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G = C2×D4.Dic7  order 448 = 26·7

Direct product of C2 and D4.Dic7

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

Aliases: C2×D4.Dic7, C28.75C24, C14⋊3(C8○D4), C7⋊C8.37C23, C4○D4.44D14, (D4×C14).12C4, (Q8×C14).12C4, C4○D4.4Dic7, D4.8(C2×Dic7), Q8.9(C2×Dic7), C4.74(C23×D7), C28.97(C22×C4), C14.48(C23×C4), (C2×D4).12Dic7, (C2×Q8).10Dic7, (C2×C28).553C23, (C22×C4).386D14, C4.Dic7⋊34C22, C2.10(C23×Dic7), C4.40(C22×Dic7), C23.19(C2×Dic7), (C22×C28).288C22, C22.33(C22×Dic7), C7⋊4(C2×C8○D4), (C2×C7⋊C8)⋊41C22, (C22×C7⋊C8)⋊14C2, (C7×C4○D4).3C4, (C14×C4○D4).9C2, (C2×C4○D4).13D7, (C7×D4).25(C2×C4), (C7×Q8).27(C2×C4), (C2×C28).135(C2×C4), (C2×C4.Dic7)⋊28C2, (C2×C4).56(C2×Dic7), (C22×C14).80(C2×C4), (C2×C14).28(C22×C4), (C7×C4○D4).48C22, (C2×C4).831(C22×D7), SmallGroup(448,1271)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — C2×D4.Dic7
Chief series
C1 — C7 — C14 — C28 — C7⋊C8 — C2×C7⋊C8 — C22×C7⋊C8 — C2×D4.Dic7
Lower central
C7 — C14 — C2×D4.Dic7

Subgroups: 596 in 266 conjugacy classes, 191 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×6], C7, C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C14, C14 [×2], C14 [×6], C2×C8 [×16], M4(2) [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C28 [×2], C28 [×6], C2×C14, C2×C14 [×6], C2×C14 [×6], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C7⋊C8 [×8], C2×C28, C2×C28 [×15], C7×D4 [×12], C7×Q8 [×4], C22×C14 [×3], C2×C8○D4, C2×C7⋊C8, C2×C7⋊C8 [×15], C4.Dic7 [×12], C22×C28 [×3], D4×C14 [×3], Q8×C14, C7×C4○D4 [×8], C22×C7⋊C8 [×3], C2×C4.Dic7 [×3], D4.Dic7 [×8], C14×C4○D4, C2×D4.Dic7

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D7, C22×C4 [×14], C24, Dic7 [×8], D14 [×7], C8○D4 [×2], C23×C4, C2×Dic7 [×28], C22×D7 [×7], C2×C8○D4, C22×Dic7 [×14], C23×D7, D4.Dic7 [×2], C23×Dic7, C2×D4.Dic7

Generators and relations
 G = < a,b,c,d,e | a2=b4=c2=1, d14=b2, e2=b2d7, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d13 >

Smallest permutation representation
►On 224 points
Generators in S224
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(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)(113 148)(114 149)(115 150)(116 151)(117 152)(118 153)(119 154)(120 155)(121 156)(122 157)(123 158)(124 159)(125 160)(126 161)(127 162)(128 163)(129 164)(130 165)(131 166)(132 167)(133 168)(134 141)(135 142)(136 143)(137 144)(138 145)(139 146)(140 147)(169 220)(170 221)(171 222)(172 223)(173 224)(174 197)(175 198)(176 199)(177 200)(178 201)(179 202)(180 203)(181 204)(182 205)(183 206)(184 207)(185 208)(186 209)(187 210)(188 211)(189 212)(190 213)(191 214)(192 215)(193 216)(194 217)(195 218)(196 219)

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

(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(13 83)(14 84)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(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)(113 176)(114 177)(115 178)(116 179)(117 180)(118 181)(119 182)(120 183)(121 184)(122 185)(123 186)(124 187)(125 188)(126 189)(127 190)(128 191)(129 192)(130 193)(131 194)(132 195)(133 196)(134 169)(135 170)(136 171)(137 172)(138 173)(139 174)(140 175)(141 220)(142 221)(143 222)(144 223)(145 224)(146 197)(147 198)(148 199)(149 200)(150 201)(151 202)(152 203)(153 204)(154 205)(155 206)(156 207)(157 208)(158 209)(159 210)(160 211)(161 212)(162 213)(163 214)(164 215)(165 216)(166 217)(167 218)(168 219)

(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)

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

Matrix representation ►G ⊆ GL4(𝔽113) generated by

112000
011200
0010
0001
,
1000
0100
00980
00815
,
1000
0100
009885
00815
,
3411200
608800
00980
00098
,
746800
843900
00950
00095
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,98,8,0,0,0,15],[1,0,0,0,0,1,0,0,0,0,98,8,0,0,85,15],[34,60,0,0,112,88,0,0,0,0,98,0,0,0,0,98],[74,84,0,0,68,39,0,0,0,0,95,0,0,0,0,95] >;
 

100 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J7A7B7C8A···8H8I···8T14A···14I14J···14AA28A···28L28M···28AD
order12222···244444···47778···88···814···1414···1428···2828···28
size11112···211112···22227···714···142···24···42···24···4

100 irreducible representations

dim1111111122222224
type+++++++---+
imageC1C2C2C2C2C4C4C4D7D14Dic7Dic7Dic7D14C8○D4D4.Dic7
kernelC2×D4.Dic7C22×C7⋊C8C2×C4.Dic7D4.Dic7C14×C4○D4D4×C14Q8×C14C7×C4○D4C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C4○D4C14C2
# reps1338162839931212812

In GAP, Magma, Sage, TeX 

C_2\times D_4.Dic_7
 
% in TeX
 
G:=Group("C2xD4.Dic7");
 
// GroupNames label
 
G:=SmallGroup(448,1271);
 
// by ID
 
G=gap.SmallGroup(448,1271);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,297,102,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^14=b^2,e^2=b^2*d^7,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^13>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