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## G = C22×D4⋊2D7order 448 = 26·7

### Direct product of C22 and D4⋊2D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C22×D4⋊2D7
 Chief series C1 — C7 — C14 — D14 — C22×D7 — C23×D7 — D7×C22×C4 — C22×D4⋊2D7
 Lower central C7 — C14 — C22×D4⋊2D7
 Upper central C1 — C23 — C22×D4

Generators and relations for C22×D42D7
G = < a,b,c,d,e,f | a2=b2=c4=d2=e7=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, fdf=c2d, fef=e-1 >

Subgroups: 3092 in 890 conjugacy classes, 463 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, D7, C14, C14, C14, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, Dic14, C4×D7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C22×C4○D4, C2×Dic14, C2×C4×D7, D42D7, C22×Dic7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C23×C14, C22×Dic14, D7×C22×C4, C2×D42D7, C23×Dic7, C22×C7⋊D4, D4×C2×C14, C22×D42D7
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, C25, C22×D7, C22×C4○D4, D42D7, C23×D7, C2×D42D7, D7×C24, C22×D42D7

Smallest permutation representation of C22×D42D7
On 224 points
Generators in S224
(1 120)(2 121)(3 122)(4 123)(5 124)(6 125)(7 126)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 134)(16 135)(17 136)(18 137)(19 138)(20 139)(21 140)(22 127)(23 128)(24 129)(25 130)(26 131)(27 132)(28 133)(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)(43 162)(44 163)(45 164)(46 165)(47 166)(48 167)(49 168)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 161)(57 176)(58 177)(59 178)(60 179)(61 180)(62 181)(63 182)(64 169)(65 170)(66 171)(67 172)(68 173)(69 174)(70 175)(71 190)(72 191)(73 192)(74 193)(75 194)(76 195)(77 196)(78 183)(79 184)(80 185)(81 186)(82 187)(83 188)(84 189)(85 204)(86 205)(87 206)(88 207)(89 208)(90 209)(91 210)(92 197)(93 198)(94 199)(95 200)(96 201)(97 202)(98 203)(99 218)(100 219)(101 220)(102 221)(103 222)(104 223)(105 224)(106 211)(107 212)(108 213)(109 214)(110 215)(111 216)(112 217)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 105)(113 176)(114 177)(115 178)(116 179)(117 180)(118 181)(119 182)(120 169)(121 170)(122 171)(123 172)(124 173)(125 174)(126 175)(127 190)(128 191)(129 192)(130 193)(131 194)(132 195)(133 196)(134 183)(135 184)(136 185)(137 186)(138 187)(139 188)(140 189)(141 204)(142 205)(143 206)(144 207)(145 208)(146 209)(147 210)(148 197)(149 198)(150 199)(151 200)(152 201)(153 202)(154 203)(155 218)(156 219)(157 220)(158 221)(159 222)(160 223)(161 224)(162 211)(163 212)(164 213)(165 214)(166 215)(167 216)(168 217)
(1 148 8 141)(2 149 9 142)(3 150 10 143)(4 151 11 144)(5 152 12 145)(6 153 13 146)(7 154 14 147)(15 162 22 155)(16 163 23 156)(17 164 24 157)(18 165 25 158)(19 166 26 159)(20 167 27 160)(21 168 28 161)(29 113 36 120)(30 114 37 121)(31 115 38 122)(32 116 39 123)(33 117 40 124)(34 118 41 125)(35 119 42 126)(43 127 50 134)(44 128 51 135)(45 129 52 136)(46 130 53 137)(47 131 54 138)(48 132 55 139)(49 133 56 140)(57 204 64 197)(58 205 65 198)(59 206 66 199)(60 207 67 200)(61 208 68 201)(62 209 69 202)(63 210 70 203)(71 218 78 211)(72 219 79 212)(73 220 80 213)(74 221 81 214)(75 222 82 215)(76 223 83 216)(77 224 84 217)(85 169 92 176)(86 170 93 177)(87 171 94 178)(88 172 95 179)(89 173 96 180)(90 174 97 181)(91 175 98 182)(99 183 106 190)(100 184 107 191)(101 185 108 192)(102 186 109 193)(103 187 110 194)(104 188 111 195)(105 189 112 196)
(1 155)(2 156)(3 157)(4 158)(5 159)(6 160)(7 161)(8 162)(9 163)(10 164)(11 165)(12 166)(13 167)(14 168)(15 141)(16 142)(17 143)(18 144)(19 145)(20 146)(21 147)(22 148)(23 149)(24 150)(25 151)(26 152)(27 153)(28 154)(29 127)(30 128)(31 129)(32 130)(33 131)(34 132)(35 133)(36 134)(37 135)(38 136)(39 137)(40 138)(41 139)(42 140)(43 113)(44 114)(45 115)(46 116)(47 117)(48 118)(49 119)(50 120)(51 121)(52 122)(53 123)(54 124)(55 125)(56 126)(57 211)(58 212)(59 213)(60 214)(61 215)(62 216)(63 217)(64 218)(65 219)(66 220)(67 221)(68 222)(69 223)(70 224)(71 197)(72 198)(73 199)(74 200)(75 201)(76 202)(77 203)(78 204)(79 205)(80 206)(81 207)(82 208)(83 209)(84 210)(85 183)(86 184)(87 185)(88 186)(89 187)(90 188)(91 189)(92 190)(93 191)(94 192)(95 193)(96 194)(97 195)(98 196)(99 169)(100 170)(101 171)(102 172)(103 173)(104 174)(105 175)(106 176)(107 177)(108 178)(109 179)(110 180)(111 181)(112 182)
(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 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 84)(16 83)(17 82)(18 81)(19 80)(20 79)(21 78)(22 77)(23 76)(24 75)(25 74)(26 73)(27 72)(28 71)(29 91)(30 90)(31 89)(32 88)(33 87)(34 86)(35 85)(36 98)(37 97)(38 96)(39 95)(40 94)(41 93)(42 92)(43 112)(44 111)(45 110)(46 109)(47 108)(48 107)(49 106)(50 105)(51 104)(52 103)(53 102)(54 101)(55 100)(56 99)(113 175)(114 174)(115 173)(116 172)(117 171)(118 170)(119 169)(120 182)(121 181)(122 180)(123 179)(124 178)(125 177)(126 176)(127 196)(128 195)(129 194)(130 193)(131 192)(132 191)(133 190)(134 189)(135 188)(136 187)(137 186)(138 185)(139 184)(140 183)(141 203)(142 202)(143 201)(144 200)(145 199)(146 198)(147 197)(148 210)(149 209)(150 208)(151 207)(152 206)(153 205)(154 204)(155 224)(156 223)(157 222)(158 221)(159 220)(160 219)(161 218)(162 217)(163 216)(164 215)(165 214)(166 213)(167 212)(168 211)

