C2^4.6D14 - GroupNames

G = C₂⁴.₆D₁₄ order 448 = 2⁶·7

6^th non-split extension by C₂⁴ of D₁₄ acting via D₁₄/C₇=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴.₆D₁₄, C₂³.₄Dic₁₄, (C₂×C₂₈).₄₉D₄, C₂.₆(C₂₈⋊₂D₄), (C₂×Dic₇).₅₄D₄, C₂².₂₃₈(D₄×D₇), (C₂²×C₄).₂₈D₁₄, (C₂²×C₁₄).₁₂Q₈, C₁₄.₂₉(C₄⋊D₄), C₇⋊₂(C₂³.Q₈), C₁₄.₅₆(C₂²⋊Q₈), C₂.₃₂(D₁₄⋊D₄), C₁₄.C₄²⋊₁₃C₂, C₂.₈(Dic₇⋊D₄), C₂.₇(C₂₈.₄₈D₄), C₂².₉₅(C₄○D₂₈), (C₂³×C₁₄).₃₁C₂², (C₂²×C₂₈).₅₇C₂², C₂².₄₅(C₂×Dic₁₄), C₂³.₃₆₇(C₂²×D₇), C₁₄.₁₃(C₄²⋊₂C₂), C₂².₉₃(D₄⋊₂D₇), (C₂²×C₁₄).₃₂₃C₂³, C₂.₂₁(C₂²⋊Dic₁₄), C₂.₁₂(C₂³.D₁₄), (C₂²×Dic₇).₃₉C₂², (C₂×C₄⋊Dic₇)⋊₈C₂, (C₂×C₁₄).₃₃(C₂×Q₈), (C₂×Dic₇⋊C₄)⋊₂₀C₂, (C₂×C₁₄).₃₁₇(C₂×D₄), (C₂×C₄).₂₈(C₇⋊D₄), (C₂×C₂²⋊C₄).₁₁D₇, (C₂×C₁₄).₇₇(C₄○D₄), (C₁₄×C₂²⋊C₄).₁₃C₂, C₂².₁₂₃(C₂×C₇⋊D₄), (C₂×C₂³.D₇).₁₀C₂, SmallGroup(448,482)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₄ — C₂⁴.₆D₁₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×C₁₄ — C₂²×Dic₇ — C₂×Dic₇⋊C₄ — C₂⁴.₆D₁₄

Lower central

C₇ — C₂²×C₁₄ — C₂⁴.₆D₁₄

Upper central

C₁ — C₂³ — C₂×C₂²⋊C₄

Generators and relations for C₂⁴.₆D₁₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e¹⁴=f²=dc=cd, ab=ba, ac=ca, eae^-1=ad=da, faf^-1=abd, bc=cb, bd=db, be=eb, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef^-1=ce¹³ >

Subgroups: 756 in 186 conjugacy classes, 63 normal (51 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₁₄, C₁₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, Dic₇, C₂₈, C₂×C₁₄, C₂×C₁₄, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₂²×C₁₄, C₂²×C₁₄, C₂²×C₁₄, C₂³.Q₈, Dic₇⋊C₄, C₄⋊Dic₇, C₂³.D₇, C₇×C₂²⋊C₄, C₂²×Dic₇, C₂²×C₂₈, C₂³×C₁₄, C₁₄.C₄², C₂×Dic₇⋊C₄, C₂×C₄⋊Dic₇, C₂×C₂³.D₇, C₁₄×C₂²⋊C₄, C₂⁴.₆D₁₄
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₇, C₂×D₄, C₂×Q₈, C₄○D₄, D₁₄, C₄⋊D₄, C₂²⋊Q₈, C₄²⋊₂C₂, Dic₁₄, C₇⋊D₄, C₂²×D₇, C₂³.Q₈, C₂×Dic₁₄, C₄○D₂₈, D₄×D₇, D₄⋊₂D₇, C₂×C₇⋊D₄, C₂²⋊Dic₁₄, C₂³.D₁₄, D₁₄⋊D₄, C₂₈.₄₈D₄, C₂₈⋊₂D₄, Dic₇⋊D₄, C₂⁴.₆D₁₄

Smallest permutation representation of C₂⁴.₆D₁₄
►On 224 points

Generators in S₂₂₄

(2 205)(4 207)(6 209)(8 211)(10 213)(12 215)(14 217)(16 219)(18 221)(20 223)(22 197)(24 199)(26 201)(28 203)(30 84)(32 58)(34 60)(36 62)(38 64)(40 66)(42 68)(44 70)(46 72)(48 74)(50 76)(52 78)(54 80)(56 82)(85 180)(86 133)(87 182)(88 135)(89 184)(90 137)(91 186)(92 139)(93 188)(94 113)(95 190)(96 115)(97 192)(98 117)(99 194)(100 119)(101 196)(102 121)(103 170)(104 123)(105 172)(106 125)(107 174)(108 127)(109 176)(110 129)(111 178)(112 131)(114 161)(116 163)(118 165)(120 167)(122 141)(124 143)(126 145)(128 147)(130 149)(132 151)(134 153)(136 155)(138 157)(140 159)(142 171)(144 173)(146 175)(148 177)(150 179)(152 181)(154 183)(156 185)(158 187)(160 189)(162 191)(164 193)(166 195)(168 169)

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

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

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

(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)

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

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

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

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

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

Matrix representation of C₂⁴.₆D₁₄ ►in GL₆(𝔽₂₉)

1	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	20	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	28	0
0	0	0	0	0	28

28	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	1	0
0	0	0	0	0	1

1	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	28	0
0	0	0	0	0	28

8	0	0	0	0	0
0	11	0	0	0	0
0	0	1	27	0	0
0	0	1	28	0	0
0	0	0	0	27	6
0	0	0	0	4	2

0	18	0	0	0	0
8	0	0	0	0	0
0	0	9	22	0	0
0	0	20	20	0	0
0	0	0	0	17	0
0	0	0	0	0	17

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,20,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[8,0,0,0,0,0,0,11,0,0,0,0,0,0,1,1,0,0,0,0,27,28,0,0,0,0,0,0,27,4,0,0,0,0,6,2],[0,8,0,0,0,0,18,0,0,0,0,0,0,0,9,20,0,0,0,0,22,20,0,0,0,0,0,0,17,0,0,0,0,0,0,17] >;

C₂⁴.₆D₁₄ in GAP, Magma, Sage, TeX

C_2^4._6D_{14}

% in TeX

G:=Group("C2^4.6D14");

// GroupNames label

G:=SmallGroup(448,482);

// by ID

G=gap.SmallGroup(448,482);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,344,254,387,18822]);

// Polycyclic

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

G = C24.6D14 order 448 = 26·7

6th non-split extension by C24 of D14 acting via D14/C7=C22

G = C₂⁴.₆D₁₄ order 448 = 2⁶·7

6^th non-split extension by C₂⁴ of D₁₄ acting via D₁₄/C₇=C₂²