direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22⋊C4×Dic7, C24.43D14, (C2×C14)⋊C42, C14.40(C4×D4), C23.D7⋊7C4, C2.1(D4×Dic7), C22⋊1(C4×Dic7), C22.93(D4×D7), C23.48(C4×D7), C14.19(C2×C42), (C22×Dic7)⋊6C4, (C2×Dic7).210D4, (C22×C4).306D14, (C23×Dic7).1C2, C23.12(C2×Dic7), C14.C42⋊37C2, C2.4(Dic7⋊4D4), (C23×C14).24C22, C23.274(C22×D7), C14.23(C42⋊C2), C22.40(D4⋊2D7), (C22×C28).340C22, (C22×C14).316C23, C22.19(C22×Dic7), C2.4(C23.11D14), (C22×Dic7).182C22, C7⋊3(C4×C22⋊C4), (C2×C28)⋊20(C2×C4), C2.8(C2×C4×Dic7), (C7×C22⋊C4)⋊9C4, (C2×C4×Dic7)⋊20C2, (C2×C4)⋊6(C2×Dic7), C2.4(D7×C22⋊C4), C22.54(C2×C4×D7), (C2×Dic7)⋊14(C2×C4), (C2×C14).313(C2×D4), C14.26(C2×C22⋊C4), (C2×C22⋊C4).19D7, (C2×C23.D7).3C2, (C14×C22⋊C4).22C2, (C22×C14).44(C2×C4), (C2×C14).137(C4○D4), (C2×C14).102(C22×C4), SmallGroup(448,475)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4×Dic7
G = < a,b,c,d,e | a2=b2=c4=d14=1, e2=d7, cac-1=ab=ba, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 916 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C42, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C24, Dic7, Dic7, C28, C2×C14, C2×C14, C2×C14, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C4×C22⋊C4, C4×Dic7, C23.D7, C7×C22⋊C4, C22×Dic7, C22×Dic7, C22×Dic7, C22×C28, C23×C14, C14.C42, C2×C4×Dic7, C2×C23.D7, C14×C22⋊C4, C23×Dic7, C22⋊C4×Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, Dic7, D14, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×D7, C2×Dic7, C22×D7, C4×C22⋊C4, C4×Dic7, C2×C4×D7, D4×D7, D4⋊2D7, C22×Dic7, C23.11D14, D7×C22⋊C4, Dic7⋊4D4, C2×C4×Dic7, D4×Dic7, C22⋊C4×Dic7
►On 224 points
Generators in S224
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 127)(22 128)(23 129)(24 130)(25 131)(26 132)(27 133)(28 134)(29 48)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 43)(39 44)(40 45)(41 46)(42 47)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 160)(100 161)(101 162)(102 163)(103 164)(104 165)(105 166)(106 167)(107 168)(108 155)(109 156)(110 157)(111 158)(112 159)(113 206)(114 207)(115 208)(116 209)(117 210)(118 197)(119 198)(120 199)(121 200)(122 201)(123 202)(124 203)(125 204)(126 205)(141 148)(142 149)(143 150)(144 151)(145 152)(146 153)(147 154)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(183 190)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)(211 218)(212 219)(213 220)(214 221)(215 222)(216 223)(217 224)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 57)(12 58)(13 59)(14 60)(15 128)(16 129)(17 130)(18 131)(19 132)(20 133)(21 134)(22 135)(23 136)(24 137)(25 138)(26 139)(27 140)(28 127)(29 55)(30 56)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 85)(81 86)(82 87)(83 88)(84 89)(99 167)(100 168)(101 155)(102 156)(103 157)(104 158)(105 159)(106 160)(107 161)(108 162)(109 163)(110 164)(111 165)(112 166)(113 199)(114 200)(115 201)(116 202)(117 203)(118 204)(119 205)(120 206)(121 207)(122 208)(123 209)(124 210)(125 197)(126 198)(141 217)(142 218)(143 219)(144 220)(145 221)(146 222)(147 223)(148 224)(149 211)(150 212)(151 213)(152 214)(153 215)(154 216)(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)
