direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C8⋊C4, C8⋊3C28, C56⋊7C4, C14.7C42, C42.1C14, C14.7M4(2), (C2×C4).2C28, (C2×C8).7C14, (C4×C28).1C2, C2.2(C4×C28), (C2×C28).5C4, (C2×C56).17C2, C4.11(C2×C28), C28.48(C2×C4), C22.8(C2×C28), C2.1(C7×M4(2)), (C2×C28).134C22, (C2×C14).37(C2×C4), (C2×C4).30(C2×C14), SmallGroup(224,46)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8⋊C4
G = < a,b,c | a7=b8=c4=1, ab=ba, ac=ca, cbc-1=b5 >
Smallest permutation representation of C7×C8⋊C4
►Regular action on 224 points
Generators in S224
(1 130 74 122 66 114 58)(2 131 75 123 67 115 59)(3 132 76 124 68 116 60)(4 133 77 125 69 117 61)(5 134 78 126 70 118 62)(6 135 79 127 71 119 63)(7 136 80 128 72 120 64)(8 129 73 121 65 113 57)(9 163 195 25 187 17 179)(10 164 196 26 188 18 180)(11 165 197 27 189 19 181)(12 166 198 28 190 20 182)(13 167 199 29 191 21 183)(14 168 200 30 192 22 184)(15 161 193 31 185 23 177)(16 162 194 32 186 24 178)(33 176 219 49 211 41 203)(34 169 220 50 212 42 204)(35 170 221 51 213 43 205)(36 171 222 52 214 44 206)(37 172 223 53 215 45 207)(38 173 224 54 216 46 208)(39 174 217 55 209 47 201)(40 175 218 56 210 48 202)(81 108 156 100 148 92 139)(82 109 157 101 149 93 140)(83 110 158 102 150 94 141)(84 111 159 103 151 95 142)(85 112 160 104 152 96 143)(86 105 153 97 145 89 144)(87 106 154 98 146 90 137)(88 107 155 99 147 91 138)
(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 89 33 163)(2 94 34 168)(3 91 35 165)(4 96 36 162)(5 93 37 167)(6 90 38 164)(7 95 39 161)(8 92 40 166)(9 58 145 203)(10 63 146 208)(11 60 147 205)(12 57 148 202)(13 62 149 207)(14 59 150 204)(15 64 151 201)(16 61 152 206)(17 66 153 211)(18 71 154 216)(19 68 155 213)(20 65 156 210)(21 70 157 215)(22 67 158 212)(23 72 159 209)(24 69 160 214)(25 74 86 219)(26 79 87 224)(27 76 88 221)(28 73 81 218)(29 78 82 223)(30 75 83 220)(31 80 84 217)(32 77 85 222)(41 179 114 97)(42 184 115 102)(43 181 116 99)(44 178 117 104)(45 183 118 101)(46 180 119 98)(47 177 120 103)(48 182 113 100)(49 187 122 105)(50 192 123 110)(51 189 124 107)(52 186 125 112)(53 191 126 109)(54 188 127 106)(55 185 128 111)(56 190 121 108)(129 139 175 198)(130 144 176 195)(131 141 169 200)(132 138 170 197)(133 143 171 194)(134 140 172 199)(135 137 173 196)(136 142 174 193)
G:=sub<Sym(224)| (1,130,74,122,66,114,58)(2,131,75,123,67,115,59)(3,132,76,124,68,116,60)(4,133,77,125,69,117,61)(5,134,78,126,70,118,62)(6,135,79,127,71,119,63)(7,136,80,128,72,120,64)(8,129,73,121,65,113,57)(9,163,195,25,187,17,179)(10,164,196,26,188,18,180)(11,165,197,27,189,19,181)(12,166,198,28,190,20,182)(13,167,199,29,191,21,183)(14,168,200,30,192,22,184)(15,161,193,31,185,23,177)(16,162,194,32,186,24,178)(33,176,219,49,211,41,203)(34,169,220,50,212,42,204)(35,170,221,51,213,43,205)(36,171,222,52,214,44,206)(37,172,223,53,215,45,207)(38,173,224,54,216,46,208)(39,174,217,55,209,47,201)(40,175,218,56,210,48,202)(81,108,156,100,148,92,139)(82,109,157,101,149,93,140