direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×C7⋊C8, C42.205D14, C7⋊4(C8×D4), C28⋊3(C2×C8), (C7×D4)⋊2C8, (C4×D4).15D7, (D4×C28).5C2, C28⋊C8⋊20C2, (D4×C14).8C4, C4.214(D4×D7), C14.90(C4×D4), C2.3(D4×Dic7), C28.373(C2×D4), C4⋊C4.11Dic7, C14.38(C8○D4), C14.24(C22×C8), (C4×C28).81C22, (C2×D4).11Dic7, C22⋊C4.7Dic7, C28.306(C4○D4), C28.55D4⋊24C2, (C2×C28).848C23, (C22×C4).309D14, C2.2(Q8.Dic7), C4.133(D4⋊2D7), C23.16(C2×Dic7), (C22×C28).348C22, C22.22(C22×Dic7), C4⋊1(C2×C7⋊C8), (C4×C7⋊C8)⋊6C2, C22⋊1(C2×C7⋊C8), (C2×C14)⋊3(C2×C8), C2.5(C22×C7⋊C8), (C7×C4⋊C4).10C4, (C22×C7⋊C8)⋊18C2, (C7×C22⋊C4).8C4, (C2×C28).162(C2×C4), (C2×C7⋊C8).315C22, (C2×C4).33(C2×Dic7), (C22×C14).59(C2×C4), (C2×C4).790(C22×D7), (C2×C14).185(C22×C4), SmallGroup(448,544)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C7⋊C8
G = < a,b,c,d | a4=b2=c7=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 340 in 134 conjugacy classes, 77 normal (33 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C28, C28, C28, C2×C14, C2×C14, C2×C14, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C7⋊C8, C7⋊C8, C2×C28, C2×C28, C2×C28, C7×D4, C22×C14, C8×D4, C2×C7⋊C8, C2×C7⋊C8, C2×C7⋊C8, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, C4×C7⋊C8, C28⋊C8, C28.55D4, C22×C7⋊C8, D4×C28, D4×C7⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, D7, C2×C8, C22×C4, C2×D4, C4○D4, Dic7, D14, C4×D4, C22×C8, C8○D4, C7⋊C8, C2×Dic7, C22×D7, C8×D4, C2×C7⋊C8, D4×D7, D4⋊2D7, C22×Dic7, C22×C7⋊C8, D4×Dic7, Q8.Dic7, D4×C7⋊C8
►On 224 points
Generators in S224
(1 109 55 170)(2 110 56 171)(3 111 49 172)(4 112 50 173)(5 105 51 174)(6 106 52 175)(7 107 53 176)(8 108 54 169)(9 86 38 122)(10 87 39 123)(11 88 40 124)(12 81 33 125)(13 82 34 126)(14 83 35 127)(15 84 36 128)(16 85 37 121)(17 184 210 31)(18 177 211 32)(19 178 212 25)(20 179 213 26)(21 180 214 27)(22 181 215 28)(23 182 216 29)(24 183 209 30)(41 164 198 150)(42 165 199 151)(43 166 200 152)(44 167 193 145)(45 168 194 146)(46 161 195 147)(47 162 196 148)(48 163 197 149)(57 102 222 66)(58 103 223 67)(59 104 224 68)(60 97 217 69)(61 98 218 70)(62 99 219 71)(63 100 220 72)(64 101 221 65)(73 137 155 208)(74 138 156 201)(75 139 157 202)(76 140 158 203)(77 141 159 204)(78 142 160 205)(79 143 153 206)(80 144 154 207)(89 186 117 134)(90 187 118 135)(91 188 119 136)(92 189 120 129)(93 190 113 130)(94 191 114 131)(95 192 115 132)(96 185 116 133)
(9 38)(10 39)(11 40)(12 33)(13 34)(14 35)(15 36)(16 37)(25 178)(26 179)(27 180)(28 181)(29 182)(30 183)(31 184)(32 177)(57 222)(58 223)(59 224)(60 217)(61 218)(62 219)(63 220)(64 221)(89 117)(90 118)(91 119)(92 120)(93 113)(94 114)(95 115)(96 116)(105 174)(106 175)(107 176)(108 169)(109 170)(110 171)(111 172)(112 173)(137 208)(138 201)(139 202)(140 203)(141 204)(142 205)(143 206)(144 207)(145 167)(146 168)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)