(1 163 178 125)(2 164 179 126)(3 165 180 113)(4 166 181 114)(5 167 182 115)(6 168 169 116)(7 155 170 117)(8 156 171 118)(9 157 172 119)(10 158 173 120)(11 159 174 121)(12 160 175 122)(13 161 176 123)(14 162 177 124)(15 92 53 222)(16 93 54 223)(17 94 55 224)(18 95 56 211)(19 96 43 212)(20 97 44 213)(21 98 45 214)(22 85 46 215)(23 86 47 216)(24 87 48 217)(25 88 49 218)(26 89 50 219)(27 90 51 220)(28 91 52 221)(29 148 130 75)(30 149 131 76)(31 150 132 77)(32 151 133 78)(33 152 134 79)(34 153 135 80)(35 154 136 81)(36 141 137 82)(37 142 138 83)(38 143 139 84)(39 144 140 71)(40 145 127 72)(41 146 128 73)(42 147 129 74)(57 105 188 207)(58 106 189 208)(59 107 190 209)(60 108 191 210)(61 109 192 197)(62 110 193 198)(63 111 194 199)(64 112 195 200)(65 99 196 201)(66 100 183 202)(67 101 184 203)(68 102 185 204)(69 103 186 205)(70 104 187 206)
(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)
(1 211 8 218)(2 224 9 217)(3 223 10 216)(4 222 11 215)(5 221 12 214)(6 220 13 213)(7 219 14 212)(15 159 22 166)(16 158 23 165)(17 157 24 164)(18 156 25 163)(19 155 26 162)(20 168 27 161)(21 167 28 160)(29 205 36 198)(30 204 37 197)(31 203 38 210)(32 202 39 209)(33 201 40 208)(34 200 41 207)(35 199 42 206)(43 117 50 124)(44 116 51 123)(45 115 52 122)(46 114 53 121)(47 113 54 120)(48 126 55 119)(49 125 56 118)(57 153 64 146)(58 152 65 145)(59 151 66 144)(60 150 67 143)(61 149 68 142)(62 148 69 141)(63 147 70 154)(71 190 78 183)(72 189 79 196)(73 188 80 195)(74 187 81 194)(75 186 82 193)(76 185 83 192)(77 184 84 191)(85 181 92 174)(86 180 93 173)(87 179 94 172)(88 178 95 171)(89 177 96 170)(90 176 97 169)(91 175 98 182)(99 127 106 134)(100 140 107 133)(101 139 108 132)(102 138 109 131)(103 137 110 130)(104 136 111 129)(105 135 112 128)
G:=sub<Sym(224)| (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,127)(22,128)(23,129)(24,130)(25,131)(26,132)(27,133)(28,134)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,43)(39,44)(40,45)(41,46)(42,47)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,160)(100,161)(101,162)(102,163)(103,164)(104,165)(105,166)(106,167)(107,168)(108,155)(109,156)(110,157)(111,158)(112,159)(113,206)(114,207)(115,208)(116,209)(117,210)(118,197)(119,198)(120,199)(121,200)(122,201)(123,202)(124,203)(125,204)(126,205)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,57)(12,58)(13,59)(14,60)(15,128)(16,129)(17,130)(18,131)(19,132)(20,133)(21,134)(22,135)(23,136)(24,137)(25,138)(26,139)(27,140)(28,127)(29,55)(30,56)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,85)(81,86)(82,87)(83,88)(84,89)(99,167)(100,168)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,161)(108,162)(109,163)(110,164)(111,165)(112,166)(113,199)(114,200)(115,201)(116,202)(117,203)(118,204)(119,205)(120,206)(121,207)(122,208)(123,209)(124,210)(125,197)(126,198)(141,217)(142,218)(143,219)(144,220)(145,221)(146,222)(147,223)(148,224)(149,211)(150,212)(151,213)(152,214)(153,215)(154,216)(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), (1,163,178,125)(2,164,179,126)(3,165,180,113)(4,166,181,114)(5,167,182,115)(6,168,169,116)(7,155,170,117)(8,156,171,118)(9,157,172,119)(10,158,173,120)(11,159,174,121)(12,160,175,122)(13,161,176,123)(14,162,177,124)(15,92,53,222)(16,93,54,223)(17,94,55,224)(18,95,56,211)(19,96,43,212)(20,97,44,213)(21