metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.7(C4⋊C4), (C2×C28).5Q8, (C2×C4).12D28, C4.Dic7⋊2C4, (C2×C28).103D4, (C22×C28).3C4, (C2×C4).1Dic14, C4.32(D14⋊C4), C28.8(C22⋊C4), C4.7(Dic7⋊C4), (C2×C14).18C42, (C22×C4).56D14, C22.9(C4×Dic7), (C22×C4).2Dic7, C7⋊1(C22.C42), C14.7(C4.D4), C2.2(C28.D4), C22.9(C4⋊Dic7), C23.23(C2×Dic7), C2.2(C28.10D4), C14.7(C4.10D4), C14.6(C2.C42), C2.7(C14.C42), (C22×C28).119C22, C22.27(C23.D7), (C2×C4⋊C4).3D7, (C14×C4⋊C4).2C2, (C2×C4).18(C4×D7), (C2×C28).55(C2×C4), (C2×C14).36(C4⋊C4), (C2×C4.Dic7).6C2, (C2×C4).175(C7⋊D4), (C2×C14).88(C22⋊C4), (C22×C14).125(C2×C4), SmallGroup(448,87)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.(C4⋊C4)
G = < a,b,c | a28=c4=1, b4=a14, bab-1=a-1, cac-1=a15, cbc-1=a21b3 >
Subgroups: 292 in 98 conjugacy classes, 51 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C28, C28, C2×C14, C2×C14, C2×C4⋊C4, C2×M4(2), C7⋊C8, C2×C28, C2×C28, C2×C28, C22×C14, C22.C42, C2×C7⋊C8, C4.Dic7, C4.Dic7, C7×C4⋊C4, C22×C28, C22×C28, C2×C4.Dic7, C14×C4⋊C4, C28.(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, C4.D4, C4.10D4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C22.C42, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C14.C42, C28.D4, C28.10D4, C28.(C4⋊C4)
►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)
(1 165 22 144 15 151 8 158)(2 164 23 143 16 150 9 157)(3 163 24 142 17 149 10 156)(4 162 25 141 18 148 11 155)(5 161 26 168 19 147 12 154)(6 160 27 167 20 146 13 153)(7 159 28 166 21 145 14 152)(29 221 50 200 43 207 36 214)(30 220 51 199 44 206 37 213)(31 219 52 198 45 205 38 212)(32 218 53 197 46 204 39 211)(33 217 54 224 47 203 40 210)(34 216 55 223 48 202 41 209)(35 215 56 222 49 201 42 208)(57 172 64 193 71 186 78 179)(58 171 65 192 72 185 79 178)(59 170 66 191 73 184 80 177)(60 169 67 190 74 183 81 176)(61 196 68 189 75 182 82 175)(62 195 69 188 76 181 83 174)(63 194 70 187 77 180 84 173)(85 119 106 126 99 133 92 140)(86 118 107 125 100 132 93 139)(87 117 108 124 101 131 94 138)(88 116 109 123 102 130 95 137)(89 115 110 122 103 129 96 136)(90 114 111 121 104 128 97 135)(91 113 112 120 105 127 98 134)
(1 103 56 192)(2 90 29 179)(3 105 30 194)(4 92 31 181)(5 107 32 196)(6 94 33 183)(7 109 34 170)(8 96 35 185)(9 111 36 172)(10 98 37 187)(11 85 38 174)(12 100 39 189)(13 87 40 176)(14 102 41 191)(15 89 42 178)(16 104 43 193)(17 91 44 180)(18 106 45 195)(19 93 46 182)(20 108 47 169)(21 95 48 184)(22 110 49 171)(23 97 50 186)(24 112 51 173)(25 99 52 188)(26 86 53 175)(27 101 54 190)(28 88 55 177)(57 150 114 207)(58 165 115 222)(59 152 116 209)(60 167 117 224)(61 154 118 211)(62 141 119 198)(63 156 120 213)(64 143 121 200)(65 158 122 215)(66 145 123 202)(67 160 124 217)(68 147 125 204)(69 162 126 219)(70 149 127 206)(71 164 128 221)(72 151 129 208)(73 166 130 223)(74 153 131 210)(75 168 132 197)(76 155 133 212)(77 142 134 199)(78 157 135 214)(79 144 136 201)(80 159 137 216)(81 146 138 203)(82 161 139 218)(83 148 140 205)(84 163 113 220)
