metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.7C42, M4(2)⋊5Dic7, C56.40(C2×C4), C56⋊C4⋊27C2, C4.7(C4×Dic7), (C8×Dic7)⋊30C2, (C2×C8).275D14, (C7×M4(2))⋊3C4, C4⋊Dic7.21C4, (C2×C14).7C42, C8.12(C2×Dic7), C23.28(C4×D7), C14.31(C8○D4), C7⋊5(C8○2M4(2)), C14.26(C2×C42), C2.5(D28.C4), C23.D7.11C4, C22.7(C4×Dic7), (C2×C56).233C22, (C2×C28).865C23, C28.177(C22×C4), (C22×C4).347D14, (C14×M4(2)).6C2, (C2×M4(2)).19D7, C4.35(C22×Dic7), (C22×C28).181C22, (C4×Dic7).285C22, C23.21D14.18C2, (C2×C7⋊C8)⋊9C4, C7⋊C8.24(C2×C4), C4.115(C2×C4×D7), (C2×C4).83(C4×D7), C2.14(C2×C4×Dic7), C22.63(C2×C4×D7), (C22×C7⋊C8).11C2, (C2×C28).101(C2×C4), (C2×C7⋊C8).334C22, (C2×C4).47(C2×Dic7), (C22×C14).65(C2×C4), (C2×Dic7).67(C2×C4), (C2×C4).807(C22×D7), (C2×C14).135(C22×C4), SmallGroup(448,656)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.7C42
G = < a,b,c | a28=1, b4=c4=a14, bab-1=a13, cac-1=a15, bc=cb >
Subgroups: 356 in 130 conjugacy classes, 87 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C7⋊C8, C56, C2×Dic7, C2×C28, C2×C28, C22×C14, C8○2M4(2), C2×C7⋊C8, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C7×M4(2), C22×C28, C8×Dic7, C56⋊C4, C22×C7⋊C8, C23.21D14, C14×M4(2), C28.7C42
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, D28.C4, C2×C4×Dic7, C28.7C42
►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)
(1 148 33 115 15 162 47 129)(2 161 34 128 16 147 48 114)(3 146 35 113 17 160 49 127)(4 159 36 126 18 145 50 140)(5 144 37 139 19 158 51 125)(6 157 38 124 20 143 52 138)(7 142 39 137 21 156 53 123)(8 155 40 122 22 141 54 136)(9 168 41 135 23 154 55 121)(10 153 42 120 24 167 56 134)(11 166 43 133 25 152 29 119)(12 151 44 118 26 165 30 132)(13 164 45 131 27 150 31 117)(14 149 46 116 28 163 32 130)(57 216 105 193 71 202 91 179)(58 201 106 178 72 215 92 192)(59 214 107 191 73 200 93 177)(60 199 108 176 74 213 94 190)(61 212 109 189 75 198 95 175)(62 197 110 174 76 211 96 188)(63 210 111 187 77 224 97 173)(64 223 112 172 78 209 98 186)(65 208 85 185 79 222 99 171)(66 221 86 170 80 207 100 184)(67 206 87 183 81 220 101 169)(68 219 88 196 82 205 102 182)(69 204 89 181 83 218 103 195)(70 217 90 194 84 203 104 180)
(1 70 8 63 15 84 22 77)(2 57 9 78 16 71 23 64)(3 72 10 65 17 58 24 79)(4 59 11 80 18 73 25 66)(5 74 12 67 19 60 26 81)(6 61 13 82 20 75 27 68)(7 76 14 69 21 62 28 83)(29 86 36 107 43 100 50 93)(30 101 37 94 44 87 51 108)(31 88 38 109 45 102 52 95)(32 103 39 96 46 89 53 110)(33 90 40 111 47 104 54 97)(34 105 41 98 48 91 55 112)(35 92 42 85 49 106 56 99)(113 192 120 185 127 178 134 171)(114 179 121 172 128 193 135 186)(115 194 122 187 129 180 136 173)(116 181 123 174 130 195 137 188)(117 196 124 189 131 182 138 175)(118 183 125 176 132 169 139 190)(119 170 126 191 133 184 140 177)(141 224 148 217 155 210 162 203)(142 211 149 204 156 197 163 218)(143 198 150 219 157 212 164 205)(144 213 151 206 158 199 165 220)(145 200 152 221 159 214 166 207)(146 215 153 208 160 201 167 222)(147 202 154 223 161 216 168 209)