,98,45,214)(22,85,46,215)(23,86,47,216)(24,87,48,217)(25,88,49,218)(26,89,50,219)(27,90,51,220)(28,91,52,221)(29,148,130,75)(30,149,131,76)(31,150,132,77)(32,151,133,78)(33,152,134,79)(34,153,135,80)(35,154,136,81)(36,141,137,82)(37,142,138,83)(38,143,139,84)(39,144,140,71)(40,145,127,72)(41,146,128,73)(42,147,129,74)(57,105,188,207)(58,106,189,208)(59,107,190,209)(60,108,191,210)(61,109,192,197)(62,110,193,198)(63,111,194,199)(64,112,195,200)(65,99,196,201)(66,100,183,202)(67,101,184,203)(68,102,185,204)(69,103,186,205)(70,104,187,206), (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), (1,211,8,218)(2,224,9,217)(3,223,10,216)(4,222,11,215)(5,221,12,214)(6,220,13,213)(7,219,14,212)(15,159,22,166)(16,158,23,165)(17,157,24,164)(18,156,25,163)(19,155,26,162)(20,168,27,161)(21,167,28,160)(29,205,36,198)(30,204,37,197)(31,203,38,210)(32,202,39,209)(33,201,40,208)(34,200,41,207)(35,199,42,206)(43,117,50,124)(44,116,51,123)(45,115,52,122)(46,114,53,121)(47,113,54,120)(48,126,55,119)(49,125,56,118)(57,153,64,146)(58,152,65,145)(59,151,66,144)(60,150,67,143)(61,149,68,142)(62,148,69,141)(63,147,70,154)(71,190,78,183)(72,189,79,196)(73,188,80,195)(74,187,81,194)(75,186,82,193)(76,185,83,192)(77,184,84,191)(85,181,92,174)(86,180,93,173)(87,179,94,172)(88,178,95,171)(89,177,96,170)(90,176,97,169)(91,175,98,182)(99,127,106,134)(100,140,107,133)(101,139,108,132)(102,138,109,131)(103,137,110,130)(104,136,111,129)(105,135,112,128)>;
G:=Group( (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,127)(22,128)(23,129)(24,130)(25,131)(26,132)(27,133)(28,134)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,43)(39,44)(40,45)(41,46)(42,47)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,160)(100,161)(101,162)(102,163)(103,164)(104,165)(105,166)(106,167)(107,168)(108,155)(109,156)(110,157)(111,158)(112,159)(113,206)(114,207)(115,208)(116,209)(117,210)(118,197)(119,198)(120,199)(121,200)(122,201)(123,202)(124,203)(125,204)(126,205)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,57)(12,58)(13,59)(14,60)(15,128)(16,129)(17,130)(18,131)(19,132)(20,133)(21,134)(22,135)(23,136)(24,137)(25,138)(26,139)(27,140)(28,127)(29,55)(30,56)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,85)(81,86)(82,87)(83,88)(84,89)(99,167)(100,168)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,161)(108,162)(109,163)(110,164)(111,165)(112,166)(113,199)(114,200)(115,201)(116,202)(117,203)(118,204)(119,205)(120,206)(121,207)(122,208)(123,209)(124,210)(125,197)(126,198)(141,217)(142,218)(143,219)(144,220)(145,221)(146,222)(147,223)(148,224)(149,211)(150,212)(151,213)(152,214)(153,215)(154,216)(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), (1,163,178,125)(2,164,179,126)(3,165,180,113)(4,166,181,114)(5,167,182,115)(6,168,169,116)(7,155,170,117)(8,156,171,118)(9,157,172,119)(10,158,173,120)(11,159,174,121)(12,160,175,122)(13,161,176,123)(14,162,177,124)(15,92,53,222)(16,93,54,223)(17,94,55,224)(18,95,56,211)(19,96,43,212)(20,97,44,213)(21,98,45,214)(22,85,46,215)(23,86,47,216)(24,87,48,217)(25,88,49,218)(26,89,50,219)(27,90,51,220)(28,91,52,221)(29,148,130,75)(30,149,131,76)(31,150,132,77)(32,151,133,78)(33,152,134,79)(34,153,135,80)(35,154,136,81