metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7⋊D4⋊C8, C7⋊2(C8×D4), C7⋊C8⋊29D4, D14⋊2(C2×C8), C22⋊1(C8×D7), D14⋊C8⋊16C2, D14⋊C4.9C4, C22⋊C8⋊16D7, Dic7⋊1(C2×C8), C4.194(D4×D7), C14.26(C4×D4), Dic7⋊C8⋊18C2, (C8×Dic7)⋊15C2, C28.353(C2×D4), (C2×C8).194D14, Dic7⋊C4.9C4, C14.7(C22×C8), C23.23(C4×D7), C23.D7.4C4, C14.24(C8○D4), C2.2(D28.C4), C28.297(C4○D4), (C2×C56).170C22, (C2×C28).820C23, (C22×C4).303D14, C4.123(D4⋊2D7), C2.3(Dic7⋊4D4), (C22×C28).337C22, (C4×Dic7).271C22, C2.9(D7×C2×C8), (D7×C2×C8)⋊13C2, (C2×C14)⋊2(C2×C8), (C22×C7⋊C8)⋊16C2, (C2×C4).63(C4×D7), (C2×C7⋊D4).3C4, C22.44(C2×C4×D7), (C7×C22⋊C8)⋊14C2, (C4×C7⋊D4).13C2, (C2×C28).153(C2×C4), (C2×C7⋊C8).300C22, (C2×C4×D7).272C22, (C22×C14).38(C2×C4), (C2×C14).75(C22×C4), (C2×Dic7).49(C2×C4), (C22×D7).35(C2×C4), (C2×C4).762(C22×D7), SmallGroup(448,259)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7⋊D4⋊C8
G = < a,b,c,d | a7=b4=c2=d8=1, bab-1=cac=a-1, ad=da, cbc=dbd-1=b-1, cd=dc >
Subgroups: 508 in 134 conjugacy classes, 61 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C4×C8, C22⋊C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C7⋊C8, C7⋊C8, C56, C4×D7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C8×D4, C8×D7, C2×C7⋊C8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, D14⋊C4, C23.D7, C2×C56, C2×C4×D7, C2×C7⋊D4, C22×C28, C8×Dic7, Dic7⋊C8, D14⋊C8, C7×C22⋊C8, D7×C2×C8, C22×C7⋊C8, C4×C7⋊D4, C7⋊D4⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, D7, C2×C8, C22×C4, C2×D4, C4○D4, D14, C4×D4, C22×C8, C8○D4, C4×D7, C22×D7, C8×D4, C8×D7, C2×C4×D7, D4×D7, D4⋊2D7, Dic7⋊4D4, D7×C2×C8, D28.C4, C7⋊D4⋊C8
►On 224 points
Generators in S224
(1 102 18 134 179 14 42)(2 103 19 135 180 15 43)(3 104 20 136 181 16 44)(4 97 21 129 182 9 45)(5 98 22 130 183 10 46)(6 99 23 131 184 11 47)(7 100 24 132 177 12 48)(8 101 17 133 178 13 41)(25 92 205 218 83 121 118)(26 93 206 219 84 122 119)(27 94 207 220 85 123 120)(28 95 208 221 86 124 113)(29 96 201 222 87 125 114)(30 89 202 223 88 126 115)(31 90 203 224 81 127 116)(32 91 204 217 82 128 117)(33 196 185 157 214 149 58)(34 197 186 158 215 150 59)(35 198 187 159 216 151 60)(36 199 188 160 209 152 61)(37 200 189 153 210 145 62)(38 193 190 154 211 146 63)(39 194 191 155 212 147 64)(40 195 192 156 213 148 57)(49 111 141 72 78 174 163)(50 112 142 65 79 175 164)(51 105 143 66 80 176 165)(52 106 144 67 73 169 166)(53 107 137 68 74 170 167)(54 108 138 69 75 171 168)(55 109 139 70 76 172 161)(56 110 140 71 77 173 162)
