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G = C22×C4.Dic7  order 448 = 26·7

Direct product of C22 and C4.Dic7

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22×C4.Dic7, C28.74C24, C24.5Dic7, C7⋊C8⋊13C23, (C23×C4).9D7, C14⋊3(C2×M4(2)), (C2×C14)⋊9M4(2), C4.73(C23×D7), C7⋊3(C22×M4(2)), (C22×C28).28C4, (C23×C28).16C2, C14.41(C23×C4), (C23×C14).11C4, C2.3(C23×Dic7), C28.180(C22×C4), (C2×C28).799C23, (C22×C4).472D14, C23.35(C2×Dic7), C4.38(C22×Dic7), (C22×C4).18Dic7, (C22×C28).546C22, C22.28(C22×Dic7), (C2×C7⋊C8)⋊48C22, (C22×C7⋊C8)⋊23C2, (C2×C28).300(C2×C4), (C2×C4).86(C2×Dic7), (C2×C4).827(C22×D7), (C22×C14).140(C2×C4), (C2×C14).205(C22×C4), SmallGroup(448,1234)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — C22×C4.Dic7
Chief series
C1 — C7 — C14 — C28 — C7⋊C8 — C2×C7⋊C8 — C22×C7⋊C8 — C22×C4.Dic7
Lower central
C7 — C14 — C22×C4.Dic7

Subgroups: 644 in 298 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×4], C4, C4 [×7], C22 [×11], C22 [×12], C7, C8 [×8], C2×C4 [×28], C23, C23 [×6], C23 [×4], C14, C14 [×6], C14 [×4], C2×C8 [×12], M4(2) [×16], C22×C4 [×2], C22×C4 [×12], C24, C28, C28 [×7], C2×C14 [×11], C2×C14 [×12], C22×C8 [×2], C2×M4(2) [×12], C23×C4, C7⋊C8 [×8], C2×C28 [×28], C22×C14, C22×C14 [×6], C22×C14 [×4], C22×M4(2), C2×C7⋊C8 [×12], C4.Dic7 [×16], C22×C28 [×2], C22×C28 [×12], C23×C14, C22×C7⋊C8 [×2], C2×C4.Dic7 [×12], C23×C28, C22×C4.Dic7

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D7, M4(2) [×4], C22×C4 [×14], C24, Dic7 [×8], D14 [×7], C2×M4(2) [×6], C23×C4, C2×Dic7 [×28], C22×D7 [×7], C22×M4(2), C4.Dic7 [×4], C22×Dic7 [×14], C23×D7, C2×C4.Dic7 [×6], C23×Dic7, C22×C4.Dic7

Generators and relations
 G = < a,b,c,d,e | a2=b2=c4=1, d14=c2, e2=c2d7, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d13 >

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

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

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

(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)

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

Matrix representation ►G ⊆ GL5(𝔽113)

10000
0112000
0011200
00010
00001
,
1120000
01000
00100
0001120
0000112
,
1120000
015000
079800
000980
0003715
,
1120000
015000
001500
0001110
0006957
,
980000
0624100
0835100
0001448
000899

G:=sub<GL(5,GF(113))| [1,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,1,0,0,0,0,0,1],[112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,112,0,0,0,0,0,112],[112,0,0,0,0,0,15,7,0,0,0,0,98,0,0,0,0,0,98,37,0,0,0,0,15],[112,0,0,0,0,0,15,0,0,0,0,0,15,0,0,0,0,0,111,69,0,0,0,0,57],[98,0,0,0,0,0,62,83,0,0,0,41,51,0,0,0,0,0,14,8,0,0,0,48,99] >;
 

136 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I4J4K4L7A7B7C8A···8P14A···14AS28A···28AV
order12···222224···444447778···814···1428···28
size11···122221···1222222214···142···22···2

136 irreducible representations

dim111111222222
type+++++-+-
imageC1C2C2C2C4C4D7M4(2)Dic7D14Dic7C4.Dic7
kernelC22×C4.Dic7C22×C7⋊C8C2×C4.Dic7C23×C28C22×C28C23×C14C23×C4C2×C14C22×C4C22×C4C24C22
# reps12121142382121348

In GAP, Magma, Sage, TeX 

C_2^2\times C_4.Dic_7
 
% in TeX
 
G:=Group("C2^2xC4.Dic7");
 
// GroupNames label
 
G:=SmallGroup(448,1234);
 
// by ID
 
G=gap.SmallGroup(448,1234);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,1123,102,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^4=1,d^14=c^2,e^2=c^2*d^7,a*b=b*a,a*c=c*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,e*c*e^-1=c^-1,e*d*e^-1=d^13>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
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