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## G = C2×D56⋊7C2order 448 = 26·7

### Direct product of C2 and D56⋊7C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C2×D56⋊7C2
 Chief series C1 — C7 — C14 — C28 — D28 — C2×D28 — C2×C4○D28 — C2×D56⋊7C2
 Lower central C7 — C14 — C28 — C2×D56⋊7C2
 Upper central C1 — C2×C4 — C22×C4 — C22×C8

Generators and relations for C2×D567C2
G = < a,b,c,d | a2=b56=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd=b28c >

Subgroups: 1380 in 266 conjugacy classes, 111 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C2×C4○D8, C56⋊C2, D56, Dic28, C2×C56, C2×C56, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C4○D28, C2×C7⋊D4, C22×C28, C2×C56⋊C2, C2×D56, D567C2, C2×Dic28, C22×C56, C2×C4○D28, C2×D567C2
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C4○D8, C22×D4, D28, C22×D7, C2×C4○D8, C2×D28, C23×D7, D567C2, C22×D28, C2×D567C2

Smallest permutation representation of C2×D567C2
On 224 points
Generators in S224
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(49 97)(50 98)(51 99)(52 100)(53 101)(54 102)(55 103)(56 104)(113 169)(114 170)(115 171)(116 172)(117 173)(118 174)(119 175)(120 176)(121 177)(122 178)(123 179)(124 180)(125 181)(126 182)(127 183)(128 184)(129 185)(130 186)(131 187)(132 188)(133 189)(134 190)(135 191)(136 192)(137 193)(138 194)(139 195)(140 196)(141 197)(142 198)(143 199)(144 200)(145 201)(146 202)(147 203)(148 204)(149 205)(150 206)(151 207)(152 208)(153 209)(154 210)(155 211)(156 212)(157 213)(158 214)(159 215)(160 216)(161 217)(162 218)(163 219)(164 220)(165 221)(166 222)(167 223)(168 224)
(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)
(1 168)(2 167)(3 166)(4 165)(5 164)(6 163)(7 162)(8 161)(9 160)(10 159)(11 158)(12 157)(13 156)(14 155)(15 154)(16 153)(17 152)(18 151)(19 150)(20 149)(21 148)(22 147)(23 146)(24 145)(25 144)(26 143)(27 142)(28 141)(29 140)(30 139)(31 138)(32 137)(33 136)(34 135)(35 134)(36 133)(37 132)(38 131)(39 130)(40 129)(41 128)(42 127)(43 126)(44 125)(45 124)(46 123)(47 122)(48 121)(49 120)(50 119)(51 118)(52 117)(53 116)(54 115)(55 114)(56 113)(57 216)(58 215)(59 214)(60 213)(61 212)(62 211)(63 210)(64 209)(65 208)(66 207)(67 206)(68 205)(69 204)(70 203)(71 202)(72 201)(73 200)(74 199)(75 198)(76 197)(77 196)(78 195)(79 194)(80 193)(81 192)(82 191)(83 190)(84 189)(85 188)(86 187)(87 186)(88 185)(89 184)(90 183)(91 182)(92 181)(93 180)(94 179)(95 178)(96 177)(97 176)(98 175)(99 174)(100 173)(101 172)(102 171)(103 170)(104 169)(105 224)(106 223)(107 222)(108 221)(109 220)(110 219)(111 218)(112 217)
(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(169 197)(170 198)(171 199)(172 200)(173 201)(174 202)(175 203)(176 204)(177 205)(178 206)(179 207)(180 208)(181 209)(182 210)(183 211)(184 212)(185 213)(186 214)(187 215)(188 216)(189 217)(190 218)(191 219)(192 220)(193 221)(194 222)(195 223)(196 224)