(1 204 51 159)(2 160 52 205)(3 206 53 153)(4 154 54 207)(5 208 55 155)(6 156 56 201)(7 202 49 157)(8 158 50 203)(9 146 171 85)(10 86 172 147)(11 148 173 87)(12 88 174 149)(13 150 175 81)(14 82 176 151)(15 152 169 83)(16 84 170 145)(17 197 142 31)(18 32 143 198)(19 199 144 25)(20 26 137 200)(21 193 138 27)(22 28 139 194)(23 195 140 29)(24 30 141 196)(33 132 115 72)(34 65 116 133)(35 134 117 66)(36 67 118 135)(37 136 119 68)(38 69 120 129)(39 130 113 70)(40 71 114 131)(41 215 164 224)(42 217 165 216)(43 209 166 218)(44 219 167 210)(45 211 168 220)(46 221 161 212)(47 213 162 222)(48 223 163 214)(57 77 125 184)(58 177 126 78)(59 79 127 178)(60 179 128 80)(61 73 121 180)(62 181 122 74)(63 75 123 182)(64 183 124 76)(89 111 185 100)(90 101 186 112)(91 105 187 102)(92 103 188 106)(93 107 189 104)(94 97 190 108)(95 109 191 98)(96 99 192 110)
(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)(25 152)(26 145)(27 146)(28 147)(29 148)(30 149)(31 150)(32 151)(33 126)(34 127)(35 128)(36 121)(37 122)(38 123)(39 124)(40 125)(41 101)(42 102)(43 103)(44 104)(45 97)(46 98)(47 99)(48 100)(57 114)(58 115)(59 116)(60 117)(61 118)(62 119)(63 120)(64 113)(65 79)(66 80)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)(81 197)(82 198)(83 199)(84 200)(85 193)(86 194)(87 195)(88 196)(89 214)(90 215)(91 216)(92 209)(93 210)(94 211)(95 212)(96 213)(105 165)(106 166)(107 167)(108 168)(109 161)(110 162)(111 163)(112 164)(129 182)(130 183)(131 184)(132 177)(133 178)(134 179)(135 180)(136 181)(137 170)(138 171)(139 172)(140 173)(141 174)(142 175)(143 176)(144 169)(153 206)(154 207)(155 208)(156 201)(157 202)(158 203)(159 204)(160 205)(185 223)(186 224)(187 217)(188 218)(189 219)(190 220)(191 221)(192 222)
(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)| (1,102,18,134,179,14,42)(2,103,19,135,180,15,43)(3,104,20,136,181,16,44)(4,97,21,129,182,9,45)(5,98,22,130,183,10,46)(6,99,23,131,184,11,47)(7,100,24,132,177,12,48)(8,101,17,133,178,13,41)(25,92,205,218,83,121,118)(26,93,206,219,84,122,119)(27,94,207,220,85,123,120)(28,95,208,221,86,124,113)(29,96,201,222,87,125,114)(30,89,202,223,88,126,115)(31,90,203,224,81,127,116)(32,91,204,217,82,128,117)(33,196,185,157,214,149,58)(34,197,186,158,215,150,59)(35,198,187,159,216,151,60)(36,199,188,160,209,152,61)(37,200,189,153,210,145,62)(38,193,190,154,211,146,63)(39,194,191,155,212,147,64)(40,195,192,156,213,148,57)(49,111,141,72,78,174,163)(50,112,142,65,79,175,164)(51,105,143,66,80,176,165)(52,106,144,67,73,169,166)(53,107,137,68,74,170,167)(54,108,138,69,75,171,168)(55,109,139,70,76,172,161)(56,110,140,71,77,173,162), (1,204,51,159)(2,160,52,205)(3,206,53,153)(4,154,54,207)(5,208,55,155)(6,156,56,201)(7,202,49,157)(8,158,50,203)(9,146,171,85)(10,86,172,147)(11,148,173,87)(12,88,174,149)(13,150,175,81)(14,82,176,151)(15,152,169,83)(16,84,170,145)(17,197,142,31)(18,32,143,198)(19,199,144,25)(20,26,137,200)(21,193,138,27)(22,28,139,194)(23,195,140,29)(24,30,141,196)(33,132,115,72)(34,65,116,133)(35,134,117,66)(36,67,118,135)(37,136,119,68)(38,69,120,129)(39,130,113,70)(40,71,114,131)(41,215,164,224