direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C28.C8, C14⋊2M5(2), C56.69C23, (C2×C28).8C8, C7⋊3(C2×M5(2)), C7⋊C16⋊12C22, (C2×C56).27C4, C28.45(C2×C8), C56.79(C2×C4), C23.3(C7⋊C8), (C2×C8).326D14, (C22×C14).7C8, (C2×C8).15Dic7, C8.26(C2×Dic7), C8.63(C22×D7), (C22×C8).14D7, (C22×C28).27C4, C14.27(C22×C8), (C22×C56).22C2, C28.176(C22×C4), (C2×C56).412C22, (C22×C4).17Dic7, C4.30(C22×Dic7), C4.9(C2×C7⋊C8), (C2×C7⋊C16)⋊12C2, (C2×C4).6(C7⋊C8), C22.6(C2×C7⋊C8), C2.7(C22×C7⋊C8), (C2×C14).37(C2×C8), (C2×C28).305(C2×C4), (C2×C4).101(C2×Dic7), SmallGroup(448,631)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C28.C8
G = < a,b,c | a2=b56=1, c4=b42, ab=ba, ac=ca, cbc-1=b13 >
Subgroups: 164 in 90 conjugacy classes, 71 normal (29 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, C14, C14, C14, C16, C2×C8, C2×C8, C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C2×C16, M5(2), C22×C8, C56, C56, C2×C28, C2×C28, C22×C14, C2×M5(2), C7⋊C16, C2×C56, C2×C56, C22×C28, C2×C7⋊C16, C28.C8, C22×C56, C2×C28.C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D7, C2×C8, C22×C4, Dic7, D14, M5(2), C22×C8, C7⋊C8, C2×Dic7, C22×D7, C2×M5(2), C2×C7⋊C8, C22×Dic7, C28.C8, C22×C7⋊C8, C2×C28.C8
►On 224 points
Generators in S224
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 85)(23 86)(24 87)(25 88)(26 89)(27 90)(28 91)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 99)(37 100)(38 101)(39 102)(40 103)(41 104)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)(113 184)(114 185)(115 186)(116 187)(117 188)(118 189)(119 190)(120 191)(121 192)(122 193)(123 194)(124 195)(125 196)(126 197)(127 198)(128 199)(129 200)(130 201)(131 202)(132 203)(133 204)(134 205)(135 206)(136 207)(137 208)(138 209)(139 210)(140 211)(141 212)(142 213)(143 214)(144 215)(145 216)(146 217)(147 218)(148 219)(149 220)(150 221)(151 222)(152 223)(153 224)(154 169)(155 170)(156 171)(157 172)(158 173)(159 174)(160 175)(161 176)(162 177)(163 178)(164 179)(165 180)(166 181)(167 182)(168 183)
(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)
(1 196 57 146 43 182 99 132 29 224 85 118 15 210 71 160)(2 209 58 159 44 195 100 145 30 181 86 131 16 223 72 117)(3 222 59 116 45 208 101 158 31 194 87 144 17 180 73 130)(4 179 60 129 46 221 102 115 32 207 88 157 18 193 74 143)(5 192 61 142 47 178 103 128 33 220 89 114 19 206 75 156)(6 205 62 155 48 191 104 141 34 177 90 127 20 219 76 113)(7 218 63 168 49 204 105 154 35 190 91 140 21 176 77 126)(8 175 64 125 50 217 106 167 36 203 92 153 22 189 78 139)(9 188 65 138 51 174 107 124 37 216 93 166 23 202 79 152)(10 201 66 151 52 187 108 137 38 173 94 123 24 215 80 165)(11 214 67 164 53 200 109 150 39 186 95 136 25 172 81 122)(12 171 68 121 54 213 110 163 40 199 96 149 26 185 82 135)(13 184 69 134 55 170 111 120 41 212 97 162 27 198 83 148)(14 197 70 147 56 183 112 133 42 169 98 119 28 211 84 161)
G:=sub<Sym(224)| (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(113,184)(114,185)(115,186)(116,187)(117,188)(118,189)(119,190)(120,191)(121,192)(122,193)(123,194)(124,195)(125,196)(126,197)(127,198)(128,199)(129,200)(130,201)(131,202)(132,203)(133,204)(134,205)(135,206)(136,207)(137,208)(138,209)(139,210)(140,211)(141,212)(142,213)(143,214)(144,215)(145,216)(146,217)(147,218)(148,219)(149,220)(150,221)(151,222)(152,223)(153,224)(154,169)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)(161,176)(162,177)(163,178)(164,179)(165,180)(166,181)(167,182)(168,183), (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), (1,196,57,146,43,182,99,132,29,224,85,118,15,210,71,160)(2,209,58,159,44,195,100,145,30,181,86,131,16,223,72,117)(3,222,59,116,45,208,101,158,31,194,87,144,17,180,73,130)(4,179,60,129,46,221,102,115,32,207,88,157,18,193,74,143)(5,192,61,142,47,178,103,128,33,220,89,114,19,206,75,156)(6,205,62,155,48,191,104,141,34,177,90,127,20,219,76,113)(7,218,63,168,49,204,105,154,35,190,91,140,21,176,77,126)(8,175,64,125,50,217,106,167,36,203,92,153,22,189,78,139)(9,188,65,138,51,174,107,124,37,216,93,166,23,202,79,152)(10,201,66,151,52,187,108,137,38,173,94,123,24,215,80,165)(11,214,67,164,53,200,109,150,39,186,95,136,25,172,81,122)(12,171,68,121,54,213,110,163,40,199,96,149,26,185,82,135)(13,184,69,134,55,170,111,120,41,212,97,162,27,198,83,148)(14,197,70,147,56,183,112,133,42,169,98,119,28,211,84,161)>;
