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## G = C56.31C23order 448 = 26·7

### 24th non-split extension by C56 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — C56.31C23
 Chief series C1 — C7 — C14 — C28 — C56 — Dic28 — C2×Dic28 — C56.31C23
 Lower central C7 — C14 — C28 — C56 — C56.31C23
 Upper central C1 — C2 — C2×C4 — C2×C8 — C4○D8

Generators and relations for C56.31C23
G = < a,b,c,d | a56=c2=d2=1, b2=a28, bab-1=a-1, ac=ca, dad=a15, bc=cb, dbd=a49b, dcd=a28c >

Subgroups: 372 in 82 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C14, C14, C16, C2×C8, D8, SD16, Q16, Q16, C2×Q8, C4○D4, Dic7, C28, C28, C2×C14, C2×C14, M5(2), SD32, Q32, C2×Q16, C4○D8, C56, Dic14, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×Q8, Q32⋊C2, C7⋊C16, Dic28, Dic28, C2×C56, C7×D8, C7×SD16, C7×Q16, C2×Dic14, C7×C4○D4, C28.C8, D8.D7, C7⋊Q32, C2×Dic28, C7×C4○D8, C56.31C23
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, C7⋊D4, C22×D7, Q32⋊C2, D4⋊D7, C2×C7⋊D4, C2×D4⋊D7, C56.31C23

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

58 conjugacy classes

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

58 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 D4 D4 D7 D8 D8 D14 D14 D14 C7⋊D4 C7⋊D4 Q32⋊C2 D4⋊D7 D4⋊D7 C56.31C23 kernel C56.31C23 C28.C8 D8.D7 C7⋊Q32 C2×Dic28 C7×C4○D8 C56 C2×C28 C4○D8 C28 C2×C14 C2×C8 D8 Q16 C8 C2×C4 C7 C4 C22 C1 # reps 1 1 2 2 1 1 1 1 3 2 2 3 3 3 6 6 2 3 3 12

Matrix representation of C56.31C23 in GL4(𝔽113) generated by

 22 44 0 0 69 61 0 0 0 0 91 7 0 0 106 33
,
 6 107 0 0 25 107 0 0 0 0 23 56 0 0 43 90
,
 1 0 0 0 0 1 0 0 0 0 112 0 0 0 0 112
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
`G:=sub<GL(4,GF(113))| [22,69,0,0,44,61,0,0,0,0,91,106,0,0,7,33],[6,25,0,0,107,107,0,0,0,0,23,43,0,0,56,90],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;`

C56.31C23 in GAP, Magma, Sage, TeX

`C_{56}._{31}C_2^3`
`% in TeX`

`G:=Group("C56.31C2^3");`
`// GroupNames label`

`G:=SmallGroup(448,729);`
`// by ID`

`G=gap.SmallGroup(448,729);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,254,387,675,185,192,1684,438,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^56=c^2=d^2=1,b^2=a^28,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^15,b*c=c*b,d*b*d=a^49*b,d*c*d=a^28*c>;`
`// generators/relations`

׿
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