)(83,110,158,102,150,94,141)(84,111,159,103,151,95,142)(85,112,160,104,152,96,143)(86,105,153,97,145,89,144)(87,106,154,98,146,90,137)(88,107,155,99,147,91,138), (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), (1,89,33,163)(2,94,34,168)(3,91,35,165)(4,96,36,162)(5,93,37,167)(6,90,38,164)(7,95,39,161)(8,92,40,166)(9,58,145,203)(10,63,146,208)(11,60,147,205)(12,57,148,202)(13,62,149,207)(14,59,150,204)(15,64,151,201)(16,61,152,206)(17,66,153,211)(18,71,154,216)(19,68,155,213)(20,65,156,210)(21,70,157,215)(22,67,158,212)(23,72,159,209)(24,69,160,214)(25,74,86,219)(26,79,87,224)(27,76,88,221)(28,73,81,218)(29,78,82,223)(30,75,83,220)(31,80,84,217)(32,77,85,222)(41,179,114,97)(42,184,115,102)(43,181,116,99)(44,178,117,104)(45,183,118,101)(46,180,119,98)(47,177,120,103)(48,182,113,100)(49,187,122,105)(50,192,123,110)(51,189,124,107)(52,186,125,112)(53,191,126,109)(54,188,127,106)(55,185,128,111)(56,190,121,108)(129,139,175,198)(130,144,176,195)(131,141,169,200)(132,138,170,197)(133,143,171,194)(134,140,172,199)(135,137,173,196)(136,142,174,193)>;
G:=Group( (1,130,74,122,66,114,58)(2,131,75,123,67,115,59)(3,132,76,124,68,116,60)(4,133,77,125,69,117,61)(5,134,78,126,70,118,62)(6,135,79,127,71,119,63)(7,136,80,128,72,120,64)(8,129,73,121,65,113,57)(9,163,195,25,187,17,179)(10,164,196,26,188,18,180)(11,165,197,27,189,19,181)(12,166,198,28,190,20,182)(13,167,199,29,191,21,183)(14,168,200,30,192,22,184)(15,161,193,31,185,23,177)(16,162,194,32,186,24,178)(33,176,219,49,211,41,203)(34,169,220,50,212,42,204)(35,170,221,51,213,43,205)(36,171,222,52,214,44,206)(37,172,223,53,215,45,207)(38,173,224,54,216,46,208)(39,174,217,55,209,47,201)(40,175,218,56,210,48,202)(81,108,156,100,148,92,139)(82,109,157,101,149,93,140)(83,110,158,102,150,94,141)(84,111,159,103,151,95,142)(85,112,160,104,152,96,143)(86,105,153,97,145,89,144)(87,106,154,98,146,90,137)(88,107,155,99,147,91,138), (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), (1,89,33,163)(2,94,34,168)(3,91,35,165)(4,96,36,162)(5,93,37,167)(6,90,38,164)(7,95,39,161)(8,92,40,166)(9,58,145,203)(10,63,146,208)(11,60,147,205)(12,57,148,202)(13,62,149,207)(14,59,150,204)(15,64,151,201)(16,61,152,206)(17,66,153,211)(18,71,154,216)(19,68,155,213)(20,65,156,210)(21,70,157,215)(22,67,158,212)(23,72,159,209)(24,69,160,214)(25,74,86,219)(26,79,87,224)(27,76,88,221)(28,73,81,218)(29,78,82,223)(30,75,83,220)(31,80,84,217)(32,77,85,222)(41,179,114,97)(42,184,115,102)(43,181,116,99)(44,178,117,104)(45,183,118,101)(46,180,119,98)(47,177,120,103)(48,182,113,100)(49,187,122,105)(50,192,123,110)(51,189,124,107)(52,186,125,112)(53,191,126,109)(54,188,127,106)(55,185,128,111)(56,190,121,108)(129,139,175,198)(130,144,176,195)(131,141,169,200)(132,138,170,197)(133,143,171,194)(134,140,172,199)(135,137,173,196)(136,142,174,193) );