G:=sub<Sym(224)| (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104)(113,169)(114,170)(115,171)(116,172)(117,173)(118,174)(119,175)(120,176)(121,177)(122,178)(123,179)(124,180)(125,181)(126,182)(127,183)(128,184)(129,185)(130,186)(131,187)(132,188)(133,189)(134,190)(135,191)(136,192)(137,193)(138,194)(139,195)(140,196)(141,197)(142,198)(143,199)(144,200)(145,201)(146,202)(147,203)(148,204)(149,205)(150,206)(151,207)(152,208)(153,209)(154,210)(155,211)(156,212)(157,213)(158,214)(159,215)(160,216)(161,217)(162,218)(163,219)(164,220)(165,221)(166,222)(167,223)(168,224), (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), (1,168)(2,167)(3,166)(4,165)(5,164)(6,163)(7,162)(8,161)(9,160)(10,159)(11,158)(12,157)(13,156)(14,155)(15,154)(16,153)(17,152)(18,151)(19,150)(20,149)(21,148)(22,147)(23,146)(24,145)(25,144)(26,143)(27,142)(28,141)(29,140)(30,139)(31,138)(32,137)(33,136)(34,135)(35,134)(36,133)(37,132)(38,131)(39,130)(40,129)(41,128)(42,127)(43,126)(44,125)(45,124)(46,123)(47,122)(48,121)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,216)(58,215)(59,214)(60,213)(61,212)(62,211)(63,210)(64,209)(65,208)(66,207)(67,206)(68,205)(69,204)(70,203)(71,202)(72,201)(73,200)(74,199)(75,198)(76,197)(77,196)(78,195)(79,194)(80,193)(81,192)(82,191)(83,190)(84,189)(85,188)(86,187)(87,186)(88,185)(89,184)(90,183)(91,182)(92,181)(93,180)(94,179)(95,178)(96,177)(97,176)(98,175)(99,174)(100,173)(101,172)(102,171)(103,170)(104,169)(105,224)(106,223)(107,222)(108,221)(109,220)(110,219)(111,218)(112,217), (113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224)>;

G:=Group( (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104)(113,169)(114,170)(115,171)(116,172)(117,173)(118,174)(119,175)(120,176)(121,177)(122,178)(123,179)(124,180)(125,181)(126,182)(127,183)(128,184)(129,185)(130,186)(131,187)(132,188)(133,189)(134,190)(135,191)(136,192)(137,193)(138,194)(139,195)(140,196)(141,197)(142,198)(143,199)(144,200)(145,201)(146,202)(147,203)(148,204)(149,205)(150,206)(151,207)(152,208)(153,209)(154,210)(155,211)(156,212)(157,213)(158,214)(159,215)(160,216)(161,217)(162,218)(163,219)(164,220)(165,221)(166,222)(167,223)(168,224), (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), (1,168)(2,167)(3,166)(4,165)(5,164)(6,163)(7,162)(8,161)(9,160)(10,159)(11,158)(12,157)(13,156)(14,155)(15,154)(16,153)(17,152)(18,151)(19,150)(20,149)(21,148)(22,147)(23,146)(24,145)(25,144)(26,143)(27,142)(28,141)(29,140)(30,139)(31,138)(32,137)(33,136)(34,135)(35,134)(36,133)(37,132)(38,131)(39,130)(40,129)(41,128)(42,127)(43,126)(44,125)(45,124)(46,123)(47,122)(48,121)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,216)(58,215)(59,214)(60,213)(61,212)(62,211)(63,210)(64,209)(65,208)(66,207)(67,206)(68,205)(69,204)(70,203)(71,202)(72,201)(73,200)(74,199)(75,198)(76,197)(77,196)(78,195)(79,194)(80,193)(81,192)(82,191)(83,190)(84,189)(85,188)(86,187)(87,186)(88,185)(89,184)(90,183)(91,182)(92,181)(93,180)(94,179)(95,178)(96,177)(97,176)(98,175)(99,174)(100,173)(101,172)(102,171)(103,170)(104,169)(105,224)(106,223)(107,222)(108,221)(109,220)(110,219)(111,218)(112,217), (113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224) );

