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G = C2×SD16⋊3D7  order 448 = 26·7

Direct product of C2 and SD16⋊3D7

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

Aliases: C2×SD16⋊3D7, C28.8C24, SD16⋊12D14, C56.44C23, D28.4C23, Dic14.4C23, C4.45(D4×D7), C14⋊3(C4○D8), (C4×D7).29D4, C28.83(C2×D4), C7⋊C8.21C23, C4.8(C23×D7), D4⋊D7⋊11C22, (C2×SD16)⋊16D7, D14.10(C2×D4), (C2×C8).265D14, (C8×D7)⋊18C22, (C7×D4).6C23, D4.6(C22×D7), C7⋊Q16⋊7C22, C8.41(C22×D7), (C14×SD16)⋊11C2, (C2×D4).184D14, Q8.2(C22×D7), (C7×Q8).2C23, D4⋊2D7⋊7C22, C56⋊C2⋊18C22, (C2×Q8).151D14, Dic7.69(C2×D4), Q8⋊2D7⋊6C22, (C22×D7).62D4, (C4×D7).26C23, C22.141(D4×D7), (C2×C56).166C22, (C2×C28).525C23, (C2×Dic7).216D4, (C7×SD16)⋊13C22, C14.109(C22×D4), (D4×C14).166C22, (C2×D28).178C22, (Q8×C14).148C22, (C2×Dic14).197C22, C7⋊3(C2×C4○D8), (D7×C2×C8)⋊10C2, C2.82(C2×D4×D7), (C2×D4⋊D7)⋊28C2, (C2×C56⋊C2)⋊32C2, (C2×C7⋊Q16)⋊26C2, (C2×D4⋊2D7)⋊25C2, (C2×Q8⋊2D7)⋊15C2, (C2×C14).398(C2×D4), (C2×C7⋊C8).284C22, (C2×C4×D7).258C22, (C2×C4).614(C22×D7), SmallGroup(448,1214)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C28 — C2×SD16⋊3D7
Chief series
C1 — C7 — C14 — C28 — C4×D7 — C2×C4×D7 — C2×D4⋊2D7 — C2×SD16⋊3D7
Lower central
C7 — C14 — C28 — C2×SD16⋊3D7

Subgroups: 1284 in 266 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×12], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4 [×2], D4 [×12], Q8 [×2], Q8 [×4], C23 [×3], D7 [×4], C14, C14 [×2], C14 [×2], C2×C8, C2×C8 [×5], D8 [×4], SD16 [×4], SD16 [×4], Q16 [×4], C22×C4 [×3], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8, C4○D4 [×12], Dic7 [×2], Dic7 [×2], C28 [×2], C28 [×2], D14 [×2], D14 [×6], C2×C14, C2×C14 [×4], C22×C8, C2×D8, C2×SD16, C2×SD16, C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C7⋊C8 [×2], C56 [×2], Dic14 [×2], Dic14, C4×D7 [×4], C4×D7 [×4], D28 [×2], D28 [×5], C2×Dic7, C2×Dic7 [×5], C7⋊D4 [×4], C2×C28, C2×C28, C7×D4 [×2], C7×D4, C7×Q8 [×2], C7×Q8, C22×D7, C22×D7, C22×C14, C2×C4○D8, C8×D7 [×4], C56⋊C2 [×4], C2×C7⋊C8, D4⋊D7 [×4], C7⋊Q16 [×4], C2×C56, C7×SD16 [×4], C2×Dic14, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, D4⋊2D7 [×4], D4⋊2D7 [×2], Q8⋊2D7 [×4], Q8⋊2D7 [×2], C22×Dic7, C2×C7⋊D4, D4×C14, Q8×C14, D7×C2×C8, C2×C56⋊C2, SD16⋊3D7 [×8], C2×D4⋊D7, C2×C7⋊Q16, C14×SD16, C2×D4⋊2D7, C2×Q8⋊2D7, C2×SD16⋊3D7

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C4○D8 [×2], C22×D4, C22×D7 [×7], C2×C4○D8, D4×D7 [×2], C23×D7, SD16⋊3D7 [×2], C2×D4×D7, C2×SD16⋊3D7

Generators and relations
 G = < a,b,c,d,e | a2=b8=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b3, bd=db, be=eb, cd=dc, ece=b4c, ede=d-1 >

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

(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)

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

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

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

Matrix representation ►G ⊆ GL4(𝔽113) generated by

112000
011200
001120
000112
,
69000
01800
001120
000112
,
06900
95000
001120
000112
,
1000
0100
0079112
0010
,
1000
011200
0079112
002534
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[69,0,0,0,0,18,0,0,0,0,112,0,0,0,0,112],[0,95,0,0,69,0,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,79,1,0,0,112,0],[1,0,0,0,0,112,0,0,0,0,79,25,0,0,112,34] >;
 

70 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J7A7B7C8A8B8C8D8E8F8G8H14A···14I14J···14O28A···28F28G···28L56A···56L
order122222222244444444447778888888814···1414···1428···2828···2856···56
size111144141428282244777728282222222141414142···28···84···48···84···4

70 irreducible representations

dim111111111222222222444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4D7D14D14D14D14C4○D8D4×D7D4×D7SD16⋊3D7
kernelC2×SD16⋊3D7D7×C2×C8C2×C56⋊C2SD16⋊3D7C2×D4⋊D7C2×C7⋊Q16C14×SD16C2×D4⋊2D7C2×Q8⋊2D7C4×D7C2×Dic7C22×D7C2×SD16C2×C8SD16C2×D4C2×Q8C14C4C22C2
# reps11181111121133123383312

In GAP, Magma, Sage, TeX 

C_2\times SD_{16}\rtimes_3D_7
 
% in TeX
 
G:=Group("C2xSD16:3D7");
 
// GroupNames label
 
G:=SmallGroup(448,1214);
 
// by ID
 
G=gap.SmallGroup(448,1214);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,1123,185,136,438,235,102,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^3,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
 
// generators/relations
 

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