(1 69 20 80 188 123 46)(2 47 124 189 73 21 70)(3 71 22 74 190 125 48)(4 41 126 191 75 23 72)(5 65 24 76 192 127 42)(6 43 128 185 77 17 66)(7 67 18 78 186 121 44)(8 45 122 187 79 19 68)(9 118 143 178 59 108 168)(10 161 109 60 179 144 119)(11 120 137 180 61 110 162)(12 163 111 62 181 138 113)(13 114 139 182 63 112 164)(14 165 105 64 183 140 115)(15 116 141 184 57 106 166)(16 167 107 58 177 142 117)(25 224 169 146 38 90 206)(26 207 91 39 147 170 217)(27 218 171 148 40 92 208)(28 201 93 33 149 172 219)(29 220 173 150 34 94 202)(30 203 95 35 151 174 221)(31 222 175 152 36 96 204)(32 205 89 37 145 176 223)(49 99 215 156 130 81 197)(50 198 82 131 157 216 100)(51 101 209 158 132 83 199)(52 200 84 133 159 210 102)(53 103 211 160 134 85 193)(54 194 86 135 153 212 104)(55 97 213 154 136 87 195)(56 196 88 129 155 214 98)
(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)| (1,109,55,170)(2,110,56,171)(3,111,49,172)(4,112,50,173)(5,105,51,174)(6,106,52,175)(7,107,53,176)(8,108,54,169)(9,86,38,122)(10,87,39,123)(11,88,40,124)(12,81,33,125)(13,82,34,126)(14,83,35,127)(15,84,36,128)(16,85,37,121)(17,184,210,31)(18,177,211,32)(19,178,212,25)(20,179,213,26)(21,180,214,27)(22,181,215,28)(23,182,216,29)(24,183,209,30)(41,164,198,150)(42,165,199,151)(43,166,200,152)(44,167,193,145)(45,168,194,146)(46,161,195,147)(47,162,196,148)(48,163,197,149)(57,102,222,66)(58,103,223,67)(59,104,224,68)(60,97,217,69)(61,98,218,70)(62,99,219,71)(63,100,220,72)(64,101,221,65)(73,137,155,208)(74,138,156,201)(75,139,157,202)(76,140,158,203)(77,141,159,204)(78,142,160,205)(79,143,153,206)(80,144,154,207)(89,186,117,134)(90,187,118,135)(91,188,119,136)(92,189,120,129)(93,190,113,130)(94,191,114,131)(95,192,115,132)(96,185,116,133), (9,38)(10,39)(11,40)(12,33)(13,34)(14,35)(15,36)(16,37)(25,178)(26,179)(27,180)(28,181)(29,182)(30,183)(31,184)(32,177)(57,222)(58,223)(59,224)(60,217)(61,218)(62,219)(63,220)(64,221)(89,117)(90,118)(91,119)(92,120)(93,113)(94,114)(95,115)(96,116)(105,174)(106,175)(107,176)(108,169)(109,170)(110,171)(111,172)(112,173)(137,208)(138,201)(139,202)(140,203)(141,204)(142,205)(143,206)(144,207)(145,167)(146,168)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166), (1,69,20,80,188,123,46)(2,47,124,189,73,21,70)(3,71,22,74,190,125,48)(4,41,126,191,75,23,72)(5,65,24,76,192,127,42)(6,43,128,185,77,17,66)(7,67,18,78,186,121,44)(8,45,122,187,79,19,68)(9,118,143,178,59,108,168)(10,161,109,60,179,144,119)(11,120,137,180,61,110,162)(12,163,111,62,181,138,113)(13,114,139,182,63,112,164)(14,165,105,64,183,140,115)(15,116,141,184,57,106,166)(16,167,107,58,177,142,117)(25,224,169,146,38,90,206)(26,207,91,39,147,170,217)(27,218,171,148,40,92,208)(28,201,93,33,149,172,219)(29,220,173,150,34,94,202)(30,203,95,35,151,174,221)(31,222,175,152,36,96,204)(32,205,89,37,145,176,223)(49,99,215,156,130,81,197)(50,198,82,131,157,216,100)(51,101,209,158,132,83,199)(52,200,84,133,159,210,102)(53,103,211,160,134,85,193)(54,194,86,135,153,212,104)(55,97,213,154,136,87,195)(56,196,88,129,155,214,98), (