Copied to
clipboard

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×C14).287C24, (C2×C28).704C23, Dic7⋊C4⋊44C22, (C22×C28)⋊56C22, C14.133(C22×D4), (C22×C14).205D4, 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, C14.62(C2×C4○D4), C2.70(C2×C4○D28), 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
Upper central
C1 — C23 — C23×C4

Generators and relations for C2×C23.23D14
 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 >

Subgroups: 1412 in 342 conjugacy classes, 127 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C23×C4, C22×D4, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C2×C22.D4, Dic7⋊C4, D14⋊C4, C23.D7, C22×Dic7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, C22×C28, C23×D7, C23×C14, C2×Dic7⋊C4, C2×D14⋊C4, C23.23D14, C2×C23.D7, C22×C7⋊D4, C23×C28, C2×C23.23D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22.D4, C22×D4, C2×C4○D4, C7⋊D4, C22×D7, C2×C22.D4, C4○D28, C2×C7⋊D4, C23×D7, C23.23D14, C2×C4○D28, C22×C7⋊D4, C2×C23.23D14

Smallest permutation representation of C2×C23.23D14
►On 224 points
Generators in S224
(1 187)(2 188)(3 189)(4 190)(5 191)(6 192)(7 193)(8 194)(9 195)(10 196)(11 169)(12 170)(13 171)(14 172)(15 173)(16 174)(17 175)(18 176)(19 177)(20 178)(21 179)(22 180)(23 181)(24 182)(25 183)(26 184)(27 185)(28 186)(29 132)(30 133)(31 134)(32 135)(33 136)(34 137)(35 138)(36 139)(37 140)(38 113)(39 114)(40 115)(41 116)(42 117)(43 118)(44 119)(45 120)(46 121)(47 122)(48 123)(49 124)(50 125)(51 126)(52 127)(53 128)(54 129)(55 130)(56 131)(57 165)(58 166)(59 167)(60 168)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)(81 161)(82 162)(83 163)(84 164)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)

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

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

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

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

Matrix representation of C2×C23.23D14 ►in 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] >;
 

C2×C23.23D14 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
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