G = C28.45(C4⋊C4) order 448 = 26·7
23rd non-split extension by C28 of C4⋊C4 acting via C4⋊C4/C2×C4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.45(C4⋊C4), (C2×C28).22Q8, C28.65(C2×Q8), C4⋊C4.234D14, (C2×C28).495D4, C14.84(C4○D8), C28.Q8⋊44C2, C28.63(C22×C4), C4.Dic14⋊43C2, (C2×C4).32Dic14, C4.30(C2×Dic14), (C22×C14).73D4, C42⋊C2.6D7, C4.32(Dic7⋊C4), (C2×C28).327C23, (C22×C4).337D14, C23.37(C7⋊D4), C7⋊3(C23.25D4), C2.1(D4.8D14), C4⋊Dic7.325C22, C22.9(Dic7⋊C4), (C22×C28).148C22, C23.21D14.13C2, (C2×C7⋊C8)⋊8C4, C4.88(C2×C4×D7), C7⋊C8.19(C2×C4), C14.40(C2×C4⋊C4), (C22×C7⋊C8).7C2, (C2×C28).89(C2×C4), (C2×C4).155(C4×D7), (C2×C14).12(C4⋊C4), (C2×C14).456(C2×D4), (C2×C7⋊C8).242C22, C2.14(C2×Dic7⋊C4), C22.71(C2×C7⋊D4), (C2×C4).273(C7⋊D4), (C7×C4⋊C4).265C22, (C7×C42⋊C2).7C2, (C2×C4).427(C22×D7), SmallGroup(448,532)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.45(C4⋊C4)
G = < a,b,c | a28=c4=1, b4=a14, bab-1=a13, ac=ca, cbc-1=a14b3 >
Subgroups: 372 in 114 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, Dic7, C28, C28, C28, C2×C14, C2×C14, C2×C14, C4.Q8, C2.D8, C42⋊C2, C42⋊C2, C22×C8, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C2×C28, C22×C14, C23.25D4, C2×C7⋊C8, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, C23.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C28.Q8, C4.Dic14, C22×C7⋊C8, C23.21D14, C7×C42⋊C2, C28.45(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, D14, C2×C4⋊C4, C4○D8, Dic14, C4×D7, C7⋊D4, C22×D7, C23.25D4, Dic7⋊C4, C2×Dic14, C2×C4×D7, C2×C7⋊D4, C2×Dic7⋊C4, D4.8D14, C28.45(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)
(1 188 122 166 15 174 136 152)(2 173 123 151 16 187 137 165)(3 186 124 164 17 172 138 150)(4 171 125 149 18 185 139 163)(5 184 126 162 19 170 140 148)(6 169 127 147 20 183 113 161)(7 182 128 160 21 196 114 146)(8 195 129 145 22 181 115 159)(9 180 130 158 23 194 116 144)(10 193 131 143 24 179 117 157)(11 178 132 156 25 192 118 142)(12 191 133 141 26 177 119 155)(13 176 134 154 27 190 120 168)(14 189 135 167 28 175 121 153)(29 67 222 103 43 81 208 89)(30 80 223 88 44 66 209 102)(31 65 224 101 45 79 210 87)(32 78 197 86 46 64 211 100)(33 63 198 99 47 77 212 85)(34 76 199 112 48 62 213 98)(35 61 200 97 49 75 214 111)(36 74 201 110 50 60 215 96)(37 59 202 95 51 73 216 109)(38 72 203 108 52 58 217 94)(39 57 204 93 53 71 218 107)(40 70 205 106 54 84 219 92)(41 83 206 91 55 69 220 105)(42 68 207 104 56 82 221 90)
(1 61 129 90)(2 62 130 91)(3 63 131 92)(4 64 132 93)(5 65 133 94)(6 66 134 95)(7 67 135 96)(8 68 136 97)(9 69 137 98)(10 70 138 99)(11 71 139 100)(12 72 140 101)(13 73 113 102)(14 74 114 103)(15 75 115 104)(16 76 116 105)(17 77 117 106)(18 78 118 107)(19 79 119 108)(20 80 120 109)(21 81 121 110)(22 82 122 111)(23 83 123 112)(24 84 124 85)(25 57 125 86)(26 58 126 87)(27 59 127 88)(28 60 128 89)(29 167 215 182)(30 168 216 183)(31 141 217 184)(32 142 218 185)(33 143 219 186)(34 144 220 187)(35 145 221 188)(36 146 222 189)(37 147 223 190)(38 148 224 191)(39 149 197 192)(40 150 198 193)(41 151 199 194)(42 152 200 195)(43 153 201 196)(44 154 202 169)(45 155 203 170)(46 156 204 171)(47 157 205 172)(48 158 206 173)(49 159 207 174)(50 160 208 175)(51 161 209 176)(52 162 210 177)(53 163 211 178)(54 164 212 179)(55 165 213 180)(56 166 214 181)
