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G = C24.7D14  order 448 = 26·7

7th non-split extension by C24 of D14 acting via D14/C7=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.7D14, C23.5Dic14, (C2×C28).50D4, C2.5(C28⋊D4), (C2×Dic7).55D4, (C22×C4).29D14, C22.239(D4×D7), (C22×C14).13Q8, C14.11(C4⋊1D4), C7⋊2(C23.4Q8), C14.57(C22⋊Q8), C14.C42⋊29C2, C2.8(C28.48D4), C22.96(C4○D28), (C22×C28).58C22, (C23×C14).32C22, C22.46(C2×Dic14), C23.368(C22×D7), C2.20(D14.D4), C22.94(D4⋊2D7), (C22×C14).324C23, C2.22(C22⋊Dic14), C14.29(C22.D4), C2.6(C23.18D14), (C22×Dic7).40C22, (C2×C4⋊Dic7)⋊9C2, (C2×Dic7⋊C4)⋊9C2, (C2×C14).34(C2×Q8), (C2×C14).318(C2×D4), (C2×C4).29(C7⋊D4), (C2×C22⋊C4).12D7, (C2×C14).78(C4○D4), (C14×C22⋊C4).14C2, C22.124(C2×C7⋊D4), (C2×C23.D7).11C2, SmallGroup(448,483)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C14 — C24.7D14
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C2×C4⋊Dic7 — C24.7D14
Lower central
C7 — C22×C14 — C24.7D14
Upper central
C1 — C23 — C2×C22⋊C4

Generators and relations for C24.7D14
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=dc=cd, f2=c, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde13 >

Subgroups: 756 in 186 conjugacy classes, 63 normal (27 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic7, C28, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C23.4Q8, Dic7⋊C4, C4⋊Dic7, C23.D7, C7×C22⋊C4, C22×Dic7, C22×C28, C23×C14, C14.C42, C2×Dic7⋊C4, C2×C4⋊Dic7, C2×C23.D7, C14×C22⋊C4, C24.7D14
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C22.D4, C4⋊1D4, Dic14, C7⋊D4, C22×D7, C23.4Q8, C2×Dic14, C4○D28, D4×D7, D4⋊2D7, C2×C7⋊D4, C22⋊Dic14, D14.D4, C28.48D4, C23.18D14, C28⋊D4, C24.7D14

Smallest permutation representation of C24.7D14
►On 224 points
Generators in S224
(2 66)(4 68)(6 70)(8 72)(10 74)(12 76)(14 78)(16 80)(18 82)(20 84)(22 58)(24 60)(26 62)(28 64)(29 193)(30 102)(31 195)(32 104)(33 169)(34 106)(35 171)(36 108)(37 173)(38 110)(39 175)(40 112)(41 177)(42 86)(43 179)(44 88)(45 181)(46 90)(47 183)(48 92)(49 185)(50 94)(51 187)(52 96)(53 189)(54 98)(55 191)(56 100)(85 153)(87 155)(89 157)(91 159)(93 161)(95 163)(97 165)(99 167)(101 141)(103 143)(105 145)(107 147)(109 149)(111 151)(114 212)(116 214)(118 216)(120 218)(122 220)(124 222)(126 224)(128 198)(130 200)(132 202)(134 204)(136 206)(138 208)(140 210)(142 194)(144 196)(146 170)(148 172)(150 174)(152 176)(154 178)(156 180)(158 182)(160 184)(162 186)(164 188)(166 190)(168 192)

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

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

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

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

82 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E···4L7A7B7C14A···14U14V···14AG28A···28X
order12···22244444···477714···1414···1428···28
size11···144444428···282222···24···44···4

82 irreducible representations

dim111111222222222244
type++++++++-+++-+-
imageC1C2C2C2C2C2D4D4Q8D7C4○D4D14D14C7⋊D4Dic14C4○D28D4×D7D4⋊2D7
kernelC24.7D14C14.C42C2×Dic7⋊C4C2×C4⋊Dic7C2×C23.D7C14×C22⋊C4C2×Dic7C2×C28C22×C14C2×C22⋊C4C2×C14C22×C4C24C2×C4C23C22C22C22
# reps112121422366312121266

Matrix representation of C24.7D14 ►in GL6(𝔽29)

100000
010000
001000
00112800
000010
0000128
,
100000
010000
0028000
0002800
0000280
0000028
,
2800000
0280000
001000
000100
000010
000001
,
100000
010000
001000
000100
0000280
0000028
,
10170000
1670000
001000
000100
0000127
0000128
,
5170000
7240000
0016500
0071300
0000280
0000281

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,11,0,0,0,0,0,28,0,0,0,0,0,0,1,1,0,0,0,0,0,28],[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],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[10,16,0,0,0,0,17,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,27,28],[5,7,0,0,0,0,17,24,0,0,0,0,0,0,16,7,0,0,0,0,5,13,0,0,0,0,0,0,28,28,0,0,0,0,0,1] >;
 

C24.7D14 in GAP, Magma, Sage, TeX 

C_2^4._7D_{14}
 
% in TeX
 
G:=Group("C2^4.7D14");
 
// GroupNames label
 
G:=SmallGroup(448,483);
 
// by ID
 
G=gap.SmallGroup(448,483);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,701,344,254,387,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=d*c=c*d,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*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^13>;
 
// generators/relations
 

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