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G = C2×Q8.10D14  order 448 = 26·7

Direct product of C2 and Q8.10D14

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

Aliases: C2×Q8.10D14, C14.10C25, C28.45C24, D14.5C24, D28.36C23, C14⋊12- (1+4), Dic7.5C24, Dic14.36C23, (C2×Q8)⋊35D14, (Q8×D7)⋊13C22, (C22×Q8)⋊10D7, C4.45(C23×D7), C2.11(D7×C24), C7⋊D4.6C23, C7⋊1(C2×2- (1+4)), C4○D28⋊23C22, (Q8×C14)⋊44C22, (C4×D7).18C23, (C7×Q8).29C23, Q8.29(C22×D7), C22.9(C23×D7), (C2×C14).330C24, (C2×C28).566C23, Q8⋊2D7⋊12C22, (C22×C4).291D14, (C2×D28).283C22, C23.242(C22×D7), (C22×C28).302C22, (C22×C14).437C23, (C2×Dic7).298C23, (C22×D7).248C23, (C2×Dic14).312C22, (C2×Q8×D7)⋊20C2, (Q8×C2×C14)⋊11C2, (C2×C4○D28)⋊35C2, (C2×Q8⋊2D7)⋊20C2, (C2×C4×D7).172C22, (C2×C4).252(C22×D7), (C2×C7⋊D4).150C22, SmallGroup(448,1374)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — C2×Q8.10D14
Chief series
C1 — C7 — C14 — D14 — C22×D7 — C2×C4×D7 — C2×Q8×D7 — C2×Q8.10D14
Lower central
C7 — C14 — C2×Q8.10D14

Subgroups: 2868 in 794 conjugacy classes, 447 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×12], C4 [×8], C22, C22 [×2], C22 [×18], C7, C2×C4 [×18], C2×C4 [×52], D4 [×40], Q8 [×16], Q8 [×24], C23, C23 [×4], D7 [×8], C14, C14 [×2], C14 [×2], C22×C4 [×3], C22×C4 [×12], C2×D4 [×10], C2×Q8 [×12], C2×Q8 [×38], C4○D4 [×80], Dic7 [×8], C28 [×12], D14 [×8], D14 [×8], C2×C14, C2×C14 [×2], C2×C14 [×2], C22×Q8, C22×Q8 [×4], C2×C4○D4 [×10], 2- (1+4) [×16], Dic14 [×24], C4×D7 [×48], D28 [×24], C2×Dic7 [×4], C7⋊D4 [×16], C2×C28 [×18], C7×Q8 [×16], C22×D7 [×4], C22×C14, C2×2- (1+4), C2×Dic14 [×6], C2×C4×D7 [×12], C2×D28 [×6], C4○D28 [×48], Q8×D7 [×32], Q8⋊2D7 [×32], C2×C7⋊D4 [×4], C22×C28 [×3], Q8×C14 [×12], C2×C4○D28 [×6], C2×Q8×D7 [×4], C2×Q8⋊2D7 [×4], Q8.10D14 [×16], Q8×C2×C14, C2×Q8.10D14

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], D7, C24 [×31], D14 [×15], 2- (1+4) [×2], C25, C22×D7 [×35], C2×2- (1+4), C23×D7 [×15], Q8.10D14 [×2], D7×C24, C2×Q8.10D14

Generators and relations
 G = < a,b,c,d,e | a2=b4=1, c2=d14=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=ece-1=b2c, ede-1=d13 >

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

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

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

(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)

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

Matrix representation ►G ⊆ GL6(𝔽29)

2800000
0280000
001000
000100
000010
000001
,
100000
010000
00002413
0000165
0051600
00132400
,
100000
010000
00271100
0018200
0000218
00001127
,
19100000
1970000
0000268
00002121
0032100
008800
,
28220000
010000
0072000
00122200
0000720
00001222

G:=sub<GL(6,GF(29))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,13,0,0,0,0,16,24,0,0,24,16,0,0,0,0,13,5,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,18,0,0,0,0,11,2,0,0,0,0,0,0,2,11,0,0,0,0,18,27],[19,19,0,0,0,0,10,7,0,0,0,0,0,0,0,0,3,8,0,0,0,0,21,8,0,0,26,21,0,0,0,0,8,21,0,0],[28,0,0,0,0,0,22,1,0,0,0,0,0,0,7,12,0,0,0,0,20,22,0,0,0,0,0,0,7,12,0,0,0,0,20,22] >;
 

94 conjugacy classes

class 1 2A2B2C2D2E2F···2M4A···4L4M···4T7A7B7C14A···14U28A···28AJ
order1222222···24···44···477714···1428···28
size11112214···142···214···142222···24···4

94 irreducible representations

dim11111122244
type+++++++++-
imageC1C2C2C2C2C2D7D14D142- (1+4)Q8.10D14
kernelC2×Q8.10D14C2×C4○D28C2×Q8×D7C2×Q8⋊2D7Q8.10D14Q8×C2×C14C22×Q8C22×C4C2×Q8C14C2
# reps16441613936212

In GAP, Magma, Sage, TeX 

C_2\times Q_8._{10}D_{14}
 
% in TeX
 
G:=Group("C2xQ8.10D14");
 
// GroupNames label
 
G:=SmallGroup(448,1374);
 
// by ID
 
G=gap.SmallGroup(448,1374);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,297,136,1684,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=d^14=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=e*c*e^-1=b^2*c,e*d*e^-1=d^13>;
 
// generators/relations
 

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