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## G = C24.12D14order 448 = 26·7

### 12nd non-split extension by C24 of D14 acting via D14/C7=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C24.12D14
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C23×D7 — C22×C7⋊D4 — C24.12D14
 Lower central C7 — C2×C14 — C24.12D14
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.12D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=f2=b, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >

Subgroups: 1428 in 286 conjugacy classes, 77 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C22×D4, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C23.23D4, D14⋊C4, C23.D7, C7×C22⋊C4, C2×C4×D7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, C23×D7, C23×C14, C14.C42, C2×D14⋊C4, C2×C23.D7, C14×C22⋊C4, D7×C22×C4, C22×C7⋊D4, C24.12D14
Quotients:

Smallest permutation representation of C24.12D14
On 224 points
Generators in S224
```(1 194)(2 62)(3 196)(4 64)(5 170)(6 66)(7 172)(8 68)(9 174)(10 70)(11 176)(12 72)(13 178)(14 74)(15 180)(16 76)(17 182)(18 78)(19 184)(20 80)(21 186)(22 82)(23 188)(24 84)(25 190)(26 58)(27 192)(28 60)(29 105)(30 198)(31 107)(32 200)(33 109)(34 202)(35 111)(36 204)(37 85)(38 206)(39 87)(40 208)(41 89)(42 210)(43 91)(44 212)(45 93)(46 214)(47 95)(48 216)(49 97)(50 218)(51 99)(52 220)(53 101)(54 222)(55 103)(56 224)(57 163)(59 165)(61 167)(63 141)(65 143)(67 145)(69 147)(71 149)(73 151)(75 153)(77 155)(79 157)(81 159)(83 161)(86 121)(88 123)(90 125)(92 127)(94 129)(96 131)(98 133)(100 135)(102 137)(104 139)(106 113)(108 115)(110 117)(112 119)(114 199)(116 201)(118 203)(120 205)(122 207)(124 209)(126 211)(128 213)(130 215)(132 217)(134 219)(136 221)(138 223)(140 197)(142 169)(144 171)(146 173)(148 175)(150 177)(152 179)(154 181)(156 183)(158 185)(160 187)(162 189)(164 191)(166 193)(168 195)
(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)
(1 222)(2 223)(3 224)(4 197)(5 198)(6 199)(7 200)(8 201)(9 202)(10 203)(11 204)(12 205)(13 206)(14 207)(15 208)(16 209)(17 210)(18 211)(19 212)(20 213)(21 214)(22 215)(23 216)(24 217)(25 218)(26 219)(27 220)(28 221)(29 169)(30 170)(31 171)(32 172)(33 173)(34 174)(35 175)(36 176)(37 177)(38 178)(39 179)(40 180)(41 181)(42 182)(43 183)(44 184)(45 185)(46 186)(47 187)(48 188)(49 189)(50 190)(51 191)(52 192)(53 193)(54 194)(55 195)(56 196)(57 133)(58 134)(59 135)(60 136)(61 137)(62 138)(63 139)(64 140)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)(81 129)(82 130)(83 131)(84 132)(85 150)(86 151)(87 152)(88 153)(89 154)(90 155)(91 156)(92 157)(93 158)(94 159)(95 160)(96 161)(97 162)(98 163)(99 164)(100 165)(101 166)(102 167)(103 168)(104 141)(105 142)(106 143)(107 144)(108 145)(109 146)(110 147)(111 148)(112 149)
