,193,79,98)(43,180,80,85)(44,195,81,100)(45,182,82,87)(46,169,83,102)(47,184,84,89)(48,171,57,104)(49,186,58,91)(50,173,59,106)(51,188,60,93)(52,175,61,108)(53,190,62,95)(54,177,63,110)(55,192,64,97)(56,179,65,112)>;
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)(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), (1,101,152,189,15,87,166,175)(2,86,153,174,16,100,167,188)(3,99,154,187,17,85,168,173)(4,112,155,172,18,98,141,186)(5,97,156,185,19,111,142,171)(6,110,157,170,20,96,143,184)(7,95,158,183,21,109,144,169)(8,108,159,196,22,94,145,182)(9,93,160,181,23,107,146,195)(10,106,161,194,24,92,147,180)(11,91,162,179,25,105,148,193)(12,104,163,192,26,90,149,178)(13,89,164,177,27,103,150,191)(14,102,165,190,28,88,151,176)(29,137,73,200,43,123,59,214)(30,122,74,213,44,136,60,199)(31,135,75,198,45,121,61,212)(32,120,76,211,46,134,62,197)(33,133,77,224,47,119,63,210)(34,118,78,209,48,132,64,223)(35,131,79,222,49,117,65,208)(36,116,80,207,50,130,66,221)(37,129,81,220,51,115,67,206)(38,114,82,205,52,128,68,219)(39,127,83,218,53,113,69,204)(40,140,84,203,54,126,70,217)(41,125,57,216,55,139,71,202)(42,138,58,201,56,124,72,215), (1,212,159,128)(2,199,160,115)(3,214,161,130)(4,201,162,117)(5,216,163,132)(6,203,164,119)(7,218,165,134)(8,205,166,121)(9,220,167,136)(10,207,168,123)(11,222,141,138)(12,209,142,125)(13,224,143,140)(14,211,144,127)(15,198,145,114)(16,213,146,129)(17,200,147,116)(18,215,148,131)(19,202,149,118)(20,217,150,133)(21,204,151,120)(22,219,152,135)(23,206,153,122)(24,221,154,137)(25,208,155,124)(26,223,156,139)(27,210,157,126)(28,197,158,113)(29,194,66,99)(30,181,67,86)(31,196,68,101)(32,183,69,88)(33,170,70,103)(34,185,71,90)(35,172,72,105)(36,187,73,92)(37,174,74,107)(38,189,75,94)(39,176,76,109)(40,191,77,96)(41,178,78,111)(42,193,79,98)(43,180,80,85)(44,195,81,100)(45,182,82,87)(46,169,83,102)(47,184,84,89)(48,171,57,104)(49,186,58,91)(50,173,59,106)(51,188,60,93)(52,175,61,108)(53,190,62,95)(54,177,63,110)(55,192,64,97)(56,179,65,112) );
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),(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)], [(1,101,152,189,15,87,166,175),(2,86,153,174,16,100,167,188),(3,99,154,187,17,85,168,173),(4,112,155,172,18,98,141,186),(5,97,156,185,19,111,142,171),(6,110,157,170,20,96,143,184),(7,95,158,183,21,109,144,169),(8,108,159,196,22,94,145,182),(9,93,160,181,23,107,146,195),(10,106,161,194,24,92,147,180),(11,91,162,179,25,105,148,193),(12,104,163,192,26,90,149,178),(13,89,164,177,27,103,150,191),(14,102,165,190,28,88,151,176),(29,137,73,200,43,123,59,214),(30,122,74,213,44,136,60,199),(31,135,75,198,45,121,61,212),(32,120,76,211,46,134,62,197),(33,133,77,224,47,119,63,210),(34,118,78,209,48,132,64,223),(35,131,79,222,49,117,65,208),(36,116,80,207,50,130,66,221),(37,129,81,220,51,115,67,206),(38,114,82,205,52,128,68,219),(39,127,83,218,53,113,69,204),(40,140,84,203,54,126,70,217),(41,125,57,216,55,139,71,202),(42,138,58,201,56,124,72,215)], [(1,212,159,128),(2,199,160,115),(3,214,161,130),(4,201,162,117),(5,216,163,132),(6,203,164,119),(7,218,165,134),(8,205,166,121),(9,220,167,136),(10,207,168,123),(11,222,141,138),(12,209,142,125),(13,224,143,140),(14,211,144,127),(15,198,145,114),(16,213,146,129),(17,200,147,116),(18,215,148,131),(19,202,149,118),(20,217,150,133),(21,204,151,120),(22,219,152,135),(23,206,153,122),(24,221,154,137),(25,208,155,124),(26,223,156,139),(27,210,157,126),(28,197,158,113),(29,194,66,99),(30,181,67,86),(31,196,68,101),(32,183,69,88),(33,170,70,103),(34,185,71,90),(35,172,72,105),(36,187,73,92),(37,174,74,107),(38,189,75,94),(39,176,76,109),(40,191,77,96),(41,178,78,111),(42,193,79,98),(43,180,80,85),(44,195,81,100),(45,182,82,87),(46,169,83,102),(47,184,84,89),(48,171,57,104),(49,186,58,91),(50,173,59,106),(51,188,60,93),(52,175,61,108),(53,190,62,95),(54,177,63,110),(55,192,64,97),(56,179,65,112)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8T | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D7 | D14 | Dic7 | Dic7 | D14 | C8○D4 | C4×D7 | Q8.Dic7 |
| kernel | C28.5C42 | C4×C7⋊C8 | C42.D7 | C22×C7⋊C8 | C2×C4.Dic7 | C7×C42⋊C2 | C2×C7⋊C8 | C4.Dic7 | C7×C22⋊C4 | C7×C4⋊C4 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C14 | C2×C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 8 | 4 | 4 | 3 | 6 | 6 | 6 | 3 | 8 | 24 | 12 |
Matrix representation of C28.5C42 ►in GL5(𝔽113)
| 112 | 0 | 0 | 0 | 0 |
| 0 | 112 | 24 | 0 | 0 |
| 0 | 89 | 10 | 0 | 0 |
| 0 | 0 | 0 | 98 | 44 |
| 0 | 0 | 0 | 0 | 15 |
| 15 | 0 | 0 | 0 | 0 |
| 0 | 66 | 76 | 0 | 0 |
| 0 | 78 | 47 | 0 | 0 |
| 0 | 0 | 0 | 69 | 0 |
| 0 | 0 | 0 | 0 | 69 |
| 15 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 77 | 98 |
G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,112,89,0,0,0,24,10,0,0,0,0,0,98,0,0,0,0,44,15],[15,0,0,0,0,0,66,78,0,0,0,76,47,0,0,0,0,0,69,0,0,0,0,0,69],[15,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,15,77,0,0,0,0,98] >;
C28.5C42 in GAP, Magma, Sage, TeX
C_{28}._5C_4^2 % in TeX
G:=Group("C28.5C4^2"); // GroupNames label
G:=SmallGroup(448,531);
// by ID
G=gap.SmallGroup(448,531);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,387,100,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=c^4=1,b^4=a^14,b*a*b^-1=a^13,c*a*c^-1=a^15,b*c=c*b>;
// generators/relations