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G = C24.14D14  order 448 = 26·7

14th non-split extension by C24 of D14 acting via D14/C7=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.14D14, (C2×Dic7)⋊4D4, C14.39C22≀C2, (C22×D7).31D4, (C22×C14).69D4, C22.241(D4×D7), (C22×C4).34D14, C2.7(C23⋊D14), C14.32(C4⋊D4), C23.18(C7⋊D4), C7⋊2(C23.10D4), C2.33(D14⋊D4), C14.C42⋊17C2, C14.35(C4.4D4), C22.99(C4○D28), (C22×C28).27C22, (C23×C14).42C22, (C23×D7).15C22, C23.371(C22×D7), C2.10(Dic7⋊D4), C2.22(D14.D4), C22.97(D4⋊2D7), (C22×C14).333C23, C2.22(Dic7.D4), C2.7(C23.23D14), C14.34(C22.D4), (C22×Dic7).45C22, (C2×D14⋊C4)⋊7C2, (C2×C22⋊C4)⋊6D7, (C14×C22⋊C4)⋊4C2, (C2×C23.D7)⋊5C2, (C2×Dic7⋊C4)⋊12C2, (C2×C14).434(C2×D4), (C22×C7⋊D4).5C2, C22.127(C2×C7⋊D4), (C2×C14).148(C4○D4), SmallGroup(448,493)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C14 — C24.14D14
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C23×D7 — C22×C7⋊D4 — C24.14D14
Lower central
C7 — C22×C14 — C24.14D14
Upper central
C1 — C23 — C2×C22⋊C4

Generators and relations for C24.14D14
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=c, f2=cb=bc, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be13 >

Subgroups: 1268 in 238 conjugacy classes, 61 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, D4, C23, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic7, C28, D14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C23.10D4, Dic7⋊C4, D14⋊C4, C23.D7, C7×C22⋊C4, C22×Dic7, C2×C7⋊D4, C22×C28, C23×D7, C23×C14, C14.C42, C2×Dic7⋊C4, C2×D14⋊C4, C2×C23.D7, C14×C22⋊C4, C22×C7⋊D4, C24.14D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C7⋊D4, C22×D7, C23.10D4, C4○D28, D4×D7, D4⋊2D7, C2×C7⋊D4, D14.D4, D14⋊D4, Dic7.D4, C23.23D14, C23⋊D14, Dic7⋊D4, C24.14D14

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

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

(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)

(1 167)(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 68)(30 69)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)(45 84)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(85 185)(86 186)(87 187)(88 188)(89 189)(90 190)(91 191)(92 192)(93 193)(94 194)(95 195)(96 196)(97 169)(98 170)(99 171)(100 172)(101 173)(102 174)(103 175)(104 176)(105 177)(106 178)(107 179)(108 180)(109 181)(110 182)(111 183)(112 184)(113 214)(114 215)(115 216)(116 217)(117 218)(118 219)(119 220)(120 221)(121 222)(122 223)(123 224)(124 197)(125 198)(126 199)(127 200)(128 201)(129 202)(130 203)(131 204)(132 205)(133 206)(134 207)(135 208)(136 209)(137 210)(138 211)(139 212)(140 213)

(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)

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

82 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4J7A7B7C14A···14U14V···14AG28A···28X
order12···2222244444···477714···1414···1428···28
size11···1442828444428···282222···24···44···4

82 irreducible representations

dim111111122222222244
type++++++++++++++-
imageC1C2C2C2C2C2C2D4D4D4D7C4○D4D14D14C7⋊D4C4○D28D4×D7D4⋊2D7
kernelC24.14D14C14.C42C2×Dic7⋊C4C2×D14⋊C4C2×C23.D7C14×C22⋊C4C22×C7⋊D4C2×Dic7C22×D7C22×C14C2×C22⋊C4C2×C14C22×C4C24C23C22C22C22
# reps11121114223663122493

Matrix representation of C24.14D14 ►in GL6(𝔽29)

0160000
2000000
0028000
0002800
000001
000010
,
100000
010000
0028000
0002800
0000280
0000028
,
2800000
0280000
001000
000100
0000280
0000028
,
100000
010000
001000
000100
0000280
0000028
,
1200000
0120000
001700
00162400
000001
0000280
,
1200000
0170000
00272600
0021200
000001
000010

G:=sub<GL(6,GF(29))| [0,20,0,0,0,0,16,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[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],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,16,0,0,0,0,7,24,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[12,0,0,0,0,0,0,17,0,0,0,0,0,0,27,21,0,0,0,0,26,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

C24.14D14 in GAP, Magma, Sage, TeX 

C_2^4._{14}D_{14}
 
% in TeX
 
G:=Group("C2^4.14D14");
 
// GroupNames label
 
G:=SmallGroup(448,493);
 
// by ID
 
G=gap.SmallGroup(448,493);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,254,387,100,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=c,f^2=c*b=b*c,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,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=b*e^13>;
 
// generators/relations
 

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