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), (1,188,122,166,15,174,136,152)(2,173,123,151,16,187,137,165)(3,186,124,164,17,172,138,150)(4,171,125,149,18,185,139,163)(5,184,126,162,19,170,140,148)(6,169,127,147,20,183,113,161)(7,182,128,160,21,196,114,146)(8,195,129,145,22,181,115,159)(9,180,130,158,23,194,116,144)(10,193,131,143,24,179,117,157)(11,178,132,156,25,192,118,142)(12,191,133,141,26,177,119,155)(13,176,134,154,27,190,120,168)(14,189,135,167,28,175,121,153)(29,67,222,103,43,81,208,89)(30,80,223,88,44,66,209,102)(31,65,224,101,45,79,210,87)(32,78,197,86,46,64,211,100)(33,63,198,99,47,77,212,85)(34,76,199,112,48,62,213,98)(35,61,200,97,49,75,214,111)(36,74,201,110,50,60,215,96)(37,59,202,95,51,73,216,109)(38,72,203,108,52,58,217,94)(39,57,204,93,53,71,218,107)(40,70,205,106,54,84,219,92)(41,83,206,91,55,69,220,105)(42,68,207,104,56,82,221,90), (1,61,129,90)(2,62,130,91)(3,63,131,92)(4,64,132,93)(5,65,133,94)(6,66,134,95)(7,67,135,96)(8,68,136,97)(9,69,137,98)(10,70,138,99)(11,71,139,100)(12,72,140,101)(13,73,113,102)(14,74,114,103)(15,75,115,104)(16,76,116,105)(17,77,117,106)(18,78,118,107)(19,79,119,108)(20,80,120,109)(21,81,121,110)(22,82,122,111)(23,83,123,112)(24,84,124,85)(25,57,125,86)(26,58,126,87)(27,59,127,88)(28,60,128,89)(29,167,215,182)(30,168,216,183)(31,141,217,184)(32,142,218,185)(33,143,219,186)(34,144,220,187)(35,145,221,188)(36,146,222,189)(37,147,223,190)(38,148,224,191)(39,149,197,192)(40,150,198,193)(41,151,199,194)(42,152,200,195)(43,153,201,196)(44,154,202,169)(45,155,203,170)(46,156,204,171)(47,157,205,172)(48,158,206,173)(49,159,207,174)(50,160,208,175)(51,161,209,176)(52,162,210,177)(53,163,211,178)(54,164,212,179)(55,165,213,180)(56,166,214,181)>;
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), (1,188,122,166,15,174,136,152)(2,173,123,151,16,187,137,165)(3,186,124,164,17,172,138,150)(4,171,125,149,18,185,139,163)(5,184,126,162,19,170,140,148)(6,169,127,147,20,183,113,161)(7,182,128,160,21,196,114,146)(8,195,129,145,22,181,115,159)(9,180,130,158,23,194,116,144)(10,193,131,143,24,179,117,157)(11,178,132,156,25,192,118,142)(12,191,133,141,26,177,119,155)(13,176,134,154,27,190,120,168)(14,189,135,167,28,175,121,153)(29,67,222,103,43,81,208,89)(30,80,223,88,44,66,209,102)(31,65,224,101,45,79,210,87)(32,78,197,86,46,64,211,100)(33,63,198,99,47,77,212,85)(34,76,199,112,48,62,213,98)(35,61,200,97,49,75,214,111)(36,74,201,110,50,60,215,96)(37,59,202,95,51,73,216,109)(38,72,203,108,52,58,217,94)(39,57,204,93,53,71,218,107)(40,70,205,106,54,84,219,92)(41,83,206,91,55,69,220,105)(42,68,207,104,56,82,221,90), (1,61,129,90)(2,62,130,91)(3,63,131,92)(4,64,132,93)(5,65,133,94)(6,66,134,95)(7,67,135,96)(8,68,136,97)(9,69,137,98)(10,70,138,99)(11,71,139,100)(12,72,140,101)(13,73,113,102)(14,74,114,103)(15,75,115,104)(16,76,116,105)(17,77,117,106)(18,78,118,107)(19,79,119,108)(20,80,120,109)(21,81,121,110)(22,82,122,111)(23,83,123,112)(24,84,124,85)(25,57,125,86)(26,58,126,87)(27,59,127,88)(28,60,128,89)(29,167,215,182)(30,168,216,183)(31,141,217,184)(32,142,218,185)(33,143,219,186)(34,144,220,187)(35,145,221,188)(36,146,222,189)(37,147,223,190)(38,148,224,191)(39,149,197,192)(40,150,198,193)(41,151,199,194)(42,152,200,195)(43,153,201,196)(44,154,202,169)(45,155,203,170)(46,156,204,171)(47,157,205,172)(48,158,206,173)(49,159,207,174)(50,160,208,175)(51,161,209,176)(52,162,210,177)(53,163,211,178)(54,164,212,179)(55,165,213,180)(56,166,214,181) );
