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## G = C2.D112order 448 = 26·7

### 2nd central extension by C2 of D112

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — C2.D112
 Chief series C1 — C7 — C14 — C28 — C56 — C2×C56 — C2×D56 — C2.D112
 Lower central C7 — C14 — C28 — C56 — C2.D112
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×C16

Generators and relations for C2.D112
G = < a,b,c | a2=b112=1, c2=a, ab=ba, ac=ca, cbc-1=ab-1 >

Subgroups: 628 in 66 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, D7, C14, C16, C4⋊C4, C2×C8, D8, C2×D4, Dic7, C28, D14, C2×C14, C2.D8, C2×C16, C2×D8, C56, D28, C2×Dic7, C2×C28, C22×D7, C2.D16, C112, D56, D56, C4⋊Dic7, C2×C56, C2×D28, C561C4, C2×C112, C2×D56, C2.D112
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, D14, D4⋊C4, D16, SD32, C4×D7, D28, C7⋊D4, C2.D16, C56⋊C2, D56, D14⋊C4, D112, C112⋊C2, C2.D56, C2.D112

Smallest permutation representation of C2.D112
On 224 points
Generators in S224
```(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 131)(18 132)(19 133)(20 134)(21 135)(22 136)(23 137)(24 138)(25 139)(26 140)(27 141)(28 142)(29 143)(30 144)(31 145)(32 146)(33 147)(34 148)(35 149)(36 150)(37 151)(38 152)(39 153)(40 154)(41 155)(42 156)(43 157)(44 158)(45 159)(46 160)(47 161)(48 162)(49 163)(50 164)(51 165)(52 166)(53 167)(54 168)(55 169)(56 170)(57 171)(58 172)(59 173)(60 174)(61 175)(62 176)(63 177)(64 178)(65 179)(66 180)(67 181)(68 182)(69 183)(70 184)(71 185)(72 186)(73 187)(74 188)(75 189)(76 190)(77 191)(78 192)(79 193)(80 194)(81 195)(82 196)(83 197)(84 198)(85 199)(86 200)(87 201)(88 202)(89 203)(90 204)(91 205)(92 206)(93 207)(94 208)(95 209)(96 210)(97 211)(98 212)(99 213)(100 214)(101 215)(102 216)(103 217)(104 218)(105 219)(106 220)(107 221)(108 222)(109 223)(110 224)(111 113)(112 114)
(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)
(1 114 115 112)(2 111 116 113)(3 224 117 110)(4 109 118 223)(5 222 119 108)(6 107 120 221)(7 220 121 106)(8 105 122 219)(9 218 123 104)(10 103 124 217)(11 216 125 102)(12 101 126 215)(13 214 127 100)(14 99 128 213)(15 212 129 98)(16 97 130 211)(17 210 131 96)(18 95 132 209)(19 208 133 94)(20 93 134 207)(21 206 135 92)(22 91 136 205)(23 204 137 90)(24 89 138 203)(25 202 139 88)(26 87 140 201)(27 200 141 86)(28 85 142 199)(29 198 143 84)(30 83 144 197)(31 196 145 82)(32 81 146 195)(33 194 147 80)(34 79 148 193)(35 192 149 78)(36 77 150 191)(37 190 151 76)(38 75 152 189)(39 188 153 74)(40 73 154 187)(41 186 155 72)(42 71 156 185)(43 184 157 70)(44 69 158 183)(45 182 159 68)(46 67 160 181)(47 180 161 66)(48 65 162 179)(49 178 163 64)(50 63 164 177)(51 176 165 62)(52 61 166 175)(53 174 167 60)(54 59 168 173)(55 172 169 58)(56 57 170 171)```

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

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

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

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 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C4 D4 D4 D7 SD16 D8 D14 D16 SD32 C4×D7 C7⋊D4 D28 C56⋊C2 D56 D112 C112⋊C2 kernel C2.D112 C56⋊1C4 C2×C112 C2×D56 D56 C56 C2×C28 C2×C16 C28 C2×C14 C2×C8 C14 C14 C8 C8 C2×C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 3 2 2 3 4 4 6 6 6 12 12 24 24

Matrix representation of C2.D112 in GL3(𝔽113) generated by

 112 0 0 0 1 0 0 0 1
,
 15 0 0 0 7 74 0 78 66
,
 98 0 0 0 7 95 0 78 106
`G:=sub<GL(3,GF(113))| [112,0,0,0,1,0,0,0,1],[15,0,0,0,7,78,0,74,66],[98,0,0,0,7,78,0,95,106] >;`

C2.D112 in GAP, Magma, Sage, TeX

`C_2.D_{112}`
`% in TeX`

`G:=Group("C2.D112");`
`// GroupNames label`

`G:=SmallGroup(448,66);`
`// by ID`

`G=gap.SmallGroup(448,66);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,85,204,422,268,1684,102,18822]);`
`// Polycyclic`

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

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