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G = C2×C23.23D14  order 448 = 26·7

Direct product of C2 and C23.23D14

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

Aliases: C2×C23.23D14, C24.70D14, (C23×C4)⋊4D7, (C23×C28)⋊4C2, (C22×C4)⋊43D14, D14⋊C4⋊41C22, (C2×C28).704C23, (C2×C14).287C24, Dic7⋊C4⋊44C22, (C22×C28)⋊56C22, (C22×C14).205D4, C14.133(C22×D4), C23.91(C7⋊D4), C23.D7⋊55C22, C22.82(C4○D28), C14⋊4(C22.D4), (C23×D7).74C22, C22.302(C23×D7), C23.233(C22×D7), (C23×C14).109C22, (C22×C14).416C23, (C2×Dic7).149C23, (C22×D7).125C23, (C22×Dic7).161C22, (C2×D14⋊C4)⋊13C2, C2.70(C2×C4○D28), C14.62(C2×C4○D4), C2.6(C22×C7⋊D4), C7⋊5(C2×C22.D4), (C2×Dic7⋊C4)⋊18C2, (C2×C14).574(C2×D4), (C2×C23.D7)⋊22C2, (C2×C4).657(C22×D7), (C22×C7⋊D4).13C2, C22.103(C2×C7⋊D4), (C2×C14).113(C4○D4), (C2×C7⋊D4).136C22, SmallGroup(448,1242)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C2×C23.23D14
Chief series
C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C22×C7⋊D4 — C2×C23.23D14
Lower central
C7 — C2×C14 — C2×C23.23D14

Subgroups: 1412 in 342 conjugacy classes, 127 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×6], C4 [×10], C22, C22 [×10], C22 [×22], C7, C2×C4 [×4], C2×C4 [×24], D4 [×8], C23, C23 [×6], C23 [×12], D7 [×2], C14, C14 [×6], C14 [×4], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4 [×6], C22×C4 [×7], C2×D4 [×8], C24, C24, Dic7 [×6], C28 [×4], D14 [×10], C2×C14, C2×C14 [×10], C2×C14 [×12], C2×C22⋊C4 [×3], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, C2×Dic7 [×6], C2×Dic7 [×6], C7⋊D4 [×8], C2×C28 [×4], C2×C28 [×12], C22×D7 [×2], C22×D7 [×6], C22×C14, C22×C14 [×6], C22×C14 [×4], C2×C22.D4, Dic7⋊C4 [×8], D14⋊C4 [×8], C23.D7 [×4], C22×Dic7, C22×Dic7 [×2], C2×C7⋊D4 [×4], C2×C7⋊D4 [×4], C22×C28 [×6], C22×C28 [×4], C23×D7, C23×C14, C2×Dic7⋊C4 [×2], C2×D14⋊C4 [×2], C23.23D14 [×8], C2×C23.D7, C22×C7⋊D4, C23×C28, C2×C23.23D14

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×4], C24, D14 [×7], C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C7⋊D4 [×4], C22×D7 [×7], C2×C22.D4, C4○D28 [×4], C2×C7⋊D4 [×6], C23×D7, C23.23D14 [×4], C2×C4○D28 [×2], C22×C7⋊D4, C2×C23.23D14

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=ce13 >

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽29)

280000
028000
002800
000280
000028
,
10000
01000
00100
000280
00071
,
10000
028000
002800
00010
00001
,
10000
01000
00100
000280
000028
,
10000
002800
028000
00020
0001314
,
10000
018800
0211100
000426
0002525

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],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,7,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,0,28,0,0,0,28,0,0,0,0,0,0,2,13,0,0,0,0,14],[1,0,0,0,0,0,18,21,0,0,0,8,11,0,0,0,0,0,4,25,0,0,0,26,25] >;
 

124 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4H4I···4N7A7B7C14A···14AS28A···28AV
order12···22222224···44···477714···1428···28
size11···1222228282···228···282222···22···2

124 irreducible representations

dim11111112222222
type+++++++++++
imageC1C2C2C2C2C2C2D4D7C4○D4D14D14C7⋊D4C4○D28
kernelC2×C23.23D14C2×Dic7⋊C4C2×D14⋊C4C23.23D14C2×C23.D7C22×C7⋊D4C23×C28C22×C14C23×C4C2×C14C22×C4C24C23C22
# reps12281114381832448

In GAP, Magma, Sage, TeX 

C_2\times C_2^3._{23}D_{14}
 
% in TeX
 
G:=Group("C2xC2^3.23D14");
 
// GroupNames label
 
G:=SmallGroup(448,1242);
 
// by ID
 
G=gap.SmallGroup(448,1242);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,675,136,18822]);
 
// 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=c*e^13>;
 
// generators/relations
 

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