)(36,141,137,82)(37,142,138,83)(38,143,139,84)(39,144,140,71)(40,145,127,72)(41,146,128,73)(42,147,129,74)(57,105,188,207)(58,106,189,208)(59,107,190,209)(60,108,191,210)(61,109,192,197)(62,110,193,198)(63,111,194,199)(64,112,195,200)(65,99,196,201)(66,100,183,202)(67,101,184,203)(68,102,185,204)(69,103,186,205)(70,104,187,206), (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), (1,211,8,218)(2,224,9,217)(3,223,10,216)(4,222,11,215)(5,221,12,214)(6,220,13,213)(7,219,14,212)(15,159,22,166)(16,158,23,165)(17,157,24,164)(18,156,25,163)(19,155,26,162)(20,168,27,161)(21,167,28,160)(29,205,36,198)(30,204,37,197)(31,203,38,210)(32,202,39,209)(33,201,40,208)(34,200,41,207)(35,199,42,206)(43,117,50,124)(44,116,51,123)(45,115,52,122)(46,114,53,121)(47,113,54,120)(48,126,55,119)(49,125,56,118)(57,153,64,146)(58,152,65,145)(59,151,66,144)(60,150,67,143)(61,149,68,142)(62,148,69,141)(63,147,70,154)(71,190,78,183)(72,189,79,196)(73,188,80,195)(74,187,81,194)(75,186,82,193)(76,185,83,192)(77,184,84,191)(85,181,92,174)(86,180,93,173)(87,179,94,172)(88,178,95,171)(89,177,96,170)(90,176,97,169)(91,175,98,182)(99,127,106,134)(100,140,107,133)(101,139,108,132)(102,138,109,131)(103,137,110,130)(104,136,111,129)(105,135,112,128) );
G=PermutationGroup([[(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(21,127),(22,128),(23,129),(24,130),(25,131),(26,132),(27,133),(28,134),(29,48),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,56),(38,43),(39,44),(40,45),(41,46),(42,47),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,160),(100,161),(101,162),(102,163),(103,164),(104,165),(105,166),(106,167),(107,168),(108,155),(109,156),(110,157),(111,158),(112,159),(113,206),(114,207),(115,208),(116,209),(117,210),(118,197),(119,198),(120,199),(121,200),(122,201),(123,202),(124,203),(125,204),(126,205),(141,148),(142,149),(143,150),(144,151),(145,152),(146,153),(147,154),(169,176),(170,177),(171,178),(172,179),(173,180),(174,181),(175,182),(183,190),(184,191),(185,192),(186,193),(187,194),(188,195),(189,196),(211,218),(212,219),(213,220),(214,221),(215,222),(216,223),(217,224)], [(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,57),(12,58),(13,59),(14,60),(15,128),(16,129),(17,130),(18,131),(19,132),(20,133),(21,134),(22,135),(23,136),(24,137),(25,138),(26,139),(27,140),(28,127),(29,55),(30,56),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,51),(40,52),(41,53),(42,54),(71,90),(72,91),(73,92),(74,93),(75,94),(76,95),(77,96),(78,97),(79,98),(80,85),(81,86),(82,87),(83,88),(84,89),(99,167),(100,168),(101,155),(102,156),(103,157),(104,158),(105,159),(106,160),(107,161),(108,162),(109,163),(110,164),(111,165),(112,166),(113,199),(114,200),(115,201),(116,202),(117,203),(118,204),(119,205),(120,206),(121,207),(122,208),(123,209),(124,210),(125,197),(126,198),(141,217),(142,218),(143,219),(144,220),(145,221),(146,222),(147,223),(148,224),(149,211),(150,212),(151,213),(152,214),(153,215),(154,216),(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)], [(1,163,178,125),(2,164,179,126),(3,165,180,113),(4,166,181,114),(5,167,182,115),(6,168,169,116),(7,155,170,117),(8,156,171,118),(9,157,172,119),(10,158,173,120),(11,159,174,121),(12,160,175,122),