G:=Group( (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(113,184)(114,185)(115,186)(116,187)(117,188)(118,189)(119,190)(120,191)(121,192)(122,193)(123,194)(124,195)(125,196)(126,197)(127,198)(128,199)(129,200)(130,201)(131,202)(132,203)(133,204)(134,205)(135,206)(136,207)(137,208)(138,209)(139,210)(140,211)(141,212)(142,213)(143,214)(144,215)(145,216)(146,217)(147,218)(148,219)(149,220)(150,221)(151,222)(152,223)(153,224)(154,169)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)(161,176)(162,177)(163,178)(164,179)(165,180)(166,181)(167,182)(168,183), (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), (1,196,57,146,43,182,99,132,29,224,85,118,15,210,71,160)(2,209,58,159,44,195,100,145,30,181,86,131,16,223,72,117)(3,222,59,116,45,208,101,158,31,194,87,144,17,180,73,130)(4,179,60,129,46,221,102,115,32,207,88,157,18,193,74,143)(5,192,61,142,47,178,103,128,33,220,89,114,19,206,75,156)(6,205,62,155,48,191,104,141,34,177,90,127,20,219,76,113)(7,218,63,168,49,204,105,154,35,190,91,140,21,176,77,126)(8,175,64,125,50,217,106,167,36,203,92,153,22,189,78,139)(9,188,65,138,51,174,107,124,37,216,93,166,23,202,79,152)(10,201,66,151,52,187,108,137,38,173,94,123,24,215,80,165)(11,214,67,164,53,200,109,150,39,186,95,136,25,172,81,122)(12,171,68,121,54,213,110,163,40,199,96,149,26,185,82,135)(13,184,69,134,55,170,111,120,41,212,97,162,27,198,83,148)(14,197,70,147,56,183,112,133,42,169,98,119,28,211,84,161) );
G=PermutationGroup([[(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,73),(11,74),(12,75),(13,76),(14,77),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,85),(23,86),(24,87),(25,88),(26,89),(27,90),(28,91),(29,92),(30,93),(31,94),(32,95),(33,96),(34,97),(35,98),(36,99),(37,100),(38,101),(39,102),(40,103),(41,104),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,112),(50,57),(51,58),(52,59),(53,60),(54,61),(55,62),(56,63),(113,184),(114,185),(115,186),(116,187),(117,188),(118,189),(119,190),(120,191),(121,192),(122,193),(123,194),(124,195),(125,196),(126,197),(127,198),(128,199),(129,200),(130,201),(131,202),(132,203),(133,204),(134,205),(135,206),(136,207),(137,208),(138,209),(139,210),(140,211),(141,212),(142,213),(143,214),(144,215),(145,216),(146,217),(147,218),(148,219),(149,220),(150,221),(151,222),(152,223),(153,224),(154,169),(155,170),(156,171),(157,172),(158,173),(159,174),(160,175),(161,176),(162,177),(163,178),(164,179),(165,180),(166,181),(167,182),(168,183)], [(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)], [(1,196,57,146,43,182,99,132,29,224,85,118,15,210,71,160),(2,209,58,159,44,195,100,145,30,181,86,131,16,223,72,117),(3,222,59,116,45,208,101,158,31,194,87,144,17,180,73,130),(4,179,60,129,46,221,102,115,32,207,88,157,18,193,74,143),(5,192,61,142,47,178,103,128,33,220,89,114,19,206,75,156),(6,205,62,155,48,191,104,141,34,177,90,127,20,219,76,113),(7,218,63,168,49,204,105,154,35,190,91,140,21,176,77,126),(8,175,64,125,50,217,106,167,36,203,92,153,22,189,78,139),(9,188,65,138,51,174,107,124,37,216,93,166,23,202,79,152),(10,201,66,151,52,187,108,137,38,173,94,123,24,215,80,165),(11,214,67,164,53,200,109,150,39,186,95,136,25,172,81,122),(12,171,68,121,54,213,110,163,40,199,96,149,26,185,82,135),(13,184,69,134,55,170,111,120,41,212,97,162,27,198,83,148),(14,197,70,147,56,183,112,133,42,169,98,119,28,211,84,161)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14U | 16A | ··· | 16P | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | D7 | Dic7 | D14 | Dic7 | M5(2) | C7⋊C8 | C7⋊C8 | C28.C8 |
| kernel | C2×C28.C8 | C2×C7⋊C16 | C28.C8 | C22×C56 | C2×C56 | C22×C28 | C2×C28 | C22×C14 | C22×C8 | C2×C8 | C2×C8 | C22×C4 | C14 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 12 | 4 | 3 | 9 | 9 | 3 | 8 | 18 | 6 | 48 |
Matrix representation of C2×C28.C8 ►in GL3(𝔽113) generated by
| 112 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 62 | 78 |
| 0 | 0 | 13 |
| 112 | 0 | 0 |
| 0 | 55 | 49 |
| 0 | 77 | 58 |
G:=sub<GL(3,GF(113))| [112,0,0,0,1,0,0,0,1],[1,0,0,0,62,0,0,78,13],[112,0,0,0,55,77,0,49,58] >;
C2×C28.C8 in GAP, Magma, Sage, TeX
C_2\times C_{28}.C_8 % in TeX
G:=Group("C2xC28.C8"); // GroupNames label
G:=SmallGroup(448,631);
// by ID
G=gap.SmallGroup(448,631);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,758,80,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^2=b^56=1,c^4=b^42,a*b=b*a,a*c=c*a,c*b*c^-1=b^13>;
// generators/relations