direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C4⋊C8, C4⋊C56, C28⋊3C8, C28.67D4, C28.12Q8, C42.2C14, C14.9M4(2), C4.4(C7×Q8), C2.2(C2×C56), (C4×C28).8C2, (C2×C56).4C2, (C2×C8).2C14, (C2×C4).4C28, C4.18(C7×D4), (C2×C28).13C4, C14.12(C2×C8), C14.11(C4⋊C4), C2.3(C7×M4(2)), C22.10(C2×C28), (C2×C28).136C22, C2.2(C7×C4⋊C4), (C2×C4).32(C2×C14), (C2×C14).39(C2×C4), SmallGroup(224,54)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4⋊C8
G = < a,b,c | a7=b4=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C7×C4⋊C8
►Regular action on 224 points
Generators in S224
(1 145 33 137 25 129 17)(2 146 34 138 26 130 18)(3 147 35 139 27 131 19)(4 148 36 140 28 132 20)(5 149 37 141 29 133 21)(6 150 38 142 30 134 22)(7 151 39 143 31 135 23)(8 152 40 144 32 136 24)(9 64 168 56 160 48 125)(10 57 161 49 153 41 126)(11 58 162 50 154 42 127)(12 59 163 51 155 43 128)(13 60 164 52 156 44 121)(14 61 165 53 157 45 122)(15 62 166 54 158 46 123)(16 63 167 55 159 47 124)(65 169 193 81 185 73 177)(66 170 194 82 186 74 178)(67 171 195 83 187 75 179)(68 172 196 84 188 76 180)(69 173 197 85 189 77 181)(70 174 198 86 190 78 182)(71 175 199 87 191 79 183)(72 176 200 88 192 80 184)(89 117 217 105 209 97 201)(90 118 218 106 210 98 202)(91 119 219 107 211 99 203)(92 120 220 108 212 100 204)(93 113 221 109 213 101 205)(94 114 222 110 214 102 206)(95 115 223 111 215 103 207)(96 116 224 112 216 104 208)
(1 91 175 45)(2 46 176 92)(3 93 169 47)(4 48 170 94)(5 95 171 41)(6 42 172 96)(7 89 173 43)(8 44 174 90)(9 82 222 36)(10 37 223 83)(11 84 224 38)(12 39 217 85)(13 86 218 40)(14 33 219 87)(15 88 220 34)(16 35 221 81)(17 203 71 157)(18 158 72 204)(19 205 65 159)(20 160 66 206)(21 207 67 153)(22 154 68 208)(23 201 69 155)(24 156 70 202)(25 211 79 165)(26 166 80 212)(27 213 73 167)(28 168 74 214)(29 215 75 161)(30 162 76 216)(31 209 77 163)(32 164 78 210)(49 133 103 179)(50 180 104 134)(51 135 97 181)(52 182 98 136)(53 129 99 183)(54 184 100 130)(55 131 101 177)(56 178 102 132)(57 141 111 187)(58 188 112 142)(59 143 105 189)(60 190 106 144)(61 137 107 191)(62 192 108 138)(63 139 109 185)(64 186 110 140)(113 193 124 147)(114 148 125 194)(115 195 126 149)(116 150 127 196)(117 197 128 151)(118 152 121 198)(119 199 122 145)(120 146 123 200)
(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)
G:=sub<Sym(224)| (1,145,33,137,25,129,17)(2,146,34,138,26,130,18)(3,147,35,139,27,131,19)(4,148,36,140,28,132,20)(5,149,37,141,29,133,21)(6,150,38,142,30,134,22)(7,151,39,143,31,135,23)(8,152,40,144,32,136,24)(9,64,168,56,160,48,125)(10,57,161,49,153,41,126)(11,58,162,50,154,42,127)(12,59,163,51,155,43,128)(13,60,164,52,156,44,121)(14,61,165,53,157,45,122)(15,62,166,54,158,46,123)(16,63,167,55,159,47,124)(65,169,193,81,185,73,177)(66,170,194,82,186,74,178)(67,171,195,83,187,75,179)(68,172,196,84,188,76,180)(69,173,197,85,189,77,181)(70,174,198,86,190,78,182)(71,175,199,87,191,79,183)(72,176,200,88,192,80,184)(89,117,217,105,209,97,201)(90,118,218,106,210,98,202)(91,119,219,107,211,99,203)