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( (1,109,55,170)(2,110,56,171)(3,111,49,172)(4,112,50,173)(5,105,51,174)(6,106,52,175)(7,107,53,176)(8,108,54,169)(9,86,38,122)(10,87,39,123)(11,88,40,124)(12,81,33,125)(13,82,34,126)(14,83,35,127)(15,84,36,128)(16,85,37,121)(17,184,210,31)(18,177,211,32)(19,178,212,25)(20,179,213,26)(21,180,214,27)(22,181,215,28)(23,182,216,29)(24,183,209,30)(41,164,198,150)(42,165,199,151)(43,166,200,152)(44,167,193,145)(45,168,194,146)(46,161,195,147)(47,162,196,148)(48,163,197,149)(57,102,222,66)(58,103,223,67)(59,104,224,68)(60,97,217,69)(61,98,218,70)(62,99,219,71)(63,100,220,72)(64,101,221,65)(73,137,155,208)(74,138,156,201)(75,139,157,202)(76,140,158,203)(77,141,159,204)(78,142,160,205)(79,143,153,206)(80,144,154,207)(89,186,117,134)(90,187,118,135)(91,188,119,136)(92,189,120,129)(93,190,113,130)(94,191,114,131)(95,192,115,132)(96,185,116,133), (9,38)(10,39)(11,40)(12,33)(13,34)(14,35)(15,36)(16,37)(25,178)(26,179)(27,180)(28,181)(29,182)(30,183)(31,184)(32,177)(57,222)(58,223)(59,224)(60,217)(61,218)(62,219)(63,220)(64,221)(89,117)(90,118)(91,119)(92,120)(93,113)(94,114)(95,115)(96,116)(105,174)(106,175)(107,176)(108,169)(109,170)(110,171)(111,172)(112,173)(137,208)(138,201)(139,202)(140,203)(141,204)(142,205)(143,206)(144,207)(145,167)(146,168)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166), (1,69,20,80,188,123,46)(2,47,124,189,73,21,70)(3,71,22,74,190,125,48)(4,41,126,191,75,23,72)(5,65,24,76,192,127,42)(6,43,128,185,77,17,66)(7,67,18,78,186,121,44)(8,45,122,187,79,19,68)(9,118,143,178,59,108,168)(10,161,109,60,179,144,119)(11,120,137,180,61,110,162)(12,163,111,62,181,138,113)(13,114,139,182,63,112,164)(14,165,105,64,183,140,115)(15,116,141,184,57,106,166)(16,167,107,58,177,142,117)(25,224,169,146,38,90,206)(26,207,91,39,147,170,217)(27,218,171,148,40,92,208)(28,201,93,33,149,172,219)(29,220,173,150,34,94,202)(30,203,95,35,151,174,221)(31,222,175,152,36,96,204)(32,205,89,37,145,176,223)(49,99,215,156,130,81,197)(50,198,82,131,157,216,100)(51,101,209,158,132,83,199)(52,200,84,133,159,210,102)(53,103,211,160,134,85,193)(54,194,86,135,153,212,104)(55,97,213,154,136,87,195)(56,196,88,129,155,214,98), (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([[(1,109,55,170),(2,110,56,171),(3,111,49,172),(4,112,50,173),(5,105,51,174),(6,106,52,175),(7,107,53,176),(8,108,54,169),(9,86,38,122),(10,87,39,123),(11,88,40,124),(12,81,33,125),(13,82,34,126),(14,83,35,127),(15,84,36,128),(16,85,37,121),(17,184,210,31),(18,177,211,32),(19,178,212,25),(20,179,213,26),(21,180,214,27),(22,181,215,28),(23,182,216,29),(24,183,209,30),(41,164,198,150),(42,165,199,151),(43,166,200,152),(44,167,193,145),(45,168,194,146),(46,161,195,147),(47,162,196,148),(48,163,197,149),(57,102,222,66),(58,103,223,67),(59,104,224,68),(60,97,217,69),(61,98,218,70),(62,99,219,71),(63,100,220,72),(64,101,221,65),(73,137,155,208),(74,138,156,201),(75,139,157,202),(76,140,