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), (1,148,33,115,15,162,47,129)(2,161,34,128,16,147,48,114)(3,146,35,113,17,160,49,127)(4,159,36,126,18,145,50,140)(5,144,37,139,19,158,51,125)(6,157,38,124,20,143,52,138)(7,142,39,137,21,156,53,123)(8,155,40,122,22,141,54,136)(9,168,41,135,23,154,55,121)(10,153,42,120,24,167,56,134)(11,166,43,133,25,152,29,119)(12,151,44,118,26,165,30,132)(13,164,45,131,27,150,31,117)(14,149,46,116,28,163,32,130)(57,216,105,193,71,202,91,179)(58,201,106,178,72,215,92,192)(59,214,107,191,73,200,93,177)(60,199,108,176,74,213,94,190)(61,212,109,189,75,198,95,175)(62,197,110,174,76,211,96,188)(63,210,111,187,77,224,97,173)(64,223,112,172,78,209,98,186)(65,208,85,185,79,222,99,171)(66,221,86,170,80,207,100,184)(67,206,87,183,81,220,101,169)(68,219,88,196,82,205,102,182)(69,204,89,181,83,218,103,195)(70,217,90,194,84,203,104,180), (1,70,8,63,15,84,22,77)(2,57,9,78,16,71,23,64)(3,72,10,65,17,58,24,79)(4,59,11,80,18,73,25,66)(5,74,12,67,19,60,26,81)(6,61,13,82,20,75,27,68)(7,76,14,69,21,62,28,83)(29,86,36,107,43,100,50,93)(30,101,37,94,44,87,51,108)(31,88,38,109,45,102,52,95)(32,103,39,96,46,89,53,110)(33,90,40,111,47,104,54,97)(34,105,41,98,48,91,55,112)(35,92,42,85,49,106,56,99)(113,192,120,185,127,178,134,171)(114,179,121,172,128,193,135,186)(115,194,122,187,129,180,136,173)(116,181,123,174,130,195,137,188)(117,196,124,189,131,182,138,175)(118,183,125,176,132,169,139,190)(119,170,126,191,133,184,140,177)(141,224,148,217,155,210,162,203)(142,211,149,204,156,197,163,218)(143,198,150,219,157,212,164,205)(144,213,151,206,158,199,165,220)(145,200,152,221,159,214,166,207)(146,215,153,208,160,201,167,222)(147,202,154,223,161,216,168,209)>;
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), (1,148,33,115,15,162,47,129)(2,161,34,128,16,147,48,114)(3,146,35,113,17,160,49,127)(4,159,36,126,18,145,50,140)(5,144,37,139,19,158,51,125)(6,157,38,124,20,143,52,138)(7,142,39,137,21,156,53,123)(8,155,40,122,22,141,54,136)(9,168,41,135,23,154,55,121)(10,153,42,120,24,167,56,134)(11,166,43,133,25,152,29,119)(12,151,44,118,26,165,30,132)(13,164,45,131,27,150,31,117)(14,149,46,116,28,163,32,130)(57,216,105,193,71,202,91,179)(58,201,106,178,72,215,92,192)(59,214,107,191,73,200,93,177)(60,199,108,176,74,213,94,190)(61,212,109,189,75,198,95,175)(62,197,110,174,76,211,96,188)(63,210,111,187,77,224,97,173)(64,223,112,172,78,209,98,186)(65,208,85,185,79,222,99,171)(66,221,86,170,80,207,100,184)(67,206,87,183,81,220,101,169)(68,219,88,196,82,205,102,182)(69,204,89,181,83,218,103,195)(70,217,90,194,84,203,104,180), (1,70,8,63,15,84,22,77)(2,57,9,78,16,71,23,64)(3,72,10,65,17,58,24,79)(4,59,11,80,18,73,25,66)(5,74,12,67,19,60,26,81)(6,61,13,82,20,75,27,68)(7,76,14,69,21,62,28,83)(29,86,36,107,43,100,50,93)(30,101,37,94,44,87,51,108)(31,88,38,109,45,102,52,95)(32,103,39,96,46,89,53,110)(33,90,40,111,47,104,54,97)(34,105,41,98,48,91,55,112)(35,92,42,85,49,106,56,99)(113,192,120,185,127,178,134,171)(114,179,121,172,128,193,135,186)(115,194,122,187,129,180,136,173)(116,181,123,174,130,195,137,188)(117,196,124,189,131,182,138,175)(118,183,125,176,132,169,139,190)(119,170,126,191,133,184,140,177)(141,224,148,217,155,210,162,203)(142,211,149,204,156,197,163,218)(143,198,150,219,157,212,164,205)(144,213,151,206,158,199,165,220)(145,200,152,221,159,214,166,207)(146,215,153,208,160,201,167,222)(147,202,154,223,161,216,168,209) );