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

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

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

100 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 2P 2Q 2R 2S 4A 4B 4C 4D 4E ··· 4L 4M ··· 4T 7A 7B 7C 14A ··· 14U 14V ··· 14AS 28A ··· 28L order 1 2 ··· 2 2 ··· 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 ··· 1 2 ··· 2 14 14 14 14 2 2 2 2 7 ··· 7 14 ··· 14 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4

100 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D7 C4○D4 D14 D14 D14 D4⋊2D7 kernel C22×D4⋊2D7 C22×Dic14 D7×C22×C4 C2×D4⋊2D7 C23×Dic7 C22×C7⋊D4 D4×C2×C14 C22×D4 C2×C14 C22×C4 C2×D4 C24 C22 # reps 1 1 1 24 2 2 1 3 8 3 36 6 12

Matrix representation of C22×D42D7 in GL6(𝔽29)

 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 1 0 0 0 0 0 0 1
,
 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 28 0 0 0 0 0 0 28 0 0 0 0 0 0 17 11 0 0 0 0 0 12
,
 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 12 18 0 0 0 0 13 17
,
 28 28 0 0 0 0 5 4 0 0 0 0 0 0 25 28 0 0 0 0 2 22 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 18 26 0 0 0 0 11 11 0 0 0 0 0 0 24 25 0 0 0 0 6 5 0 0 0 0 0 0 28 13 0 0 0 0 0 1

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,0,0,0,0,0,11,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,13,0,0,0,0,18,17],[28,5,0,0,0,0,28,4,0,0,0,0,0,0,25,2,0,0,0,0,28,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,11,0,0,0,0,26,11,0,0,0,0,0,0,24,6,0,0,0,0,25,5,0,0,0,0,0,0,28,0,0,0,0,0,13,1] >;

C22×D42D7 in GAP, Magma, Sage, TeX

C_2^2\times D_4\rtimes_2D_7
% in TeX

G:=Group("C2^2xD4:2D7");
// GroupNames label

G:=SmallGroup(448,1370);
// by ID

G=gap.SmallGroup(448,1370);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,136,1684,235,18822]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=d^2=e^7=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=c^2*d,f*e*f=e^-1>;
// generators/relations

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