G=PermutationGroup([[(1,130,74,122,66,114,58),(2,131,75,123,67,115,59),(3,132,76,124,68,116,60),(4,133,77,125,69,117,61),(5,134,78,126,70,118,62),(6,135,79,127,71,119,63),(7,136,80,128,72,120,64),(8,129,73,121,65,113,57),(9,163,195,25,187,17,179),(10,164,196,26,188,18,180),(11,165,197,27,189,19,181),(12,166,198,28,190,20,182),(13,167,199,29,191,21,183),(14,168,200,30,192,22,184),(15,161,193,31,185,23,177),(16,162,194,32,186,24,178),(33,176,219,49,211,41,203),(34,169,220,50,212,42,204),(35,170,221,51,213,43,205),(36,171,222,52,214,44,206),(37,172,223,53,215,45,207),(38,173,224,54,216,46,208),(39,174,217,55,209,47,201),(40,175,218,56,210,48,202),(81,108,156,100,148,92,139),(82,109,157,101,149,93,140),(83,110,158,102,150,94,141),(84,111,159,103,151,95,142),(85,112,160,104,152,96,143),(86,105,153,97,145,89,144),(87,106,154,98,146,90,137),(88,107,155,99,147,91,138)], [(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)], [(1,89,33,163),(2,94,34,168),(3,91,35,165),(4,96,36,162),(5,93,37,167),(6,90,38,164),(7,95,39,161),(8,92,40,166),(9,58,145,203),(10,63,146,208),(11,60,147,205),(12,57,148,202),(13,62,149,207),(14,59,150,204),(15,64,151,201),(16,61,152,206),(17,66,153,211),(18,71,154,216),(19,68,155,213),(20,65,156,210),(21,70,157,215),(22,67,158,212),(23,72,159,209),(24,69,160,214),(25,74,86,219),(26,79,87,224),(27,76,88,221),(28,73,81,218),(29,78,82,223),(30,75,83,220),(31,80,84,217),(32,77,85,222),(41,179,114,97),(42,184,115,102),(43,181,116,99),(44,178,117,104),(45,183,118,101),(46,180,119,98),(47,177,120,103),(48,182,113,100),(49,187,122,105),(50,192,123,110),(51,189,124,107),(52,186,125,112),(53,191,126,109),(54,188,127,106),(55,185,128,111),(56,190,121,108),(129,139,175,198),(130,144,176,195),(131,141,169,200),(132,138,170,197),(133,143,171,194),(134,140,172,199),(135,137,173,196),(136,142,174,193)]])
C7×C8⋊C4 is a maximal subgroup of
C42.D14 C42.2D14 C28.15C42 C56⋊Q8 C8⋊Dic14 C42.14D14 C42.182D14 C8⋊9D28 Dic7.C42 D14.4C42 C42.185D14 C42.16D14 D56⋊C4 C8⋊D28 C42.19D14 C42.20D14 C8.D28 Dic28⋊C4 D56⋊4C4 M4(2)×C28
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 28A | ··· | 28X | 28Y | ··· | 28AV | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C7 | C14 | C14 | C28 | C28 | M4(2) | C7×M4(2) |
| kernel | C7×C8⋊C4 | C4×C28 | C2×C56 | C56 | C2×C28 | C8⋊C4 | C42 | C2×C8 | C8 | C2×C4 | C14 | C2 |
| # reps | 1 | 1 | 2 | 8 | 4 | 6 | 6 | 12 | 48 | 24 | 4 | 24 |
Matrix representation of C7×C8⋊C4 ►in GL3(𝔽113) generated by
| 1 | 0 | 0 |
| 0 | 30 | 0 |
| 0 | 0 | 30 |
| 98 | 0 | 0 |
| 0 | 90 | 86 |
| 0 | 86 | 23 |
| 15 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 112 | 0 |
G:=sub<GL(3,GF(113))| [1,0,0,0,30,0,0,0,30],[98,0,0,0,90,86,0,86,23],[15,0,0,0,0,112,0,1,0] >;
C7×C8⋊C4 in GAP, Magma, Sage, TeX
C_7\times C_8\rtimes C_4
% in TeX
G:=Group("C7xC8:C4"); // GroupNames label
G:=SmallGroup(224,46);
// by ID
G=gap.SmallGroup(224,46);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-2,168,1369,343,117]);
// Polycyclic
G:=Group<a,b,c|a^7=b^8=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
Export