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)], [(1,188,122,166,15,174,136,152),(2,173,123,151,16,187,137,165),(3,186,124,164,17,172,138,150),(4,171,125,149,18,185,139,163),(5,184,126,162,19,170,140,148),(6,169,127,147,20,183,113,161),(7,182,128,160,21,196,114,146),(8,195,129,145,22,181,115,159),(9,180,130,158,23,194,116,144),(10,193,131,143,24,179,117,157),(11,178,132,156,25,192,118,142),(12,191,133,141,26,177,119,155),(13,176,134,154,27,190,120,168),(14,189,135,167,28,175,121,153),(29,67,222,103,43,81,208,89),(30,80,223,88,44,66,209,102),(31,65,224,101,45,79,210,87),(32,78,197,86,46,64,211,100),(33,63,198,99,47,77,212,85),(34,76,199,112,48,62,213,98),(35,61,200,97,49,75,214,111),(36,74,201,110,50,60,215,96),(37,59,202,95,51,73,216,109),(38,72,203,108,52,58,217,94),(39,57,204,93,53,71,218,107),(40,70,205,106,54,84,219,92),(41,83,206,91,55,69,220,105),(42,68,207,104,56,82,221,90)], [(1,61,129,90),(2,62,130,91),(3,63,131,92),(4,64,132,93),(5,65,133,94),(6,66,134,95),(7,67,135,96),(8,68,136,97),(9,69,137,98),(10,70,138,99),(11,71,139,100),(12,72,140,101),(13,73,113,102),(14,74,114,103),(15,75,115,104),(16,76,116,105),(17,77,117,106),(18,78,118,107),(19,79,119,108),(20,80,120,109),(21,81,121,110),(22,82,122,111),(23,83,123,112),(24,84,124,85),(25,57,125,86),(26,58,126,87),(27,59,127,88),(28,60,128,89),(29,167,215,182),(30,168,216,183),(31,141,217,184),(32,142,218,185),(33,143,219,186),(34,144,220,187),(35,145,221,188),(36,146,222,189),(37,147,223,190),(38,148,224,191),(39,149,197,192),(40,150,198,193),(41,151,199,194),(42,152,200,195),(43,153,201,196),(44,154,202,169),(45,155,203,170),(46,156,204,171),(47,157,205,172),(48,158,206,173),(49,159,207,174),(50,160,208,175),(51,161,209,176),(52,162,210,177),(53,163,211,178),(54,164,212,179),(55,165,213,180),(56,166,214,181)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D7 | D14 | D14 | C4○D8 | Dic14 | C4×D7 | C7⋊D4 | C7⋊D4 | D4.8D14 |
| kernel | C28.45(C4⋊C4) | C28.Q8 | C4.Dic14 | C22×C7⋊C8 | C23.21D14 | C7×C42⋊C2 | C2×C7⋊C8 | C2×C28 | C2×C28 | C22×C14 | C42⋊C2 | C4⋊C4 | C22×C4 | C14 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 3 | 6 | 3 | 8 | 12 | 12 | 6 | 6 | 12 |
Matrix representation of C28.45(C4⋊C4) ►in GL4(𝔽113) generated by
| 1 | 104 | 0 | 0 |
| 9 | 33 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 79 | 68 | 0 | 0 |
| 101 | 34 | 0 | 0 |
| 0 | 0 | 95 | 0 |
| 0 | 0 | 0 | 69 |
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(113))| [1,9,0,0,104,33,0,0,0,0,98,0,0,0,0,98],[79,101,0,0,68,34,0,0,0,0,95,0,0,0,0,69],[98,0,0,0,0,98,0,0,0,0,0,1,0,0,1,0] >;
C28.45(C4⋊C4) in GAP, Magma, Sage, TeX
C_{28}._{45}(C_4\rtimes C_4) % in TeX
G:=Group("C28.45(C4:C4)"); // GroupNames label
G:=SmallGroup(448,532);
// by ID
G=gap.SmallGroup(448,532);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,422,387,58,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=c^4=1,b^4=a^14,b*a*b^-1=a^13,a*c=c*a,c*b*c^-1=a^14*b^3>;
// generators/relations