)(42,217,165,216)(43,209,166,218)(44,219,167,210)(45,211,168,220)(46,221,161,212)(47,213,162,222)(48,223,163,214)(57,77,125,184)(58,177,126,78)(59,79,127,178)(60,179,128,80)(61,73,121,180)(62,181,122,74)(63,75,123,182)(64,183,124,76)(89,111,185,100)(90,101,186,112)(91,105,187,102)(92,103,188,106)(93,107,189,104)(94,97,190,108)(95,109,191,98)(96,99,192,110), (9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20)(25,152)(26,145)(27,146)(28,147)(29,148)(30,149)(31,150)(32,151)(33,126)(34,127)(35,128)(36,121)(37,122)(38,123)(39,124)(40,125)(41,101)(42,102)(43,103)(44,104)(45,97)(46,98)(47,99)(48,100)(57,114)(58,115)(59,116)(60,117)(61,118)(62,119)(63,120)(64,113)(65,79)(66,80)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78)(81,197)(82,198)(83,199)(84,200)(85,193)(86,194)(87,195)(88,196)(89,214)(90,215)(91,216)(92,209)(93,210)(94,211)(95,212)(96,213)(105,165)(106,166)(107,167)(108,168)(109,161)(110,162)(111,163)(112,164)(129,182)(130,183)(131,184)(132,177)(133,178)(134,179)(135,180)(136,181)(137,170)(138,171)(139,172)(140,173)(141,174)(142,175)(143,176)(144,169)(153,206)(154,207)(155,208)(156,201)(157,202)(158,203)(159,204)(160,205)(185,223)(186,224)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222), (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( (1,102,18,134,179,14,42)(2,103,19,135,180,15,43)(3,104,20,136,181,16,44)(4,97,21,129,182,9,45)(5,98,22,130,183,10,46)(6,99,23,131,184,11,47)(7,100,24,132,177,12,48)(8,101,17,133,178,13,41)(25,92,205,218,83,121,118)(26,93,206,219,84,122,119)(27,94,207,220,85,123,120)(28,95,208,221,86,124,113)(29,96,201,222,87,125,114)(30,89,202,223,88,126,115)(31,90,203,224,81,127,116)(32,91,204,217,82,128,117)(33,196,185,157,214,149,58)(34,197,186,158,215,150,59)(35,198,187,159,216,151,60)(36,199,188,160,209,152,61)(37,200,189,153,210,145,62)(38,193,190,154,211,146,63)(39,194,191,155,212,147,64)(40,195,192,156,213,148,57)(49,111,141,72,78,174,163)(50,112,142,65,79,175,164)(51,105,143,66,80,176,165)(52,106,144,67,73,169,166)(53,107,137,68,74,170,167)(54,108,138,69,75,171,168)(55,109,139,70,76,172,161)(56,110,140,71,77,173,162), (1,204,51,159)(2,160,52,205)(3,206,53,153)(4,154,54,207)(5,208,55,155)(6,156,56,201)(7,202,49,157)(8,158,50,203)(9,146,171,85)(10,86,172,147)(11,148,173,87)(12,88,174,149)(13,150,175,81)(14,82,176,151)(15,152,169,83)(16,84,170,145)(17,197,142,31)(18,32,143,198)(19,199,144,25)(20,26,137,200)(21,193,138,27)(22,28,139,194)(23,195,140,29)(24,30,141,196)(33,132,115,72)(34,65,116,133)(35,134,117,66)(36,67,118,135)(37,136,119,68)(38,69,120,129)(39,130,113,70)(40,71,114,131)(41,215,164,224)(42,217,165,216)(43,209,166,218)(44,219,167,210)(45,211,168,220)(46,221,161,212)(47,213,162,222)(48,223,163,214)(57,77,125,184)(58,177,126,78)(59,79,127,178)(60,179,128