158,203),(77,141,159,204),(78,142,160,205),(79,143,153,206),(80,144,154,207),(89,186,117,134),(90,187,118,135),(91,188,119,136),(92,189,120,129),(93,190,113,130),(94,191,114,131),(95,192,115,132),(96,185,116,133)], [(9,38),(10,39),(11,40),(12,33),(13,34),(14,35),(15,36),(16,37),(25,178),(26,179),(27,180),(28,181),(29,182),(30,183),(31,184),(32,177),(57,222),(58,223),(59,224),(60,217),(61,218),(62,219),(63,220),(64,221),(89,117),(90,118),(91,119),(92,120),(93,113),(94,114),(95,115),(96,116),(105,174),(106,175),(107,176),(108,169),(109,170),(110,171),(111,172),(112,173),(137,208),(138,201),(139,202),(140,203),(141,204),(142,205),(143,206),(144,207),(145,167),(146,168),(147,161),(148,162),(149,163),(150,164),(151,165),(152,166)], [(1,69,20,80,188,123,46),(2,47,124,189,73,21,70),(3,71,22,74,190,125,48),(4,41,126,191,75,23,72),(5,65,24,76,192,127,42),(6,43,128,185,77,17,66),(7,67,18,78,186,121,44),(8,45,122,187,79,19,68),(9,118,143,178,59,108,168),(10,161,109,60,179,144,119),(11,120,137,180,61,110,162),(12,163,111,62,181,138,113),(13,114,139,182,63,112,164),(14,165,105,64,183,140,115),(15,116,141,184,57,106,166),(16,167,107,58,177,142,117),(25,224,169,146,38,90,206),(26,207,91,39,147,170,217),(27,218,171,148,40,92,208),(28,201,93,33,149,172,219),(29,220,173,150,34,94,202),(30,203,95,35,151,174,221),(31,222,175,152,36,96,204),(32,205,89,37,145,176,223),(49,99,215,156,130,81,197),(50,198,82,131,157,216,100),(51,101,209,158,132,83,199),(52,200,84,133,159,210,102),(53,103,211,160,134,85,193),(54,194,86,135,153,212,104),(55,97,213,154,136,87,195),(56,196,88,129,155,214,98)], [(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)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8T | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | D7 | C4○D4 | D14 | Dic7 | Dic7 | D14 | Dic7 | C8○D4 | C7⋊C8 | D4×D7 | D4⋊2D7 | Q8.Dic7 |
| kernel | D4×C7⋊C8 | C4×C7⋊C8 | C28⋊C8 | C28.55D4 | C22×C7⋊C8 | D4×C28 | C7×C22⋊C4 | C7×C4⋊C4 | D4×C14 | C7×D4 | C7⋊C8 | C4×D4 | C28 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | D4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 16 | 2 | 3 | 2 | 3 | 6 | 3 | 6 | 3 | 4 | 24 | 3 | 3 | 6 |
Matrix representation of D4×C7⋊C8 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 22 |
| 0 | 0 | 41 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 112 |
| 24 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 12 | 0 | 0 |
| 74 | 101 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,112,41,0,0,22,1],[1,0,0,0,0,1,0,0,0,0,1,72,0,0,0,112],[24,1,0,0,112,0,0,0,0,0,1,0,0,0,0,1],[12,74,0,0,12,101,0,0,0,0,15,0,0,0,0,15] >;
D4×C7⋊C8 in GAP, Magma, Sage, TeX
D_4\times C_7\rtimes C_8
% in TeX
G:=Group("D4xC7:C8"); // GroupNames label
G:=SmallGroup(448,544);
// by ID
G=gap.SmallGroup(448,544);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,219,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^7=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations