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G = C2×C23.11D14  order 448 = 26·7

Direct product of C2 and C23.11D14

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

Aliases: C2×C23.11D14, C24.50D14, C14.6(C23×C4), C23.57(C4×D7), (C2×C14).24C24, C14⋊2(C42⋊C2), (C2×C28).569C23, Dic7⋊C4⋊56C22, C22⋊C4.122D14, (C22×Dic7)⋊10C4, (C4×Dic7)⋊71C22, (C22×C4).311D14, (C23×Dic7).7C2, C22.16(C23×D7), (C23×C14).50C22, Dic7.22(C22×C4), C23.143(C22×D7), C23.D7.82C22, C22.63(D4⋊2D7), (C22×C28).349C22, (C22×C14).386C23, (C2×Dic7).302C23, (C22×Dic7).243C22, C2.8(D7×C22×C4), C7⋊2(C2×C42⋊C2), (C2×C4×Dic7)⋊28C2, C22.23(C2×C4×D7), C14.65(C2×C4○D4), C2.1(C2×D4⋊2D7), (C2×Dic7⋊C4)⋊33C2, (C2×Dic7)⋊21(C2×C4), (C2×C22⋊C4).20D7, (C14×C22⋊C4).25C2, (C2×C14).17(C22×C4), (C22×C14).76(C2×C4), (C2×C4).254(C22×D7), (C2×C23.D7).19C2, (C2×C14).165(C4○D4), (C7×C22⋊C4).132C22, SmallGroup(448,933)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — C2×C23.11D14
Chief series
C1 — C7 — C14 — C2×C14 — C2×Dic7 — C22×Dic7 — C23×Dic7 — C2×C23.11D14
Lower central
C7 — C14 — C2×C23.11D14
Upper central
C1 — C23 — C2×C22⋊C4

Generators and relations for C2×C23.11D14
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >

Subgroups: 1044 in 330 conjugacy classes, 167 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic7, Dic7, C28, C2×C14, C2×C14, C2×C14, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C2×C42⋊C2, C4×Dic7, Dic7⋊C4, C23.D7, C7×C22⋊C4, C22×Dic7, C22×Dic7, C22×C28, C23×C14, C23.11D14, C2×C4×Dic7, C2×Dic7⋊C4, C2×C23.D7, C14×C22⋊C4, C23×Dic7, C2×C23.11D14
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C4○D4, C24, D14, C42⋊C2, C23×C4, C2×C4○D4, C4×D7, C22×D7, C2×C42⋊C2, C2×C4×D7, D4⋊2D7, C23×D7, C23.11D14, D7×C22×C4, C2×D4⋊2D7, C2×C23.11D14

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

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

(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)

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

(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)

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

100 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P4Q···4AB7A7B7C14A···14U14V···14AG28A···28X
order12···222224···44···44···477714···1414···1428···28
size11···122222···27···714···142222···24···44···4

100 irreducible representations

dim111111112222224
type+++++++++++-
imageC1C2C2C2C2C2C2C4D7C4○D4D14D14D14C4×D7D4⋊2D7
kernelC2×C23.11D14C23.11D14C2×C4×Dic7C2×Dic7⋊C4C2×C23.D7C14×C22⋊C4C23×Dic7C22×Dic7C2×C22⋊C4C2×C14C22⋊C4C22×C4C24C23C22
# reps1822111163812632412

Matrix representation of C2×C23.11D14 ►in GL5(𝔽29)

280000
01000
00100
000280
000028
,
10000
01000
00100
00010
0002428
,
280000
028000
002800
00010
00001
,
10000
01000
00100
000280
000028
,
170000
026400
0252500
0001228
0002717
,
120000
012000
0261700
000112
0002428

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,24,0,0,0,0,28],[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],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[17,0,0,0,0,0,26,25,0,0,0,4,25,0,0,0,0,0,12,27,0,0,0,28,17],[12,0,0,0,0,0,12,26,0,0,0,0,17,0,0,0,0,0,1,24,0,0,0,12,28] >;
 

C2×C23.11D14 in GAP, Magma, Sage, TeX 

C_2\times C_2^3._{11}D_{14}
 
% in TeX
 
G:=Group("C2xC2^3.11D14");
 
// GroupNames label
 
G:=SmallGroup(448,933);
 
// by ID
 
G=gap.SmallGroup(448,933);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,297,80,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=c,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,e*b*e^-1=f*b*f^-1=b*d=d*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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