,80)(61,73,121,180)(62,181,122,74)(63,75,123,182)(64,183,124,76)(89,111,185,100)(90,101,186,112)(91,105,187,102)(92,103,188,106)(93,107,189,104)(94,97,190,108)(95,109,191,98)(96,99,192,110), (9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20)(25,152)(26,145)(27,146)(28,147)(29,148)(30,149)(31,150)(32,151)(33,126)(34,127)(35,128)(36,121)(37,122)(38,123)(39,124)(40,125)(41,101)(42,102)(43,103)(44,104)(45,97)(46,98)(47,99)(48,100)(57,114)(58,115)(59,116)(60,117)(61,118)(62,119)(63,120)(64,113)(65,79)(66,80)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78)(81,197)(82,198)(83,199)(84,200)(85,193)(86,194)(87,195)(88,196)(89,214)(90,215)(91,216)(92,209)(93,210)(94,211)(95,212)(96,213)(105,165)(106,166)(107,167)(108,168)(109,161)(110,162)(111,163)(112,164)(129,182)(130,183)(131,184)(132,177)(133,178)(134,179)(135,180)(136,181)(137,170)(138,171)(139,172)(140,173)(141,174)(142,175)(143,176)(144,169)(153,206)(154,207)(155,208)(156,201)(157,202)(158,203)(159,204)(160,205)(185,223)(186,224)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222), (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([[(1,102,18,134,179,14,42),(2,103,19,135,180,15,43),(3,104,20,136,181,16,44),(4,97,21,129,182,9,45),(5,98,22,130,183,10,46),(6,99,23,131,184,11,47),(7,100,24,132,177,12,48),(8,101,17,133,178,13,41),(25,92,205,218,83,121,118),(26,93,206,219,84,122,119),(27,94,207,220,85,123,120),(28,95,208,221,86,124,113),(29,96,201,222,87,125,114),(30,89,202,223,88,126,115),(31,90,203,224,81,127,116),(32,91,204,217,82,128,117),(33,196,185,157,214,149,58),(34,197,186,158,215,150,59),(35,198,187,159,216,151,60),(36,199,188,160,209,152,61),(37,200,189,153,210,145,62),(38,193,190,154,211,146,63),(39,194,191,155,212,147,64),(40,195,192,156,213,148,57),(49,111,141,72,78,174,163),(50,112,142,65,79,175,164),(51,105,143,66,80,176,165),(52,106,144,67,73,169,166),(53,107,137,68,74,170,167),(54,108,138,69,75,171,168),(55,109,139,70,76,172,161),(56,110,140,71,77,173,162)], [(1,204,51,159),(2,160,52,205),(3,206,53,153),(4,154,54,207),(5,208,55,155),(6,156,56,201),(7,202,49,157),(8,158,50,203),(9,146,171,85),(10,86,172,147),(11,148,173,87),(12,88,174,149),(13,150,175,81),(14,82,176,151),(15,152,169,83),(16,84,170,145),(17,197,142,31),(18,32,143,198),(19,199,144,25),(20,26,137,200),(21,193,138,27),(22,28,139,194),(23,195,140,29),(24,30,141,196),(33,132,115,72),(34,65,116,133),(35,134,117,66),(36,67,118,135),(37,136,119,68),(38,69,120,129),(39,130,113,70),(40,71,114,131),(41,215,164,224),(42,217,165,216),(43,209,166,218),(44,219,167,210),(45,211,168,220),(46,221,161,212),(47,213,162,222),(48,223,163,214),(57,77,125,184),(58,177,126,78),(59,79,127,178),(60,179,128,80),(61,73,121,180),(62,181,122,74),(63,75,123,182),(64,183,124,76),(89,111,185,100),(90,101