(92,120,220,108,212,100,204)(93,113,221,109,213,101,205)(94,114,222,110,214,102,206)(95,115,223,111,215,103,207)(96,116,224,112,216,104,208), (1,91,175,45)(2,46,176,92)(3,93,169,47)(4,48,170,94)(5,95,171,41)(6,42,172,96)(7,89,173,43)(8,44,174,90)(9,82,222,36)(10,37,223,83)(11,84,224,38)(12,39,217,85)(13,86,218,40)(14,33,219,87)(15,88,220,34)(16,35,221,81)(17,203,71,157)(18,158,72,204)(19,205,65,159)(20,160,66,206)(21,207,67,153)(22,154,68,208)(23,201,69,155)(24,156,70,202)(25,211,79,165)(26,166,80,212)(27,213,73,167)(28,168,74,214)(29,215,75,161)(30,162,76,216)(31,209,77,163)(32,164,78,210)(49,133,103,179)(50,180,104,134)(51,135,97,181)(52,182,98,136)(53,129,99,183)(54,184,100,130)(55,131,101,177)(56,178,102,132)(57,141,111,187)(58,188,112,142)(59,143,105,189)(60,190,106,144)(61,137,107,191)(62,192,108,138)(63,139,109,185)(64,186,110,140)(113,193,124,147)(114,148,125,194)(115,195,126,149)(116,150,127,196)(117,197,128,151)(118,152,121,198)(119,199,122,145)(120,146,123,200), (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)>;
G:=Group( (1,145,33,137,25,129,17)(2,146,34,138,26,130,18)(3,147,35,139,27,131,19)(4,148,36,140,28,132,20)(5,149,37,141,29,133,21)(6,150,38,142,30,134,22)(7,151,39,143,31,135,23)(8,152,40,144,32,136,24)(9,64,168,56,160,48,125)(10,57,161,49,153,41,126)(11,58,162,50,154,42,127)(12,59,163,51,155,43,128)(13,60,164,52,156,44,121)(14,61,165,53,157,45,122)(15,62,166,54,158,46,123)(16,63,167,55,159,47,124)(65,169,193,81,185,73,177)(66,170,194,82,186,74,178)(67,171,195,83,187,75,179)(68,172,196,84,188,76,180)(69,173,197,85,189,77,181)(70,174,198,86,190,78,182)(71,175,199,87,191,79,183)(72,176,200,88,192,80,184)(89,117,217,105,209,97,201)(90,118,218,106,210,98,202)(91,119,219,107,211,99,203)(92,120,220,108,212,100,204)(93,113,221,109,213,101,205)(94,114,222,110,214,102,206)(95,115,223,111,215,103,207)(96,116,224,112,216,104,208), (1,91,175,45)(2,46,176,92)(3,93,169,47)(4,48,170,94)(5,95,171,41)(6,42,172,96)(7,89,173,43)(8,44,174,90)(9,82,222,36)(10,37,223,83)(11,84,224,38)(12,39,217,85)(13,86,218,40)(14,33,219,87)(15,88,220,34)(16,35,221,81)(17,203,71,157)(18,158,72,204)(19,205,65,159)(20,160,66,206)(21,207,67,153)(22,154,68,208)(23,201,69,155)(24,156,70,202)(25,211,79,165)(26,166,80,212)(27,213,73,167)(28,168,74,214)(29,215,75,161)(30,162,76,216)(31,209,77,163)(32,164,78,210)(49,133,103,179)(50,180,104,134)(51,135,97,181)(52,182,98,136)(53,129,99,183)(54,184,100,130)(55,131,101,177)(56,178,102,132)(57,141,111,187)(58,188,112,142)(59,143,105,189)(60,190,106,144)(61,137,107,191)(62,192,108,138)(63,139,109,185)(64,186,110,140)(113,193,124,147)(114,148,125,194)(115,195,126,149)(116,150,127,196)(117,197,128,151)(118,152,121,198)(119,199,122,145)(120,146,123,200), (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) );