(1 167)(2 168)(3 141)(4 142)(5 143)(6 144)(7 145)(8 146)(9 147)(10 148)(11 149)(12 150)(13 151)(14 152)(15 153)(16 154)(17 155)(18 156)(19 157)(20 158)(21 159)(22 160)(23 161)(24 162)(25 163)(26 164)(27 165)(28 166)(29 140)(30 113)(31 114)(32 115)(33 116)(34 117)(35 118)(36 119)(37 120)(38 121)(39 122)(40 123)(41 124)(42 125)(43 126)(44 127)(45 128)(46 129)(47 130)(48 131)(49 132)(50 133)(51 134)(52 135)(53 136)(54 137)(55 138)(56 139)(57 190)(58 191)(59 192)(60 193)(61 194)(62 195)(63 196)(64 169)(65 170)(66 171)(67 172)(68 173)(69 174)(70 175)(71 176)(72 177)(73 178)(74 179)(75 180)(76 181)(77 182)(78 183)(79 184)(80 185)(81 186)(82 187)(83 188)(84 189)(85 205)(86 206)(87 207)(88 208)(89 209)(90 210)(91 211)(92 212)(93 213)(94 214)(95 215)(96 216)(97 217)(98 218)(99 219)(100 220)(101 221)(102 222)(103 223)(104 224)(105 197)(106 198)(107 199)(108 200)(109 201)(110 202)(111 203)(112 204)
(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)
(1 166 15 152)(2 151 16 165)(3 164 17 150)(4 149 18 163)(5 162 19 148)(6 147 20 161)(7 160 21 146)(8 145 22 159)(9 158 23 144)(10 143 24 157)(11 156 25 142)(12 141 26 155)(13 154 27 168)(14 167 28 153)(29 71 43 57)(30 84 44 70)(31 69 45 83)(32 82 46 68)(33 67 47 81)(34 80 48 66)(35 65 49 79)(36 78 50 64)(37 63 51 77)(38 76 52 62)(39 61 53 75)(40 74 54 60)(41 59 55 73)(42 72 56 58)(85 224 99 210)(86 209 100 223)(87 222 101 208)(88 207 102 221)(89 220 103 206)(90 205 104 219)(91 218 105 204)(92 203 106 217)(93 216 107 202)(94 201 108 215)(95 214 109 200)(96 199 110 213)(97 212 111 198)(98 197 112 211)(113 189 127 175)(114 174 128 188)(115 187 129 173)(116 172 130 186)(117 185 131 171)(118 170 132 184)(119 183 133 169)(120 196 134 182)(121 181 135 195)(122 194 136 180)(123 179 137 193)(124 192 138 178)(125 177 139 191)(126 190 140 176)```

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

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

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

88 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 7A 7B 7C 14A ··· 14U 14V ··· 14AG 28A ··· 28X order 1 2 ··· 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 ··· 1 4 4 14 14 14 14 2 2 2 2 4 4 14 14 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 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C4 D4 D4 D4 D7 C4○D4 D14 D14 C7⋊D4 C4×D7 C4○D28 D4×D7 D4⋊2D7 kernel C24.12D14 C14.C42 C2×D14⋊C4 C2×C23.D7 C14×C22⋊C4 D7×C22×C4 C22×C7⋊D4 C2×C7⋊D4 C2×Dic7 C2×C28 C22×D7 C2×C22⋊C4 C2×C14 C22×C4 C24 C2×C4 C23 C22 C22 C22 # reps 1 2 1 1 1 1 1 8 2 2 4 3 4 6 3 12 12 12 9 3

Matrix representation of C24.12D14 in GL5(𝔽29)

 28 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 24 14 0 0 0 19 5
,
 28 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 28 0 0 0 0 0 28 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 17 0 0 0 0 0 12 0 0 0 0 0 0 19 0 0 0 3 26
,
 12 0 0 0 0 0 17 0 0 0 0 0 17 0 0 0 0 0 3 19 0 0 0 24 26

`G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,24,19,0,0,0,14,5],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[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],[12,0,0,0,0,0,17,0,0,0,0,0,12,0,0,0,0,0,0,3,0,0,0,19,26],[12,0,0,0,0,0,17,0,0,0,0,0,17,0,0,0,0,0,3,24,0,0,0,19,26] >;`

C24.12D14 in GAP, Magma, Sage, TeX

`C_2^4._{12}D_{14}`
`% in TeX`

`G:=Group("C2^4.12D14");`
`// GroupNames label`

`G:=SmallGroup(448,490);`
`// by ID`

`G=gap.SmallGroup(448,490);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,387,58,18822]);`
`// Polycyclic`

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

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