(13,161,176,123),(14,162,177,124),(15,92,53,222),(16,93,54,223),(17,94,55,224),(18,95,56,211),(19,96,43,212),(20,97,44,213),(21,98,45,214),(22,85,46,215),(23,86,47,216),(24,87,48,217),(25,88,49,218),(26,89,50,219),(27,90,51,220),(28,91,52,221),(29,148,130,75),(30,149,131,76),(31,150,132,77),(32,151,133,78),(33,152,134,79),(34,153,135,80),(35,154,136,81),(36,141,137,82),(37,142,138,83),(38,143,139,84),(39,144,140,71),(40,145,127,72),(41,146,128,73),(42,147,129,74),(57,105,188,207),(58,106,189,208),(59,107,190,209),(60,108,191,210),(61,109,192,197),(62,110,193,198),(63,111,194,199),(64,112,195,200),(65,99,196,201),(66,100,183,202),(67,101,184,203),(68,102,185,204),(69,103,186,205),(70,104,187,206)], [(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)], [(1,211,8,218),(2,224,9,217),(3,223,10,216),(4,222,11,215),(5,221,12,214),(6,220,13,213),(7,219,14,212),(15,159,22,166),(16,158,23,165),(17,157,24,164),(18,156,25,163),(19,155,26,162),(20,168,27,161),(21,167,28,160),(29,205,36,198),(30,204,37,197),(31,203,38,210),(32,202,39,209),(33,201,40,208),(34,200,41,207),(35,199,42,206),(43,117,50,124),(44,116,51,123),(45,115,52,122),(46,114,53,121),(47,113,54,120),(48,126,55,119),(49,125,56,118),(57,153,64,146),(58,152,65,145),(59,151,66,144),(60,150,67,143),(61,149,68,142),(62,148,69,141),(63,147,70,154),(71,190,78,183),(72,189,79,196),(73,188,80,195),(74,187,81,194),(75,186,82,193),(76,185,83,192),(77,184,84,191),(85,181,92,174),(86,180,93,173),(87,179,94,172),(88,178,95,171),(89,177,96,170),(90,176,97,169),(91,175,98,182),(99,127,106,134),(100,140,107,133),(101,139,108,132),(102,138,109,131),(103,137,110,130),(104,136,111,129),(105,135,112,128)]])
100 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AB | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D7 | C4○D4 | Dic7 | D14 | D14 | C4×D7 | D4×D7 | D4⋊2D7 |
| kernel | C22⋊C4×Dic7 | C14.C42 | C2×C4×Dic7 | C2×C23.D7 | C14×C22⋊C4 | C23×Dic7 | C23.D7 | C7×C22⋊C4 | C22×Dic7 | C2×Dic7 | C2×C22⋊C4 | C2×C14 | C22⋊C4 | C22×C4 | C24 | C23 | C22 | C22 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 8 | 8 | 4 | 3 | 4 | 12 | 6 | 3 | 24 | 6 | 6 |
Matrix representation of C22⋊C4×Dic7 ►in GL5(𝔽29)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 28 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 28 | 27 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 17 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 25 | 3 |
| 0 | 0 | 0 | 0 | 7 |
| 17 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 6 | 1 |
G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,1,28,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,28,0,0,0,0,27,1,0,0,0,0,0,17,0,0,0,0,0,17],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,25,0,0,0,0,3,7],[17,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,6,0,0,0,0,1] >;
C22⋊C4×Dic7 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\times {\rm Dic}_7 % in TeX
G:=Group("C2^2:C4xDic7"); // GroupNames label
G:=SmallGroup(448,475);
// by ID
G=gap.SmallGroup(448,475);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,387,100,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^14=1,e^2=d^7,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations