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G = C24.21D14  order 448 = 26·7

21st non-split extension by C24 of D14 acting via D14/C7=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.21D14, (C2×C28)⋊23D4, (C2×Dic7)⋊7D4, (C22×D7)⋊5D4, (C22×D4)⋊2D7, (C22×C14)⋊8D4, C23⋊3(C7⋊D4), C7⋊3(C23⋊2D4), C14.68C22≀C2, C22.284(D4×D7), C2.27(C28⋊D4), C14.37(C4⋊1D4), C2.26(C28⋊2D4), (C22×C4).153D14, C2.7(C24⋊D7), C2.35(C23⋊D14), C14.131(C4⋊D4), C14.C42⋊46C2, (C23×C14).49C22, (C23×D7).25C22, C23.385(C22×D7), C2.35(Dic7⋊D4), (C22×C14).368C23, (C22×C28).396C22, C22.107(D4⋊2D7), (C22×Dic7).69C22, (D4×C2×C14)⋊12C2, (C2×C4)⋊5(C7⋊D4), (C2×D14⋊C4)⋊39C2, (C22×C7⋊D4)⋊2C2, (C2×C14).557(C2×D4), (C2×C23.D7)⋊12C2, C22.219(C2×C7⋊D4), (C2×C14).164(C4○D4), SmallGroup(448,757)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C14 — C24.21D14
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C23×D7 — C22×C7⋊D4 — C24.21D14
Lower central
C7 — C22×C14 — C24.21D14
Upper central
C1 — C23 — C22×D4

Generators and relations for C24.21D14
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=1, f2=c, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >

Subgroups: 1588 in 322 conjugacy classes, 69 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C14, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C22×D4, C22×D4, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C23⋊2D4, D14⋊C4, C23.D7, C22×Dic7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C23×C14, C14.C42, C2×D14⋊C4, C2×C23.D7, C22×C7⋊D4, D4×C2×C14, C24.21D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22≀C2, C4⋊D4, C4⋊1D4, C7⋊D4, C22×D7, C23⋊2D4, D4×D7, D4⋊2D7, C2×C7⋊D4, C23⋊D14, C28⋊2D4, Dic7⋊D4, C28⋊D4, C24⋊D7, C24.21D14

Smallest permutation representation of C24.21D14
►On 224 points
Generators in S224
(2 212)(4 214)(6 216)(8 218)(10 220)(12 222)(14 224)(15 85)(16 166)(17 87)(18 168)(19 89)(20 156)(21 91)(22 158)(23 93)(24 160)(25 95)(26 162)(27 97)(28 164)(30 181)(32 169)(34 171)(36 173)(38 175)(40 177)(42 179)(43 76)(44 104)(45 78)(46 106)(47 80)(48 108)(49 82)(50 110)(51 84)(52 112)(53 72)(54 100)(55 74)(56 102)(57 135)(59 137)(61 139)(63 127)(65 129)(67 131)(69 133)(71 201)(73 203)(75 205)(77 207)(79 209)(81 197)(83 199)(86 113)(88 115)(90 117)(92 119)(94 121)(96 123)(98 125)(99 202)(101 204)(103 206)(105 208)(107 210)(109 198)(111 200)(114 167)(116 155)(118 157)(120 159)(122 161)(124 163)(126 165)(141 196)(143 184)(145 186)(147 188)(149 190)(151 192)(153 194)

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

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

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

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

82 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C···4H7A7B7C14A···14U14V···14AS28A···28L
order12···2222222444···477714···1414···1428···28
size11···1444428284428···282222···24···44···4

82 irreducible representations

dim111111222222222244
type++++++++++++++-
imageC1C2C2C2C2C2D4D4D4D4D7C4○D4D14D14C7⋊D4C7⋊D4D4×D7D4⋊2D7
kernelC24.21D14C14.C42C2×D14⋊C4C2×C23.D7C22×C7⋊D4D4×C2×C14C2×Dic7C2×C28C22×D7C22×C14C22×D4C2×C14C22×C4C24C2×C4C23C22C22
# reps11122142243236122493

Matrix representation of C24.21D14 ►in GL6(𝔽29)

100000
19280000
001000
0002800
000010
00002128
,
2800000
0280000
0028000
0002800
0000280
0000028
,
100000
010000
0028000
0002800
0000280
0000028
,
100000
010000
001000
000100
0000280
0000028
,
2300000
5240000
005000
0002300
00002528
0000154
,
24280000
2450000
000600
0024000
0000120
00002017

G:=sub<GL(6,GF(29))| [1,19,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,21,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[23,5,0,0,0,0,0,24,0,0,0,0,0,0,5,0,0,0,0,0,0,23,0,0,0,0,0,0,25,15,0,0,0,0,28,4],[24,24,0,0,0,0,28,5,0,0,0,0,0,0,0,24,0,0,0,0,6,0,0,0,0,0,0,0,12,20,0,0,0,0,0,17] >;
 

C24.21D14 in GAP, Magma, Sage, TeX 

C_2^4._{21}D_{14}
 
% in TeX
 
G:=Group("C2^4.21D14");
 
// GroupNames label
 
G:=SmallGroup(448,757);
 
// by ID
 
G=gap.SmallGroup(448,757);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,254,387,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^14=1,f^2=c,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*b*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=c*e^-1>;
 
// generators/relations
 

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