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G = C4⋊D4⋊D7  order 448 = 26·7

4th semidirect product of C4⋊D4 and D7 acting via D7/C7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C7⋊C8⋊4D4, C4⋊D4⋊4D7, C7⋊3(C8⋊2D4), C4⋊C4.59D14, (C2×C28).72D4, C4.171(D4×D7), C28⋊7D4⋊24C2, (C2×D4).39D14, C28.148(C2×D4), C14.D8⋊36C2, D4⋊Dic7⋊16C2, C4.Dic14⋊35C2, (C22×C14).85D4, C28.184(C4○D4), C4.60(D4⋊2D7), C14.94(C4⋊D4), C14.91(C8⋊C22), (C2×C28).358C23, (D4×C14).55C22, (C2×D28).98C22, (C22×C4).121D14, C23.24(C7⋊D4), C2.13(D4⋊D14), C4⋊Dic7.143C22, C2.15(Dic7⋊D4), C2.12(D4.D14), (C22×C28).162C22, (C2×D4⋊D7)⋊11C2, (C7×C4⋊D4)⋊4C2, (C2×C14).489(C2×D4), (C2×C4).50(C7⋊D4), (C2×C7⋊C8).109C22, (C2×C4.Dic7)⋊11C2, (C7×C4⋊C4).106C22, (C2×C4).458(C22×D7), C22.164(C2×C7⋊D4), SmallGroup(448,573)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C28 — C4⋊D4⋊D7
Chief series
C1 — C7 — C14 — C28 — C2×C28 — C2×D28 — C28⋊7D4 — C4⋊D4⋊D7
Lower central
C7 — C14 — C2×C28 — C4⋊D4⋊D7
Upper central
C1 — C22 — C22×C4 — C4⋊D4

Generators and relations for C4⋊D4⋊D7
 G = < a,b,c,d,e | a4=b4=c2=d7=e2=1, bab-1=cac=eae=a-1, ad=da, cbc=b-1, bd=db, ebe=ab-1, cd=dc, ece=a-1c, ede=d-1 >

Subgroups: 684 in 130 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), D8, C22×C4, C2×D4, C2×D4, Dic7, C28, C28, D14, C2×C14, C2×C14, D4⋊C4, C4.Q8, C4⋊D4, C4⋊D4, C2×M4(2), C2×D8, C7⋊C8, C7⋊C8, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C22×C14, C8⋊2D4, C2×C7⋊C8, C4.Dic7, C4⋊Dic7, D14⋊C4, D4⋊D7, C7×C22⋊C4, C7×C4⋊C4, C2×D28, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, C4.Dic14, C14.D8, D4⋊Dic7, C2×C4.Dic7, C28⋊7D4, C2×D4⋊D7, C7×C4⋊D4, C4⋊D4⋊D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C8⋊C22, C7⋊D4, C22×D7, C8⋊2D4, D4×D7, D4⋊2D7, C2×C7⋊D4, D4.D14, Dic7⋊D4, D4⋊D14, C4⋊D4⋊D7

Smallest permutation representation of C4⋊D4⋊D7
►On 224 points
Generators in S224
(1 41 13 34)(2 42 14 35)(3 36 8 29)(4 37 9 30)(5 38 10 31)(6 39 11 32)(7 40 12 33)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)(57 85 64 92)(58 86 65 93)(59 87 66 94)(60 88 67 95)(61 89 68 96)(62 90 69 97)(63 91 70 98)(71 99 78 106)(72 100 79 107)(73 101 80 108)(74 102 81 109)(75 103 82 110)(76 104 83 111)(77 105 84 112)(113 141 120 148)(114 142 121 149)(115 143 122 150)(116 144 123 151)(117 145 124 152)(118 146 125 153)(119 147 126 154)(127 155 134 162)(128 156 135 163)(129 157 136 164)(130 158 137 165)(131 159 138 166)(132 160 139 167)(133 161 140 168)(169 204 176 197)(170 205 177 198)(171 206 178 199)(172 207 179 200)(173 208 180 201)(174 209 181 202)(175 210 182 203)(183 218 190 211)(184 219 191 212)(185 220 192 213)(186 221 193 214)(187 222 194 215)(188 223 195 216)(189 224 196 217)

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

(1 62)(2 63)(3 57)(4 58)(5 59)(6 60)(7 61)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(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)(141 176)(142 177)(143 178)(144 179)(145 180)(146 181)(147 182)(148 169)(149 170)(150 171)(151 172)(152 173)(153 174)(154 175)(155 190)(156 191)(157 192)(158 193)(159 194)(160 195)(161 196)(162 183)(163 184)(164 185)(165 186)(166 187)(167 188)(168 189)

(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)

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

58 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E7A7B7C8A8B8C8D14A···14I14J···14O14P···14U28A···28L28M···28R
order122222244444777888814···1414···1414···1428···2828···28
size11114856224856222282828282···24···48···84···48···8

58 irreducible representations

dim11111111222222222244444
type+++++++++++++++++-+
imageC1C2C2C2C2C2C2C2D4D4D4D7C4○D4D14D14D14C7⋊D4C7⋊D4C8⋊C22D4×D7D4⋊2D7D4.D14D4⋊D14
kernelC4⋊D4⋊D7C4.Dic14C14.D8D4⋊Dic7C2×C4.Dic7C28⋊7D4C2×D4⋊D7C7×C4⋊D4C7⋊C8C2×C28C22×C14C4⋊D4C28C4⋊C4C22×C4C2×D4C2×C4C23C14C4C4C2C2
# reps11111111211323336623366

Matrix representation of C4⋊D4⋊D7 ►in GL8(𝔽113)

10000000
01000000
00100000
00010000
000015000
0000239800
0000950980
000000015
,
00100000
00010000
1120000000
0112000000
000013800
0000011200
000010445098
00002242150
,
00100000
00010000
10000000
01000000
000013800
0000011200
00004545015
00001102980
,
01000000
11224000000
00010000
00112240000
0000106000
0000010600
000038106160
0000640016
,
0001120000
0011200000
0112000000
1120000000
00007191400
00003661067
000096818966
000051410

G:=sub<GL(8,GF(113))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,15,23,95,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,15],[0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,104,22,0,0,0,0,38,112,45,42,0,0,0,0,0,0,0,15,0,0,0,0,0,0,98,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,45,110,0,0,0,0,38,112,45,2,0,0,0,0,0,0,0,98,0,0,0,0,0,0,15,0],[0,112,0,0,0,0,0,0,1,24,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,24,0,0,0,0,0,0,0,0,106,0,38,64,0,0,0,0,0,106,106,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,0,71,3,96,5,0,0,0,0,91,66,81,1,0,0,0,0,40,106,89,41,0,0,0,0,0,7,66,0] >;
 

C4⋊D4⋊D7 in GAP, Magma, Sage, TeX 

C_4\rtimes D_4\rtimes D_7
 
% in TeX
 
G:=Group("C4:D4:D7");
 
// GroupNames label
 
G:=SmallGroup(448,573);
 
// by ID
 
G=gap.SmallGroup(448,573);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,254,555,1123,297,136,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^7=e^2=1,b*a*b^-1=c*a*c=e*a*e=a^-1,a*d=d*a,c*b*c=b^-1,b*d=d*b,e*b*e=a*b^-1,c*d=d*c,e*c*e=a^-1*c,e*d*e=d^-1>;
 
// generators/relations
 

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