C2^3:Dic14 - GroupNames

G = C₂³⋊Dic₁₄ order 448 = 2⁶·7

1^st semidirect product of C₂³ and Dic₁₄ acting via Dic₁₄/C₁₄=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴.₅D₁₄, C₂³⋊₁Dic₁₄, (C₂×C₂₈).₄₈D₄, (C₂²×C₁₄)⋊₃Q₈, C₇⋊₂(C₂³⋊Q₈), C₁₄.₃₆C₂²≀C₂, (C₂×Dic₇).₅₃D₄, C₂².₂₃₇(D₄×D₇), (C₂²×C₄).₂₇D₁₄, C₂.₆(C₂³⋊D₁₄), (C₂²×Dic₁₄)⋊₃C₂, C₁₄.₅₅(C₂²⋊Q₈), C₁₄.C₄²⋊₂₈C₂, C₂.₆(C₂₈.₄₈D₄), C₁₄.₃₁(C₄.₄D₄), C₂.₅(C₂₈.₁₇D₄), C₂².₉₄(C₄○D₂₈), (C₂³×C₁₄).₃₀C₂², (C₂²×C₂₈).₅₆C₂², C₂².₄₄(C₂×Dic₁₄), C₂³.₃₆₆(C₂²×D₇), C₂².₉₂(D₄⋊₂D₇), (C₂²×C₁₄).₃₂₂C₂³, C₂.₂₀(Dic₇.D₄), C₂.₂₀(C₂²⋊Dic₁₄), (C₂²×Dic₇).₃₈C₂², (C₂×C₁₄).₃₂(C₂×Q₈), (C₂×C₁₄).₃₁₆(C₂×D₄), (C₂×C₄).₂₇(C₇⋊D₄), (C₂×C₂²⋊C₄).₁₀D₇, (C₂×C₂³.D₇).₉C₂, (C₂×C₁₄).₇₆(C₄○D₄), (C₁₄×C₂²⋊C₄).₁₂C₂, C₂².₁₂₂(C₂×C₇⋊D₄), SmallGroup(448,481)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₄ — C₂³⋊Dic₁₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×C₁₄ — C₂²×Dic₇ — C₂²×Dic₁₄ — C₂³⋊Dic₁₄

Lower central

C₇ — C₂²×C₁₄ — C₂³⋊Dic₁₄

Upper central

C₁ — C₂³ — C₂×C₂²⋊C₄

Generators and relations for C₂³⋊Dic₁₄
G = < a,b,c,d,e | a²=b²=c²=d²⁸=1, e²=d¹⁴, eae^-1=ab=ba, dad^-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 884 in 202 conjugacy classes, 63 normal (27 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₇, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₁₄, C₁₄, C₁₄, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, Dic₇, C₂₈, C₂×C₁₄, C₂×C₁₄, C₂×C₁₄, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂²×Q₈, Dic₁₄, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₂²×C₁₄, C₂²×C₁₄, C₂²×C₁₄, C₂³⋊Q₈, C₂³.D₇, C₇×C₂²⋊C₄, C₂×Dic₁₄, C₂²×Dic₇, C₂²×C₂₈, C₂³×C₁₄, C₁₄.C₄², C₁₄.C₄², C₂×C₂³.D₇, C₁₄×C₂²⋊C₄, C₂²×Dic₁₄, C₂³⋊Dic₁₄
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₇, C₂×D₄, C₂×Q₈, C₄○D₄, D₁₄, C₂²≀C₂, C₂²⋊Q₈, C₄.₄D₄, Dic₁₄, C₇⋊D₄, C₂²×D₇, C₂³⋊Q₈, C₂×Dic₁₄, C₄○D₂₈, D₄×D₇, D₄⋊₂D₇, C₂×C₇⋊D₄, C₂²⋊Dic₁₄, Dic₇.D₄, C₂₈.₄₈D₄, C₂₈.₁₇D₄, C₂³⋊D₁₄, C₂³⋊Dic₁₄

