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## G = C23.42D28order 448 = 26·7

### 13rd non-split extension by C23 of D28 acting via D28/D14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C23.42D28
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C23×Dic7 — C23.42D28
 Lower central C7 — C2×C14 — C23.42D28
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C23.42D28
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=b, f2=bcd, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce13 >

Subgroups: 852 in 218 conjugacy classes, 79 normal (19 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C23, C23, C23, C14, C14, C14, C22⋊C4, C22×C4, C22×C4, C24, Dic7, C28, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C2×Dic7, C2×Dic7, C2×C28, C22×C14, C22×C14, C22×C14, C23.34D4, C23.D7, C7×C22⋊C4, C22×Dic7, C22×Dic7, C22×C28, C23×C14, C14.C42, C2×C23.D7, C14×C22⋊C4, C23×Dic7, C23.42D28
Quotients:

Smallest permutation representation of C23.42D28
On 224 points
Generators in S224
```(1 101)(2 183)(3 103)(4 185)(5 105)(6 187)(7 107)(8 189)(9 109)(10 191)(11 111)(12 193)(13 85)(14 195)(15 87)(16 169)(17 89)(18 171)(19 91)(20 173)(21 93)(22 175)(23 95)(24 177)(25 97)(26 179)(27 99)(28 181)(29 192)(30 112)(31 194)(32 86)(33 196)(34 88)(35 170)(36 90)(37 172)(38 92)(39 174)(40 94)(41 176)(42 96)(43 178)(44 98)(45 180)(46 100)(47 182)(48 102)(49 184)(50 104)(51 186)(52 106)(53 188)(54 108)(55 190)(56 110)(57 139)(58 144)(59 113)(60 146)(61 115)(62 148)(63 117)(64 150)(65 119)(66 152)(67 121)(68 154)(69 123)(70 156)(71 125)(72 158)(73 127)(74 160)(75 129)(76 162)(77 131)(78 164)(79 133)(80 166)(81 135)(82 168)(83 137)(84 142)(114 214)(116 216)(118 218)(120 220)(122 222)(124 224)(126 198)(128 200)(130 202)(132 204)(134 206)(136 208)(138 210)(140 212)(141 209)(143 211)(145 213)(147 215)(149 217)(151 219)(153 221)(155 223)(157 197)(159 199)(161 201)(163 203)(165 205)(167 207)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 85)(28 86)(29 178)(30 179)(31 180)(32 181)(33 182)(34 183)(35 184)(36 185)(37 186)(38 187)(39 188)(40 189)(41 190)(42 191)(43 192)(44 193)(45 194)(46 195)(47 196)(48 169)(49 170)(50 171)(51 172)(52 173)(53 174)(54 175)(55 176)(56 177)(57 125)(58 126)(59 127)(60 128)(61 129)(62 130)(63 131)(64 132)(65 133)(66 134)(67 135)(68 136)(69 137)(70 138)(71 139)(72 140)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(81 121)(82 122)(83 123)(84 124)(141 223)(142 224)(143 197)(144 198)(145 199)(146 200)(147 201)(148 202)(149 203)(150 204)(151 205)(152 206)(153 207)(154 208)(155 209)(156 210)(157 211)(158 212)(159 213)(160 214)(161 215)(162 216)(163 217)(164 218)(165 219)(166 220)(167 221)(168 222)
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(57 211)(58 212)(59 213)(60 214)(61 215)(62 216)(63 217)(64 218)(65 219)(66 220)(67 221)(68 222)(69 223)(70 224)(71 197)(72 198)(73 199)(74 200)(75 201)(76 202)(77 203)(78 204)(79 205)(80 206)(81 207)(82 208)(83 209)(84 210)(85 194)(86 195)(87 196)(88 169)(89 170)(90 171)(91 172)(92 173)(93 174)(94 175)(95 176)(96 177)(97 178)(98 179)(99 180)(100 181)(101 182)(102 183)(103 184)(104 185)(105 186)(106 187)(107 188)(108 189)(109 190)(110 191)(111 192)(112 193)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 151)(120 152)(121 153)(122 154)(123 155)(124 156)(125 157)(126 158)(127 159)(128 160)(129 161)(130 162)(131 163)(132 164)(133 165)(134 166)(135 167)(136 168)(137 141)(138 142)(139 143)(140 144)
(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)
(1 160 182 74)(2 199 183 127)(3 158 184 72)(4 197 185 125)(5 156 186 70)(6 223 187 123)(7 154 188 68)(8 221 189 121)(9 152 190 66)(10 219 191 119)(11 150 192 64)(12 217 193 117)(13 148 194 62)(14 215 195 115)(15 146 196 60)(16 213 169 113)(17 144 170 58)(18 211 171 139)(19 142 172 84)(20 209 173 137)(21 168 174 82)(22 207 175 135)(23 166 176 80)(24 205 177 133)(25 164 178 78)(26 203 179 131)(27 162 180 76)(28 201 181 129)(29 118 111 218)(30 63 112 149)(31 116 85 216)(32 61 86 147)(33 114 87 214)(34 59 88 145)(35 140 89 212)(36 57 90 143)(37 138 91 210)(38 83 92 141)(39 136 93 208)(40 81 94 167)(41 134 95 206)(42 79 96 165)(43 132 97 204)(44 77 98 163)(45 130 99 202)(46 75 100 161)(47 128 101 200)(48 73 102 159)(49 126 103 198)(50 71 104 157)(51 124 105 224)(52 69 106 155)(53 122 107 222)(54 67 108 153)(55 120 109 220)(56 65 110 151)```

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

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

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

88 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E ··· 4L 4M 4N 4O 4P 7A 7B 7C 14A ··· 14U 14V ··· 14AG 28A ··· 28X order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 ··· 1 2 2 2 2 4 4 4 4 14 ··· 14 28 28 28 28 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4

88 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C4 D4 D7 C4○D4 D14 D14 C4×D7 D28 C7⋊D4 D4⋊2D7 kernel C23.42D28 C14.C42 C2×C23.D7 C14×C22⋊C4 C23×Dic7 C22×Dic7 C22×C14 C2×C22⋊C4 C2×C14 C22×C4 C24 C23 C23 C23 C22 # reps 1 4 1 1 1 8 4 3 8 6 3 12 12 12 12

Matrix representation of C23.42D28 in GL5(𝔽29)

 1 0 0 0 0 0 1 2 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 28
,
 28 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 28 0 0 0 0 0 28
,
 1 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 1 0 0 0 0 28 28 0 0 0 0 0 22 7 0 0 0 22 3
,
 12 0 0 0 0 0 17 0 0 0 0 12 12 0 0 0 0 0 26 7 0 0 0 11 3

`G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,1,0,0,0,0,2,28,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,1,28,0,0,0,0,28,0,0,0,0,0,22,22,0,0,0,7,3],[12,0,0,0,0,0,17,12,0,0,0,0,12,0,0,0,0,0,26,11,0,0,0,7,3] >;`

C23.42D28 in GAP, Magma, Sage, TeX

`C_2^3._{42}D_{28}`
`% in TeX`

`G:=Group("C2^3.42D28");`
`// GroupNames label`

`G:=SmallGroup(448,477);`
`// by ID`

`G=gap.SmallGroup(448,477);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,422,387,58,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=b,f^2=b*c*d,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^13>;`
`// generators/relations`

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