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## G = C2×C112⋊C2order 448 = 26·7

### Direct product of C2 and C112⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — C2×C112⋊C2
 Chief series C1 — C7 — C14 — C28 — C56 — D56 — C2×D56 — C2×C112⋊C2
 Lower central C7 — C14 — C28 — C56 — C2×C112⋊C2
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×C16

Generators and relations for C2×C112⋊C2
G = < a,b,c | a2=b112=c2=1, ab=ba, ac=ca, cbc=b55 >

Subgroups: 756 in 90 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C16, C2×C8, D8, Q16, C2×D4, C2×Q8, Dic7, C28, D14, C2×C14, C2×C16, SD32, C2×D8, C2×Q16, C56, Dic14, D28, C2×Dic7, C2×C28, C22×D7, C2×SD32, C112, D56, D56, Dic28, Dic28, C2×C56, C2×Dic14, C2×D28, C112⋊C2, C2×C112, C2×D56, C2×Dic28, C2×C112⋊C2
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, SD32, C2×D8, D28, C22×D7, C2×SD32, D56, C2×D28, C112⋊C2, C2×D56, C2×C112⋊C2

Smallest permutation representation of C2×C112⋊C2
On 224 points
Generators in S224
(1 141)(2 142)(3 143)(4 144)(5 145)(6 146)(7 147)(8 148)(9 149)(10 150)(11 151)(12 152)(13 153)(14 154)(15 155)(16 156)(17 157)(18 158)(19 159)(20 160)(21 161)(22 162)(23 163)(24 164)(25 165)(26 166)(27 167)(28 168)(29 169)(30 170)(31 171)(32 172)(33 173)(34 174)(35 175)(36 176)(37 177)(38 178)(39 179)(40 180)(41 181)(42 182)(43 183)(44 184)(45 185)(46 186)(47 187)(48 188)(49 189)(50 190)(51 191)(52 192)(53 193)(54 194)(55 195)(56 196)(57 197)(58 198)(59 199)(60 200)(61 201)(62 202)(63 203)(64 204)(65 205)(66 206)(67 207)(68 208)(69 209)(70 210)(71 211)(72 212)(73 213)(74 214)(75 215)(76 216)(77 217)(78 218)(79 219)(80 220)(81 221)(82 222)(83 223)(84 224)(85 113)(86 114)(87 115)(88 116)(89 117)(90 118)(91 119)(92 120)(93 121)(94 122)(95 123)(96 124)(97 125)(98 126)(99 127)(100 128)(101 129)(102 130)(103 131)(104 132)(105 133)(106 134)(107 135)(108 136)(109 137)(110 138)(111 139)(112 140)
(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 56)(3 111)(4 54)(5 109)(6 52)(7 107)(8 50)(9 105)(10 48)(11 103)(12 46)(13 101)(14 44)(15 99)(16 42)(17 97)(18 40)(19 95)(20 38)(21 93)(22 36)(23 91)(24 34)(25 89)(26 32)(27 87)(28 30)(29 85)(31 83)(33 81)(35 79)(37 77)(39 75)(41 73)(43 71)(45 69)(47 67)(49 65)(51 63)(53 61)(55 59)(58 112)(60 110)(62 108)(64 106)(66 104)(68 102)(70 100)(72 98)(74 96)(76 94)(78 92)(80 90)(82 88)(84 86)(113 169)(114 224)(115 167)(116 222)(117 165)(118 220)(119 163)(120 218)(121 161)(122 216)(123 159)(124 214)(125 157)(126 212)(127 155)(128 210)(129 153)(130 208)(131 151)(132 206)(133 149)(134 204)(135 147)(136 202)(137 145)(138 200)(139 143)(140 198)(142 196)(144 194)(146 192)(148 190)(150 188)(152 186)(154 184)(156 182)(158 180)(160 178)(162 176)(164 174)(166 172)(168 170)(171 223)(173 221)(175 219)(177 217)(179 215)(181 213)(183 211)(185 209)(187 207)(189 205)(191 203)(193 201)(195 199)

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

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

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

118 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 16A ··· 16H 28A ··· 28L 56A ··· 56X 112A ··· 112AV order 1 2 2 2 2 2 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 16 ··· 16 28 ··· 28 56 ··· 56 112 ··· 112 size 1 1 1 1 56 56 2 2 56 56 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

118 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D7 D8 D8 D14 D14 SD32 D28 D28 D56 D56 C112⋊C2 kernel C2×C112⋊C2 C112⋊C2 C2×C112 C2×D56 C2×Dic28 C56 C2×C28 C2×C16 C28 C2×C14 C16 C2×C8 C14 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 3 2 2 6 3 8 6 6 12 12 48

Matrix representation of C2×C112⋊C2 in GL5(𝔽113)

 112 0 0 0 0 0 112 0 0 0 0 0 112 0 0 0 0 0 1 0 0 0 0 0 1
,
 112 0 0 0 0 0 60 69 0 0 0 44 60 0 0 0 0 0 28 73 0 0 0 76 65
,
 112 0 0 0 0 0 1 0 0 0 0 0 112 0 0 0 0 0 33 1 0 0 0 42 80

G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,1,0,0,0,0,0,1],[112,0,0,0,0,0,60,44,0,0,0,69,60,0,0,0,0,0,28,76,0,0,0,73,65],[112,0,0,0,0,0,1,0,0,0,0,0,112,0,0,0,0,0,33,42,0,0,0,1,80] >;

C2×C112⋊C2 in GAP, Magma, Sage, TeX

C_2\times C_{112}\rtimes C_2
% in TeX

G:=Group("C2xC112:C2");
// GroupNames label

G:=SmallGroup(448,437);
// by ID

G=gap.SmallGroup(448,437);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,142,1571,80,1684,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^2=b^112=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^55>;
// generators/relations

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