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), (1,165,22,144,15,151,8,158)(2,164,23,143,16,150,9,157)(3,163,24,142,17,149,10,156)(4,162,25,141,18,148,11,155)(5,161,26,168,19,147,12,154)(6,160,27,167,20,146,13,153)(7,159,28,166,21,145,14,152)(29,221,50,200,43,207,36,214)(30,220,51,199,44,206,37,213)(31,219,52,198,45,205,38,212)(32,218,53,197,46,204,39,211)(33,217,54,224,47,203,40,210)(34,216,55,223,48,202,41,209)(35,215,56,222,49,201,42,208)(57,172,64,193,71,186,78,179)(58,171,65,192,72,185,79,178)(59,170,66,191,73,184,80,177)(60,169,67,190,74,183,81,176)(61,196,68,189,75,182,82,175)(62,195,69,188,76,181,83,174)(63,194,70,187,77,180,84,173)(85,119,106,126,99,133,92,140)(86,118,107,125,100,132,93,139)(87,117,108,124,101,131,94,138)(88,116,109,123,102,130,95,137)(89,115,110,122,103,129,96,136)(90,114,111,121,104,128,97,135)(91,113,112,120,105,127,98,134), (1,103,56,192)(2,90,29,179)(3,105,30,194)(4,92,31,181)(5,107,32,196)(6,94,33,183)(7,109,34,170)(8,96,35,185)(9,111,36,172)(10,98,37,187)(11,85,38,174)(12,100,39,189)(13,87,40,176)(14,102,41,191)(15,89,42,178)(16,104,43,193)(17,91,44,180)(18,106,45,195)(19,93,46,182)(20,108,47,169)(21,95,48,184)(22,110,49,171)(23,97,50,186)(24,112,51,173)(25,99,52,188)(26,86,53,175)(27,101,54,190)(28,88,55,177)(57,150,114,207)(58,165,115,222)(59,152,116,209)(60,167,117,224)(61,154,118,211)(62,141,119,198)(63,156,120,213)(64,143,121,200)(65,158,122,215)(66,145,123,202)(67,160,124,217)(68,147,125,204)(69,162,126,219)(70,149,127,206)(71,164,128,221)(72,151,129,208)(73,166,130,223)(74,153,131,210)(75,168,132,197)(76,155,133,212)(77,142,134,199)(78,157,135,214)(79,144,136,201)(80,159,137,216)(81,146,138,203)(82,161,139,218)(83,148,140,205)(84,163,113,220)>;
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), (1,165,22,144,15,151,8,158)(2,164,23,143,16,150,9,157)(3,163,24,142,17,149,10,156)(4,162,25,141,18,148,11,155)(5,161,26,168,19,147,12,154)(6,160,27,167,20,146,13,153)(7,159,28,166,21,145,14,152)(29,221,50,200,43,207,36,214)(30,220,51,199,44,206,37,213)(31,219,52,198,45,205,38,212)(32,218,53,197,46,204,39,211)(33,217,54,224,47,203,40,210)(34,216,55,223,48,202,41,209)(35,215,56,222,49,201,42,208)(57,172,64,193,71,186,78,179)(58,171,65,192,72,185,79,178)(59,170,66,191,73,184,80,177)(60,169,67,190,74,183,81,176)(61,196,68,189,75,182,82,175)(62,195,69,188,76,181,83,174)(63,194,70,187,77,180,84,173)(85,119,106,126,99,133,92,140)(86,118,107,125,100,132,93,139)(87,117,108,124,101,131,94,138)(88,116,109,123,102,130,95,137)(89,115,110,122,103,129,96,136)(90,114,111,121,104,128,97,135)(91,113,112,120,105,127,98,134), (1,103,56,192)(2,90,29,179)(3,105,30,194)(4,92,31,181)(5,107,32,196)(6,94,33,183)(7,109,34,170)(8,96,35,185)(9,111,36,172)(10,98,37,187)(11,85,38,174)(12,100,39,189)(13,87,40,176)(14,102,41,191)(15,89,42,178)(16,104,43,193)(17,91,44,180)(18,106,45,195)(19,93,46,182)(20,108,47,169)(21,95,48,184)(22,110,49,171)(23,97,50,186)(24,112,51,173)(25,99,52,188)(26,86,53,175)(27,101,54,190)(28,88,55,177)(57,150,114,207)(58,165,115,222)(59,152,116,209)(60,167,117,224)(61,154,118,211)(62,141,119,198)(63,156,120,213)(64,143,121,200)(65,158,122,215)(66,145,123,202)(67,160,124,217)(68,147,125,204)(69,162,126,219)(70,149,127,206)(71,164,128,221)(72,151,129,208)(73,166,130,223)(74,153,131,210)(75,168,132,197)(76,155,133,212)(77,142,134,199)(78,157,135,214)(79,144,136,201)(80,159,137,216)(81,146,138,203)(82,161,139,218)(83,148,140,205)(84,163,113,220) );