G=PermutationGroup([[(1,105),(2,106),(3,107),(4,108),(5,109),(6,110),(7,111),(8,112),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96),(49,97),(50,98),(51,99),(52,100),(53,101),(54,102),(55,103),(56,104),(113,169),(114,170),(115,171),(116,172),(117,173),(118,174),(119,175),(120,176),(121,177),(122,178),(123,179),(124,180),(125,181),(126,182),(127,183),(128,184),(129,185),(130,186),(131,187),(132,188),(133,189),(134,190),(135,191),(136,192),(137,193),(138,194),(139,195),(140,196),(141,197),(142,198),(143,199),(144,200),(145,201),(146,202),(147,203),(148,204),(149,205),(150,206),(151,207),(152,208),(153,209),(154,210),(155,211),(156,212),(157,213),(158,214),(159,215),(160,216),(161,217),(162,218),(163,219),(164,220),(165,221),(166,222),(167,223),(168,224)], [(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)], [(1,168),(2,167),(3,166),(4,165),(5,164),(6,163),(7,162),(8,161),(9,160),(10,159),(11,158),(12,157),(13,156),(14,155),(15,154),(16,153),(17,152),(18,151),(19,150),(20,149),(21,148),(22,147),(23,146),(24,145),(25,144),(26,143),(27,142),(28,141),(29,140),(30,139),(31,138),(32,137),(33,136),(34,135),(35,134),(36,133),(37,132),(38,131),(39,130),(40,129),(41,128),(42,127),(43,126),(44,125),(45,124),(46,123),(47,122),(48,121),(49,120),(50,119),(51,118),(52,117),(53,116),(54,115),(55,114),(56,113),(57,216),(58,215),(59,214),(60,213),(61,212),(62,211),(63,210),(64,209),(65,208),(66,207),(67,206),(68,205),(69,204),(70,203),(71,202),(72,201),(73,200),(74,199),(75,198),(76,197),(77,196),(78,195),(79,194),(80,193),(81,192),(82,191),(83,190),(84,189),(85,188),(86,187),(87,186),(88,185),(89,184),(90,183),(91,182),(92,181),(93,180),(94,179),(95,178),(96,177),(97,176),(98,175),(99,174),(100,173),(101,172),(102,171),(103,170),(104,169),(105,224),(106,223),(107,222),(108,221),(109,220),(110,219),(111,218),(112,217)], [(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148),(121,149),(122,150),(123,151),(124,152),(125,153),(126,154),(127,155),(128,156),(129,157),(130,158),(131,159),(132,160),(133,161),(134,162),(135,163),(136,164),(137,165),(138,166),(139,167),(140,168),(169,197),(170,198),(171,199),(172,200),(173,201),(174,202),(175,203),(176,204),(177,205),(178,206),(179,207),(180,208),(181,209),(182,210),(183,211),(184,212),(185,213),(186,214),(187,215),(188,216),(189,217),(190,218),(191,219),(192,220),(193,221),(194,222),(195,223),(196,224)]])

124 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 7A 7B 7C 8A ··· 8H 14A ··· 14U 28A ··· 28X 56A ··· 56AV order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 7 7 7 8 ··· 8 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 28 28 28 28 1 1 1 1 2 2 28 28 28 28 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

124 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 C4○D8 D28 D28 D56⋊7C2 kernel C2×D56⋊7C2 C2×C56⋊C2 C2×D56 D56⋊7C2 C2×Dic28 C22×C56 C2×C4○D28 C2×C28 C22×C14 C22×C8 C2×C8 C22×C4 C14 C2×C4 C23 C2 # reps 1 2 1 8 1 1 2 3 1 3 18 3 8 18 6 48

Matrix representation of C2×D567C2 in GL3(𝔽113) generated by

 112 0 0 0 112 0 0 0 112
,
 1 0 0 0 22 0 0 0 36
,
 112 0 0 0 0 36 0 22 0
,
 112 0 0 0 1 0 0 0 112
G:=sub<GL(3,GF(113))| [112,0,0,0,112,0,0,0,112],[1,0,0,0,22,0,0,0,36],[112,0,0,0,0,22,0,36,0],[112,0,0,0,1,0,0,0,112] >;

C2×D567C2 in GAP, Magma, Sage, TeX

C_2\times D_{56}\rtimes_7C_2
% in TeX

G:=Group("C2xD56:7C2");
// GroupNames label

G:=SmallGroup(448,1194);
// by ID

G=gap.SmallGroup(448,1194);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,675,80,1684,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^56=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d=b^28*c>;
// generators/relations

׿
×
𝔽