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## G = C4×C56⋊C2order 448 = 26·7

### Direct product of C4 and C56⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C4×C56⋊C2
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C2×D28 — C2×C56⋊C2 — C4×C56⋊C2
 Lower central C7 — C14 — C28 — C4×C56⋊C2
 Upper central C1 — C2×C4 — C42 — C4×C8

Generators and relations for C4×C56⋊C2
G = < a,b,c | a4=b56=c2=1, ab=ba, ac=ca, cbc=b27 >

Subgroups: 676 in 122 conjugacy classes, 55 normal (39 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C42, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, C28, D14, C2×C14, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C56, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C2×C28, C22×D7, C4×SD16, C56⋊C2, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C4×C28, C2×C56, C2×Dic14, C2×C4×D7, C2×D28, C28.44D4, C8⋊Dic7, C2.D56, C4×C56, C4×Dic14, C4×D28, C2×C56⋊C2, C4×C56⋊C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, SD16, C22×C4, C2×D4, C4○D4, D14, C4×D4, C2×SD16, C4○D8, C4×D7, D28, C22×D7, C4×SD16, C56⋊C2, C2×C4×D7, C2×D28, C4○D28, C4×D28, C2×C56⋊C2, D567C2, C4×C56⋊C2

Smallest permutation representation of C4×C56⋊C2
On 224 points
Generators in S224
(1 197 114 79)(2 198 115 80)(3 199 116 81)(4 200 117 82)(5 201 118 83)(6 202 119 84)(7 203 120 85)(8 204 121 86)(9 205 122 87)(10 206 123 88)(11 207 124 89)(12 208 125 90)(13 209 126 91)(14 210 127 92)(15 211 128 93)(16 212 129 94)(17 213 130 95)(18 214 131 96)(19 215 132 97)(20 216 133 98)(21 217 134 99)(22 218 135 100)(23 219 136 101)(24 220 137 102)(25 221 138 103)(26 222 139 104)(27 223 140 105)(28 224 141 106)(29 169 142 107)(30 170 143 108)(31 171 144 109)(32 172 145 110)(33 173 146 111)(34 174 147 112)(35 175 148 57)(36 176 149 58)(37 177 150 59)(38 178 151 60)(39 179 152 61)(40 180 153 62)(41 181 154 63)(42 182 155 64)(43 183 156 65)(44 184 157 66)(45 185 158 67)(46 186 159 68)(47 187 160 69)(48 188 161 70)(49 189 162 71)(50 190 163 72)(51 191 164 73)(52 192 165 74)(53 193 166 75)(54 194 167 76)(55 195 168 77)(56 196 113 78)
(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)
(2 28)(3 55)(4 26)(5 53)(6 24)(7 51)(8 22)(9 49)(10 20)(11 47)(12 18)(13 45)(14 16)(15 43)(17 41)(19 39)(21 37)(23 35)(25 33)(27 31)(30 56)(32 54)(34 52)(36 50)(38 48)(40 46)(42 44)(57 101)(58 72)(59 99)(60 70)(61 97)(62 68)(63 95)(64 66)(65 93)(67 91)(69 89)(71 87)(73 85)(74 112)(75 83)(76 110)(77 81)(78 108)(80 106)(82 104)(84 102)(86 100)(88 98)(90 96)(92 94)(103 111)(105 109)(113 143)(115 141)(116 168)(117 139)(118 166)(119 137)(120 164)(121 135)(122 162)(123 133)(124 160)(125 131)(126 158)(127 129)(128 156)(130 154)(132 152)(134 150)(136 148)(138 146)(140 144)(145 167)(147 165)(149 163)(151 161)(153 159)(155 157)(170 196)(171 223)(172 194)(173 221)(174 192)(175 219)(176 190)(177 217)(178 188)(179 215)(180 186)(181 213)(182 184)(183 211)(185 209)(187 207)(189 205)(191 203)(193 201)(195 199)(198 224)(200 222)(202 220)(204 218)(206 216)(208 214)(210 212)

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

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

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

124 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 7A 7B 7C 8A ··· 8H 14A ··· 14I 28A ··· 28AJ 56A ··· 56AV order 1 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 28 28 1 1 1 1 2 2 2 2 28 ··· 28 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

124 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D7 SD16 C4○D4 D14 D14 C4○D8 C4×D7 D28 C56⋊C2 C4○D28 D56⋊7C2 kernel C4×C56⋊C2 C28.44D4 C8⋊Dic7 C2.D56 C4×C56 C4×Dic14 C4×D28 C2×C56⋊C2 C56⋊C2 C2×C28 C4×C8 C28 C28 C42 C2×C8 C14 C8 C2×C4 C4 C4 C2 # reps 1 1 1 1 1 1 1 1 8 2 3 4 2 3 6 4 12 12 24 12 24

Matrix representation of C4×C56⋊C2 in GL3(𝔽113) generated by

 98 0 0 0 98 0 0 0 98
,
 1 0 0 0 73 50 0 63 71
,
 112 0 0 0 1 0 0 9 112
G:=sub<GL(3,GF(113))| [98,0,0,0,98,0,0,0,98],[1,0,0,0,73,63,0,50,71],[112,0,0,0,1,9,0,0,112] >;

C4×C56⋊C2 in GAP, Magma, Sage, TeX

C_4\times C_{56}\rtimes C_2
% in TeX

G:=Group("C4xC56:C2");
// GroupNames label

G:=SmallGroup(448,225);
// by ID

G=gap.SmallGroup(448,225);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,58,1684,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^4=b^56=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^27>;
// generators/relations

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