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G = C2×C23.21D14  order 448 = 26·7

Direct product of C2 and C23.21D14

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

Aliases: C2×C23.21D14, C24.68D14, (C22×C28)⋊16C4, (C23×C4).12D7, (C23×C28).17C2, C14.43(C23×C4), C4⋊Dic7⋊83C22, (C22×C4)⋊10Dic7, C14⋊4(C42⋊C2), C2.5(C23×Dic7), (C2×C28).885C23, C28.181(C22×C4), (C2×C14).284C24, (C4×Dic7)⋊81C22, (C22×C4).448D14, C4.39(C22×Dic7), C23.36(C2×Dic7), C22.41(C23×D7), C22.80(C4○D28), C23.232(C22×D7), (C22×C28).547C22, (C22×C14).413C23, (C23×C14).106C22, (C2×Dic7).278C23, C23.D7.144C22, C22.31(C22×Dic7), (C22×Dic7).229C22, (C2×C28)⋊37(C2×C4), C7⋊5(C2×C42⋊C2), (C2×C4×Dic7)⋊37C2, C2.5(C2×C4○D28), (C2×C4⋊Dic7)⋊50C2, C14.60(C2×C4○D4), (C2×C4)⋊11(C2×Dic7), (C2×C4).829(C22×D7), (C2×C23.D7).25C2, (C2×C14).111(C4○D4), (C2×C14).208(C22×C4), (C22×C14).143(C2×C4), SmallGroup(448,1239)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — C2×C23.21D14
Chief series
C1 — C7 — C14 — C2×C14 — C2×Dic7 — C22×Dic7 — C2×C4×Dic7 — C2×C23.21D14
Lower central
C7 — C14 — C2×C23.21D14

Subgroups: 900 in 330 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×10], C22 [×12], C7, C2×C4 [×28], C2×C4 [×16], C23, C23 [×6], C23 [×4], C14, C14 [×6], C14 [×4], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×12], C22×C4 [×4], C24, Dic7 [×8], C28 [×8], C2×C14, C2×C14 [×10], C2×C14 [×12], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C23×C4, C2×Dic7 [×8], C2×Dic7 [×8], C2×C28 [×28], C22×C14, C22×C14 [×6], C22×C14 [×4], C2×C42⋊C2, C4×Dic7 [×8], C4⋊Dic7 [×8], C23.D7 [×8], C22×Dic7 [×4], C22×C28 [×2], C22×C28 [×12], C23×C14, C2×C4×Dic7 [×2], C2×C4⋊Dic7 [×2], C23.21D14 [×8], C2×C23.D7 [×2], C23×C28, C2×C23.21D14

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D7, C22×C4 [×14], C4○D4 [×4], C24, Dic7 [×8], D14 [×7], C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], C2×Dic7 [×28], C22×D7 [×7], C2×C42⋊C2, C4○D28 [×4], C22×Dic7 [×14], C23×D7, C23.21D14 [×4], C2×C4○D28 [×2], C23×Dic7, C2×C23.21D14

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >

Smallest permutation representation
►On 224 points
Generators in S224
(1 139)(2 140)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 119)(10 120)(11 121)(12 122)(13 123)(14 124)(15 125)(16 126)(17 127)(18 128)(19 129)(20 130)(21 131)(22 132)(23 133)(24 134)(25 135)(26 136)(27 137)(28 138)(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)(57 144)(58 145)(59 146)(60 147)(61 148)(62 149)(63 150)(64 151)(65 152)(66 153)(67 154)(68 155)(69 156)(70 157)(71 158)(72 159)(73 160)(74 161)(75 162)(76 163)(77 164)(78 165)(79 166)(80 167)(81 168)(82 141)(83 142)(84 143)(169 203)(170 204)(171 205)(172 206)(173 207)(174 208)(175 209)(176 210)(177 211)(178 212)(179 213)(180 214)(181 215)(182 216)(183 217)(184 218)(185 219)(186 220)(187 221)(188 222)(189 223)(190 224)(191 197)(192 198)(193 199)(194 200)(195 201)(196 202)

(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)

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

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

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

Matrix representation ►G ⊆ GL5(𝔽29)

280000
028000
002800
000280
000028
,
280000
01000
002800
00010
000028
,
280000
028000
002800
00010
00001
,
10000
028000
002800
000280
000028
,
280000
012000
001200
000140
00002
,
170000
001200
017000
00001
000280

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,28],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,14,0,0,0,0,0,2],[17,0,0,0,0,0,0,17,0,0,0,12,0,0,0,0,0,0,0,28,0,0,0,1,0] >;
 

136 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I4J4K4L4M···4AB7A7B7C14A···14AS28A···28AV
order12···222224···444444···477714···1428···28
size11···122221···1222214···142222···22···2

136 irreducible representations

dim1111111222222
type+++++++-++
imageC1C2C2C2C2C2C4D7C4○D4Dic7D14D14C4○D28
kernelC2×C23.21D14C2×C4×Dic7C2×C4⋊Dic7C23.21D14C2×C23.D7C23×C28C22×C28C23×C4C2×C14C22×C4C22×C4C24C22
# reps12282116382418348

In GAP, Magma, Sage, TeX 

C_2\times C_2^3._{21}D_{14}
 
% in TeX
 
G:=Group("C2xC2^3.21D14");
 
// GroupNames label
 
G:=SmallGroup(448,1239);
 
// by ID
 
G=gap.SmallGroup(448,1239);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,184,1123,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=d,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^13>;
 
// generators/relations
 

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