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## G = C56.43D4order 448 = 26·7

### 43rd non-split extension by C56 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — C56.43D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4×Dic7 — C28.17D4 — C56.43D4
 Lower central C7 — C14 — C2×C28 — C56.43D4
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for C56.43D4
G = < a,b,c | a56=b4=1, c2=a28, bab-1=a41, cac-1=a27, cbc-1=a28b-1 >

Subgroups: 708 in 130 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×4], D4 [×4], Q8 [×4], C23 [×2], D7, C14, C14 [×2], C14, C42, C22⋊C4 [×4], C2×C8, C2×C8, D8 [×2], SD16 [×4], Q16 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8, Dic7 [×3], C28 [×2], C28, D14 [×3], C2×C14, C2×C14 [×3], C4×C8, C4.4D4 [×2], C2×D8, C2×SD16, C2×SD16, C2×Q16, C7⋊C8 [×2], C56 [×2], Dic14 [×2], D28 [×2], C2×Dic7 [×2], C2×Dic7, C2×C28, C2×C28, C7×D4 [×2], C7×Q8 [×2], C22×D7, C22×C14, C8.12D4, C56⋊C2 [×2], C2×C7⋊C8, C4×Dic7, D14⋊C4 [×2], D4⋊D7 [×2], C7⋊Q16 [×2], C23.D7 [×2], C2×C56, C7×SD16 [×2], C2×Dic14, C2×D28, D4×C14, Q8×C14, C8×Dic7, C2×C56⋊C2, C2×D4⋊D7, C28.17D4, C2×C7⋊Q16, C28.23D4, C14×SD16, C56.43D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D7, C2×D4 [×3], D14 [×3], C41D4, C4○D8 [×2], C7⋊D4 [×2], C22×D7, C8.12D4, D4×D7 [×2], C2×C7⋊D4, SD163D7 [×2], C28⋊D4, C56.43D4

Smallest permutation representation of C56.43D4
On 224 points
Generators in S224
```(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)
(1 173 168 109)(2 214 113 94)(3 199 114 79)(4 184 115 64)(5 169 116 105)(6 210 117 90)(7 195 118 75)(8 180 119 60)(9 221 120 101)(10 206 121 86)(11 191 122 71)(12 176 123 112)(13 217 124 97)(14 202 125 82)(15 187 126 67)(16 172 127 108)(17 213 128 93)(18 198 129 78)(19 183 130 63)(20 224 131 104)(21 209 132 89)(22 194 133 74)(23 179 134 59)(24 220 135 100)(25 205 136 85)(26 190 137 70)(27 175 138 111)(28 216 139 96)(29 201 140 81)(30 186 141 66)(31 171 142 107)(32 212 143 92)(33 197 144 77)(34 182 145 62)(35 223 146 103)(36 208 147 88)(37 193 148 73)(38 178 149 58)(39 219 150 99)(40 204 151 84)(41 189 152 69)(42 174 153 110)(43 215 154 95)(44 200 155 80)(45 185 156 65)(46 170 157 106)(47 211 158 91)(48 196 159 76)(49 181 160 61)(50 222 161 102)(51 207 162 87)(52 192 163 72)(53 177 164 57)(54 218 165 98)(55 203 166 83)(56 188 167 68)
(1 8 29 36)(2 35 30 7)(3 6 31 34)(4 33 32 5)(9 56 37 28)(10 27 38 55)(11 54 39 26)(12 25 40 53)(13 52 41 24)(14 23 42 51)(15 50 43 22)(16 21 44 49)(17 48 45 20)(18 19 46 47)(57 204 85 176)(58 175 86 203)(59 202 87 174)(60 173 88 201)(61 200 89 172)(62 171 90 199)(63 198 91 170)(64 169 92 197)(65 196 93 224)(66 223 94 195)(67 194 95 222)(68 221 96 193)(69 192 97 220)(70 219 98 191)(71 190 99 218)(72 217 100 189)(73 188 101 216)(74 215 102 187)(75 186 103 214)(76 213 104 185)(77 184 105 212)(78 211 106 183)(79 182 107 210)(80 209 108 181)(81 180 109 208)(82 207 110 179)(83 178 111 206)(84 205 112 177)(113 146 141 118)(114 117 142 145)(115 144 143 116)(119 140 147 168)(120 167 148 139)(121 138 149 166)(122 165 150 137)(123 136 151 164)(124 163 152 135)(125 134 153 162)(126 161 154 133)(127 132 155 160)(128 159 156 131)(129 130 157 158)```

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 14A ··· 14I 14J ··· 14O 28A ··· 28F 28G ··· 28L 56A ··· 56L order 1 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 8 56 2 2 8 14 14 14 14 56 2 2 2 2 2 2 2 14 14 14 14 2 ··· 2 8 ··· 8 4 ··· 4 8 ··· 8 4 ··· 4

64 irreducible representations

 dim 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 D4 D4 D4 D7 D14 D14 D14 C4○D8 C7⋊D4 D4×D7 D4×D7 SD16⋊3D7 kernel C56.43D4 C8×Dic7 C2×C56⋊C2 C2×D4⋊D7 C28.17D4 C2×C7⋊Q16 C28.23D4 C14×SD16 C7⋊C8 C56 C2×Dic7 C2×SD16 C2×C8 C2×D4 C2×Q8 C14 C8 C4 C22 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 3 3 3 3 8 12 3 3 12

Matrix representation of C56.43D4 in GL4(𝔽113) generated by

 0 1 0 0 112 9 0 0 0 0 26 26 0 0 100 0
,
 79 108 0 0 28 34 0 0 0 0 15 30 0 0 98 98
,
 112 0 0 0 104 1 0 0 0 0 87 87 0 0 13 26
`G:=sub<GL(4,GF(113))| [0,112,0,0,1,9,0,0,0,0,26,100,0,0,26,0],[79,28,0,0,108,34,0,0,0,0,15,98,0,0,30,98],[112,104,0,0,0,1,0,0,0,0,87,13,0,0,87,26] >;`

C56.43D4 in GAP, Magma, Sage, TeX

`C_{56}._{43}D_4`
`% in TeX`

`G:=Group("C56.43D4");`
`// GroupNames label`

`G:=SmallGroup(448,702);`
`// by ID`

`G=gap.SmallGroup(448,702);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,701,1094,135,184,570,297,136,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^56=b^4=1,c^2=a^28,b*a*b^-1=a^41,c*a*c^-1=a^27,c*b*c^-1=a^28*b^-1>;`
`// generators/relations`

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