G=PermutationGroup([[(1,145,33,137,25,129,17),(2,146,34,138,26,130,18),(3,147,35,139,27,131,19),(4,148,36,140,28,132,20),(5,149,37,141,29,133,21),(6,150,38,142,30,134,22),(7,151,39,143,31,135,23),(8,152,40,144,32,136,24),(9,64,168,56,160,48,125),(10,57,161,49,153,41,126),(11,58,162,50,154,42,127),(12,59,163,51,155,43,128),(13,60,164,52,156,44,121),(14,61,165,53,157,45,122),(15,62,166,54,158,46,123),(16,63,167,55,159,47,124),(65,169,193,81,185,73,177),(66,170,194,82,186,74,178),(67,171,195,83,187,75,179),(68,172,196,84,188,76,180),(69,173,197,85,189,77,181),(70,174,198,86,190,78,182),(71,175,199,87,191,79,183),(72,176,200,88,192,80,184),(89,117,217,105,209,97,201),(90,118,218,106,210,98,202),(91,119,219,107,211,99,203),(92,120,220,108,212,100,204),(93,113,221,109,213,101,205),(94,114,222,110,214,102,206),(95,115,223,111,215,103,207),(96,116,224,112,216,104,208)], [(1,91,175,45),(2,46,176,92),(3,93,169,47),(4,48,170,94),(5,95,171,41),(6,42,172,96),(7,89,173,43),(8,44,174,90),(9,82,222,36),(10,37,223,83),(11,84,224,38),(12,39,217,85),(13,86,218,40),(14,33,219,87),(15,88,220,34),(16,35,221,81),(17,203,71,157),(18,158,72,204),(19,205,65,159),(20,160,66,206),(21,207,67,153),(22,154,68,208),(23,201,69,155),(24,156,70,202),(25,211,79,165),(26,166,80,212),(27,213,73,167),(28,168,74,214),(29,215,75,161),(30,162,76,216),(31,209,77,163),(32,164,78,210),(49,133,103,179),(50,180,104,134),(51,135,97,181),(52,182,98,136),(53,129,99,183),(54,184,100,130),(55,131,101,177),(56,178,102,132),(57,141,111,187),(58,188,112,142),(59,143,105,189),(60,190,106,144),(61,137,107,191),(62,192,108,138),(63,139,109,185),(64,186,110,140),(113,193,124,147),(114,148,125,194),(115,195,126,149),(116,150,127,196),(117,197,128,151),(118,152,121,198),(119,199,122,145),(120,146,123,200)], [(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)]])
C7×C4⋊C8 is a maximal subgroup of
C28.53D8 C28.39SD16 C4.Dic28 C28.47D8 D28⋊2C8 Dic14⋊2C8 C4.D56 C28.2D8 C42.27D14 Dic14.3Q8 Dic14⋊C8 C28.M4(2) C42.200D14 D28⋊C8 C42.202D14 D14⋊3M4(2) C28⋊M4(2) C28⋊2M4(2) C42.30D14 C42.31D14 C28⋊SD16 D28⋊3Q8 C4⋊D56 D28.19D4 C42.36D14 D28⋊4Q8 D28.3Q8 Dic14⋊8D4 C4⋊Dic28 C28.7Q16 Dic14⋊4Q8 D4×C56 Q8×C56
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 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C4 | C7 | C8 | C14 | C14 | C28 | C56 | D4 | Q8 | M4(2) | C7×D4 | C7×Q8 | C7×M4(2) |
| kernel | C7×C4⋊C8 | C4×C28 | C2×C56 | C2×C28 | C4⋊C8 | C28 | C42 | C2×C8 | C2×C4 | C4 | C28 | C28 | C14 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 4 | 6 | 8 | 6 | 12 | 24 | 48 | 1 | 1 | 2 | 6 | 6 | 12 |
Matrix representation of C7×C4⋊C8 ►in GL3(𝔽113) generated by
| 28 | 0 | 0 |
| 0 | 106 | 0 |
| 0 | 0 | 106 |
| 112 | 0 | 0 |
| 0 | 0 | 112 |
| 0 | 1 | 0 |
| 95 | 0 | 0 |
| 0 | 6 | 47 |
| 0 | 47 | 107 |
G:=sub<GL(3,GF(113))| [28,0,0,0,106,0,0,0,106],[112,0,0,0,0,1,0,112,0],[95,0,0,0,6,47,0,47,107] >;
C7×C4⋊C8 in GAP, Magma, Sage, TeX
C_7\times C_4\rtimes C_8
% in TeX
G:=Group("C7xC4:C8"); // GroupNames label
G:=SmallGroup(224,54);
// by ID
G=gap.SmallGroup(224,54);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,175,88]);
// Polycyclic
G:=Group<a,b,c|a^7=b^4=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export