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## G = C56.70C23order 448 = 26·7

### 16th non-split extension by C56 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C56.70C23
 Chief series C1 — C7 — C14 — C28 — C56 — C7⋊C16 — C2×C7⋊C16 — C56.70C23
 Lower central C7 — C14 — C56.70C23
 Upper central C1 — C8 — C8○D4

Generators and relations for C56.70C23
G = < a,b,c,d | a56=c2=d2=1, b2=a49, bab-1=a41, ac=ca, ad=da, bc=cb, bd=db, dcd=a28c >

Subgroups: 156 in 84 conjugacy classes, 67 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, Q8, C14, C14, C16, C2×C8, M4(2), C4○D4, C28, C28, C2×C14, C2×C16, M5(2), C8○D4, C56, C56, C2×C28, C7×D4, C7×Q8, D4○C16, C7⋊C16, C7⋊C16, C2×C56, C7×M4(2), C7×C4○D4, C2×C7⋊C16, C28.C8, C7×C8○D4, C56.70C23
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D7, C2×C8, C22×C4, Dic7, D14, C22×C8, C7⋊C8, C2×Dic7, C22×D7, D4○C16, C2×C7⋊C8, C22×Dic7, C22×C7⋊C8, C56.70C23

Smallest permutation representation of C56.70C23
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)
(1 113 50 162 43 155 36 148 29 141 22 134 15 127 8 120)(2 154 51 147 44 140 37 133 30 126 23 119 16 168 9 161)(3 139 52 132 45 125 38 118 31 167 24 160 17 153 10 146)(4 124 53 117 46 166 39 159 32 152 25 145 18 138 11 131)(5 165 54 158 47 151 40 144 33 137 26 130 19 123 12 116)(6 150 55 143 48 136 41 129 34 122 27 115 20 164 13 157)(7 135 56 128 49 121 42 114 35 163 28 156 21 149 14 142)(57 184 106 177 99 170 92 219 85 212 78 205 71 198 64 191)(58 169 107 218 100 211 93 204 86 197 79 190 72 183 65 176)(59 210 108 203 101 196 94 189 87 182 80 175 73 224 66 217)(60 195 109 188 102 181 95 174 88 223 81 216 74 209 67 202)(61 180 110 173 103 222 96 215 89 208 82 201 75 194 68 187)(62 221 111 214 104 207 97 200 90 193 83 186 76 179 69 172)(63 206 112 199 105 192 98 185 91 178 84 171 77 220 70 213)
(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)(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)
(1 93)(2 94)(3 95)(4 96)(5 97)(6 98)(7 99)(8 100)(9 101)(10 102)(11 103)(12 104)(13 105)(14 106)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(113 204)(114 205)(115 206)(116 207)(117 208)(118 209)(119 210)(120 211)(121 212)(122 213)(123 214)(124 215)(125 216)(126 217)(127 218)(128 219)(129 220)(130 221)(131 222)(132 223)(133 224)(134 169)(135 170)(136 171)(137 172)(138 173)(139 174)(140 175)(141 176)(142 177)(143 178)(144 179)(145 180)(146 181)(147 182)(148 183)(149 184)(150 185)(151 186)(152 187)(153 188)(154 189)(155 190)(156 191)(157 192)(158 193)(159 194)(160 195)(161 196)(162 197)(163 198)(164 199)(165 200)(166 201)(167 202)(168 203)```

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 8C 8D 8E ··· 8J 14A 14B 14C 14D ··· 14L 16A ··· 16H 16I ··· 16T 28A ··· 28F 28G ··· 28O 56A ··· 56L 56M ··· 56AD order 1 2 2 2 2 4 4 4 4 4 7 7 7 8 8 8 8 8 ··· 8 14 14 14 14 ··· 14 16 ··· 16 16 ··· 16 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 size 1 1 2 2 2 1 1 2 2 2 2 2 2 1 1 1 1 2 ··· 2 2 2 2 4 ··· 4 7 ··· 7 14 ··· 14 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + - - image C1 C2 C2 C2 C4 C4 C8 C8 D7 D14 Dic7 Dic7 C7⋊C8 C7⋊C8 D4○C16 C56.70C23 kernel C56.70C23 C2×C7⋊C16 C28.C8 C7×C8○D4 C7×M4(2) C7×C4○D4 C7×D4 C7×Q8 C8○D4 C2×C8 M4(2) C4○D4 D4 Q8 C7 C1 # reps 1 3 3 1 6 2 12 4 3 9 9 3 18 6 8 12

Matrix representation of C56.70C23 in GL4(𝔽113) generated by

 0 98 0 0 15 91 0 0 0 0 18 0 0 0 0 18
,
 11 86 0 0 72 102 0 0 0 0 73 0 0 0 0 73
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 112
,
 1 0 0 0 0 1 0 0 0 0 112 2 0 0 0 1
`G:=sub<GL(4,GF(113))| [0,15,0,0,98,91,0,0,0,0,18,0,0,0,0,18],[11,72,0,0,86,102,0,0,0,0,73,0,0,0,0,73],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,2,1] >;`

C56.70C23 in GAP, Magma, Sage, TeX

`C_{56}._{70}C_2^3`
`% in TeX`

`G:=Group("C56.70C2^3");`
`// GroupNames label`

`G:=SmallGroup(448,674);`
`// by ID`

`G=gap.SmallGroup(448,674);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,387,80,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^56=c^2=d^2=1,b^2=a^49,b*a*b^-1=a^41,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^28*c>;`
`// generators/relations`

׿
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