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)], [(1,165,22,144,15,151,8,158),(2,164,23,143,16,150,9,157),(3,163,24,142,17,149,10,156),(4,162,25,141,18,148,11,155),(5,161,26,168,19,147,12,154),(6,160,27,167,20,146,13,153),(7,159,28,166,21,145,14,152),(29,221,50,200,43,207,36,214),(30,220,51,199,44,206,37,213),(31,219,52,198,45,205,38,212),(32,218,53,197,46,204,39,211),(33,217,54,224,47,203,40,210),(34,216,55,223,48,202,41,209),(35,215,56,222,49,201,42,208),(57,172,64,193,71,186,78,179),(58,171,65,192,72,185,79,178),(59,170,66,191,73,184,80,177),(60,169,67,190,74,183,81,176),(61,196,68,189,75,182,82,175),(62,195,69,188,76,181,83,174),(63,194,70,187,77,180,84,173),(85,119,106,126,99,133,92,140),(86,118,107,125,100,132,93,139),(87,117,108,124,101,131,94,138),(88,116,109,123,102,130,95,137),(89,115,110,122,103,129,96,136),(90,114,111,121,104,128,97,135),(91,113,112,120,105,127,98,134)], [(1,103,56,192),(2,90,29,179),(3,105,30,194),(4,92,31,181),(5,107,32,196),(6,94,33,183),(7,109,34,170),(8,96,35,185),(9,111,36,172),(10,98,37,187),(11,85,38,174),(12,100,39,189),(13,87,40,176),(14,102,41,191),(15,89,42,178),(16,104,43,193),(17,91,44,180),(18,106,45,195),(19,93,46,182),(20,108,47,169),(21,95,48,184),(22,110,49,171),(23,97,50,186),(24,112,51,173),(25,99,52,188),(26,86,53,175),(27,101,54,190),(28,88,55,177),(57,150,114,207),(58,165,115,222),(59,152,116,209),(60,167,117,224),(61,154,118,211),(62,141,119,198),(63,156,120,213),(64,143,121,200),(65,158,122,215),(66,145,123,202),(67,160,124,217),(68,147,125,204),(69,162,126,219),(70,149,127,206),(71,164,128,221),(72,151,129,208),(73,166,130,223),(74,153,131,210),(75,168,132,197),(76,155,133,212),(77,142,134,199),(78,157,135,214),(79,144,136,201),(80,159,137,216),(81,146,138,203),(82,161,139,218),(83,148,140,205),(84,163,113,220)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 28 | ··· | 28 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D7 | Dic7 | D14 | Dic14 | C4×D7 | D28 | C7⋊D4 | C4.D4 | C4.10D4 | C28.D4 | C28.10D4 |
| kernel | C28.(C4⋊C4) | C2×C4.Dic7 | C14×C4⋊C4 | C4.Dic7 | C22×C28 | C2×C28 | C2×C28 | C2×C4⋊C4 | C22×C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C14 | C14 | C2 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 3 | 1 | 3 | 6 | 3 | 6 | 12 | 6 | 12 | 1 | 1 | 6 | 6 |
Matrix representation of C28.(C4⋊C4) ►in GL6(𝔽113)
| 4 | 97 | 0 | 0 | 0 | 0 |
| 0 | 85 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 97 | 0 | 0 |
| 0 | 0 | 60 | 49 | 0 | 0 |
| 0 | 0 | 51 | 85 | 0 | 83 |
| 0 | 0 | 14 | 72 | 30 | 0 |
| 26 | 37 | 0 | 0 | 0 | 0 |
| 52 | 87 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 74 | 111 | 0 |
| 0 | 0 | 0 | 19 | 105 | 105 |
| 0 | 0 | 16 | 31 | 31 | 43 |
| 0 | 0 | 98 | 14 | 63 | 51 |
| 98 | 15 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 97 | 108 | 0 | 0 |
| 0 | 0 | 51 | 16 | 0 | 0 |
| 0 | 0 | 79 | 15 | 77 | 93 |
| 0 | 0 | 8 | 50 | 93 | 36 |
G:=sub<GL(6,GF(113))| [4,0,0,0,0,0,97,85,0,0,0,0,0,0,64,60,51,14,0,0,97,49,85,72,0,0,0,0,0,30,0,0,0,0,83,0],[26,52,0,0,0,0,37,87,0,0,0,0,0,0,12,0,16,98,0,0,74,19,31,14,0,0,111,105,31,63,0,0,0,105,43,51],[98,0,0,0,0,0,15,15,0,0,0,0,0,0,97,51,79,8,0,0,108,16,15,50,0,0,0,0,77,93,0,0,0,0,93,36] >;
C28.(C4⋊C4) in GAP, Magma, Sage, TeX
C_{28}.(C_4\rtimes C_4) % in TeX
G:=Group("C28.(C4:C4)"); // GroupNames label
G:=SmallGroup(448,87);
// by ID
G=gap.SmallGroup(448,87);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,387,184,1684,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=c^4=1,b^4=a^14,b*a*b^-1=a^-1,c*a*c^-1=a^15,c*b*c^-1=a^21*b^3>;
// generators/relations