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)], [(1,148,33,115,15,162,47,129),(2,161,34,128,16,147,48,114),(3,146,35,113,17,160,49,127),(4,159,36,126,18,145,50,140),(5,144,37,139,19,158,51,125),(6,157,38,124,20,143,52,138),(7,142,39,137,21,156,53,123),(8,155,40,122,22,141,54,136),(9,168,41,135,23,154,55,121),(10,153,42,120,24,167,56,134),(11,166,43,133,25,152,29,119),(12,151,44,118,26,165,30,132),(13,164,45,131,27,150,31,117),(14,149,46,116,28,163,32,130),(57,216,105,193,71,202,91,179),(58,201,106,178,72,215,92,192),(59,214,107,191,73,200,93,177),(60,199,108,176,74,213,94,190),(61,212,109,189,75,198,95,175),(62,197,110,174,76,211,96,188),(63,210,111,187,77,224,97,173),(64,223,112,172,78,209,98,186),(65,208,85,185,79,222,99,171),(66,221,86,170,80,207,100,184),(67,206,87,183,81,220,101,169),(68,219,88,196,82,205,102,182),(69,204,89,181,83,218,103,195),(70,217,90,194,84,203,104,180)], [(1,70,8,63,15,84,22,77),(2,57,9,78,16,71,23,64),(3,72,10,65,17,58,24,79),(4,59,11,80,18,73,25,66),(5,74,12,67,19,60,26,81),(6,61,13,82,20,75,27,68),(7,76,14,69,21,62,28,83),(29,86,36,107,43,100,50,93),(30,101,37,94,44,87,51,108),(31,88,38,109,45,102,52,95),(32,103,39,96,46,89,53,110),(33,90,40,111,47,104,54,97),(34,105,41,98,48,91,55,112),(35,92,42,85,49,106,56,99),(113,192,120,185,127,178,134,171),(114,179,121,172,128,193,135,186),(115,194,122,187,129,180,136,173),(116,181,123,174,130,195,137,188),(117,196,124,189,131,182,138,175),(118,183,125,176,132,169,139,190),(119,170,126,191,133,184,140,177),(141,224,148,217,155,210,162,203),(142,211,149,204,156,197,163,218),(143,198,150,219,157,212,164,205),(144,213,151,206,158,199,165,220),(145,200,152,221,159,214,166,207),(146,215,153,208,160,201,167,222),(147,202,154,223,161,216,168,209)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8P | 8Q | 8R | 8S | 8T | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 7 | ··· | 7 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 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 | D14 | C8○D4 | C4×D7 | C4×D7 | D28.C4 |
| kernel | C28.7C42 | C8×Dic7 | C56⋊C4 | C22×C7⋊C8 | C23.21D14 | C14×M4(2) | C2×C7⋊C8 | C4⋊Dic7 | C23.D7 | C7×M4(2) | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C14 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 4 | 8 | 3 | 6 | 12 | 3 | 8 | 18 | 6 | 12 |
Matrix representation of C28.7C42 ►in GL5(𝔽113)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 98 | 0 | 0 |
| 0 | 0 | 0 | 112 | 79 |
| 0 | 0 | 0 | 34 | 25 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 95 | 0 | 0 | 0 |
| 0 | 0 | 95 | 0 | 0 |
| 0 | 0 | 0 | 60 | 82 |
| 0 | 0 | 0 | 76 | 53 |
| 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(5,GF(113))| [1,0,0,0,0,0,15,0,0,0,0,0,98,0,0,0,0,0,112,34,0,0,0,79,25],[112,0,0,0,0,0,95,0,0,0,0,0,95,0,0,0,0,0,60,76,0,0,0,82,53],[98,0,0,0,0,0,0,98,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,112] >;
C28.7C42 in GAP, Magma, Sage, TeX
C_{28}._7C_4^2 % in TeX
G:=Group("C28.7C4^2"); // GroupNames label
G:=SmallGroup(448,656);
// by ID
G=gap.SmallGroup(448,656);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,387,100,136,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=1,b^4=c^4=a^14,b*a*b^-1=a^13,c*a*c^-1=a^15,b*c=c*b>;
// generators/relations