Smallest permutation representation of C₂³⋊Dic₁₄
►On 224 points

Generators in S₂₂₄

(2 164)(4 166)(6 168)(8 142)(10 144)(12 146)(14 148)(16 150)(18 152)(20 154)(22 156)(24 158)(26 160)(28 162)(29 179)(30 75)(31 181)(32 77)(33 183)(34 79)(35 185)(36 81)(37 187)(38 83)(39 189)(40 57)(41 191)(42 59)(43 193)(44 61)(45 195)(46 63)(47 169)(48 65)(49 171)(50 67)(51 173)(52 69)(53 175)(54 71)(55 177)(56 73)(58 138)(60 140)(62 114)(64 116)(66 118)(68 120)(70 122)(72 124)(74 126)(76 128)(78 130)(80 132)(82 134)(84 136)(86 203)(88 205)(90 207)(92 209)(94 211)(96 213)(98 215)(100 217)(102 219)(104 221)(106 223)(108 197)(110 199)(112 201)(113 194)(115 196)(117 170)(119 172)(121 174)(123 176)(125 178)(127 180)(129 182)(131 184)(133 186)(135 188)(137 190)(139 192)

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

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

(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)

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

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

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

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

82 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	4B	4C	4D	4E	···	4L	7A	7B	7C	14A	···	14U	14V	···	14AG	28A	···	28X
order	1	2	···	2	2	2	4	4	4	4	4	···	4	7	7	7	14	···	14	14	···	14	28	···	28
size	1	1	···	1	4	4	4	4	4	4	28	···	28	2	2	2	2	···	2	4	···	4	4	···	4

82 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	-	+		+	+		-		+	-
image	C₁	C₂	C₂	C₂	C₂	D₄	D₄	Q₈	D₇	C₄○D₄	D₁₄	D₁₄	C₇⋊D₄	Dic₁₄	C₄○D₂₈	D₄×D₇	D₄⋊₂D₇
kernel	C₂³⋊Dic₁₄	C₁₄.C₄²	C₂×C₂³.D₇	C₁₄×C₂²⋊C₄	C₂²×Dic₁₄	C₂×Dic₇	C₂×C₂₈	C₂²×C₁₄	C₂×C₂²⋊C₄	C₂×C₁₄	C₂²×C₄	C₂⁴	C₂×C₄	C₂³	C₂²	C₂²	C₂²
# reps	1	3	2	1	1	4	2	2	3	6	6	3	12	12	12	6	6

Matrix representation of C₂³⋊Dic₁₄ ►in GL₆(𝔽₂₉)

1	0	0	0	0	0
28	28	0	0	0	0
0	0	1	0	0	0
0	0	16	28	0	0
0	0	0	0	1	0
0	0	0	0	11	28

1	0	0	0	0	0
0	1	0	0	0	0
0	0	28	0	0	0
0	0	0	28	0	0
0	0	0	0	28	0
0	0	0	0	0	28

28	0	0	0	0	0
0	28	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

28	27	0	0	0	0
1	1	0	0	0	0
0	0	16	0	0	0
0	0	26	20	0	0
0	0	0	0	14	0
0	0	0	0	1	27

17	0	0	0	0	0
12	12	0	0	0	0
0	0	16	27	0	0
0	0	26	13	0	0
0	0	0	0	27	3
0	0	0	0	8	2

G:=sub<GL(6,GF(29))| [1,28,0,0,0,0,0,28,0,0,0,0,0,0,1,16,0,0,0,0,0,28,0,0,0,0,0,0,1,11,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,1,0,0,0,0,27,1,0,0,0,0,0,0,16,26,0,0,0,0,0,20,0,0,0,0,0,0,14,1,0,0,0,0,0,27],[17,12,0,0,0,0,0,12,0,0,0,0,0,0,16,26,0,0,0,0,27,13,0,0,0,0,0,0,27,8,0,0,0,0,3,2] >;

C₂³⋊Dic₁₄ in GAP, Magma, Sage, TeX

C_2^3\rtimes {\rm Dic}_{14}

% in TeX

G:=Group("C2^3:Dic14");

// GroupNames label

G:=SmallGroup(448,481);

// by ID

G=gap.SmallGroup(448,481);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,120,254,387,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=1,e^2=d^14,e*a*e^-1=a*b=b*a,d*a*d^-1=a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;

// generators/relations

G = C23⋊Dic14 order 448 = 26·7

1st semidirect product of C23 and Dic14 acting via Dic14/C14=C22

G = C₂³⋊Dic₁₄ order 448 = 2⁶·7

1^st semidirect product of C₂³ and Dic₁₄ acting via Dic₁₄/C₁₄=C₂²