,186,112),(91,105,187,102),(92,103,188,106),(93,107,189,104),(94,97,190,108),(95,109,191,98),(96,99,192,110)], [(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20),(25,152),(26,145),(27,146),(28,147),(29,148),(30,149),(31,150),(32,151),(33,126),(34,127),(35,128),(36,121),(37,122),(38,123),(39,124),(40,125),(41,101),(42,102),(43,103),(44,104),(45,97),(46,98),(47,99),(48,100),(57,114),(58,115),(59,116),(60,117),(61,118),(62,119),(63,120),(64,113),(65,79),(66,80),(67,73),(68,74),(69,75),(70,76),(71,77),(72,78),(81,197),(82,198),(83,199),(84,200),(85,193),(86,194),(87,195),(88,196),(89,214),(90,215),(91,216),(92,209),(93,210),(94,211),(95,212),(96,213),(105,165),(106,166),(107,167),(108,168),(109,161),(110,162),(111,163),(112,164),(129,182),(130,183),(131,184),(132,177),(133,178),(134,179),(135,180),(136,181),(137,170),(138,171),(139,172),(140,173),(141,174),(142,175),(143,176),(144,169),(153,206),(154,207),(155,208),(156,201),(157,202),(158,203),(159,204),(160,205),(185,223),(186,224),(187,217),(188,218),(189,219),(190,220),(191,221),(192,222)], [(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 | 4F | 4G | ··· | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8P | 8Q | 8R | 8S | 8T | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 1 | 1 | 1 | 1 | 2 | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 7 | ··· | 7 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | D4 | D7 | C4○D4 | D14 | D14 | C8○D4 | C4×D7 | C4×D7 | C8×D7 | D4×D7 | D4⋊2D7 | D28.C4 |
| kernel | C7⋊D4⋊C8 | C8×Dic7 | Dic7⋊C8 | D14⋊C8 | C7×C22⋊C8 | D7×C2×C8 | C22×C7⋊C8 | C4×C7⋊D4 | Dic7⋊C4 | D14⋊C4 | C23.D7 | C2×C7⋊D4 | C7⋊D4 | C7⋊C8 | C22⋊C8 | C28 | C2×C8 | C22×C4 | C14 | C2×C4 | C23 | C22 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 16 | 2 | 3 | 2 | 6 | 3 | 4 | 6 | 6 | 24 | 3 | 3 | 6 |
Matrix representation of C7⋊D4⋊C8 ►in GL4(𝔽113) generated by
| 104 | 41 | 0 | 0 |
| 1 | 33 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 80 | 0 | 0 |
| 89 | 0 | 0 | 0 |
| 0 | 0 | 67 | 106 |
| 0 | 0 | 28 | 46 |
| 0 | 80 | 0 | 0 |
| 89 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 3 | 112 |
| 44 | 0 | 0 | 0 |
| 0 | 44 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 45 | 98 |
G:=sub<GL(4,GF(113))| [104,1,0,0,41,33,0,0,0,0,1,0,0,0,0,1],[0,89,0,0,80,0,0,0,0,0,67,28,0,0,106,46],[0,89,0,0,80,0,0,0,0,0,1,3,0,0,0,112],[44,0,0,0,0,44,0,0,0,0,15,45,0,0,0,98] >;
C7⋊D4⋊C8 in GAP, Magma, Sage, TeX
C_7\rtimes D_4\rtimes C_8
% in TeX
G:=Group("C7:D4:C8"); // GroupNames label
G:=SmallGroup(448,259);
// by ID
G=gap.SmallGroup(448,259);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,219,58,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^2=d^8=1,b*a*b^-1=c*a*c=a^-1,a*d=d*a,c*b*c=d*b*d^-1=b^-1,c*d=d*c>;
// generators/relations