C7xC4^2.29C2^2 - GroupNames

G = C₇×C₄².₂₉C₂² order 448 = 2⁶·7

Direct product of C₇ and C₄².₂₉C₂²

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C₇×C₄².₂₉C₂², C₈⋊C₄⋊₁₁C₁₄, C₄⋊₁D₄.₅C₁₄, (C₂×C₂₈).₃₄₁D₄, C₄².C₂⋊₃C₁₄, D₄⋊C₄⋊₁₉C₁₄, C₄².₂₇(C₂×C₁₄), C₂₈.₂₇₂(C₄○D₄), (C₄×C₂₈).₂₆₉C₂², (C₂×C₅₆).₃₃₆C₂², (C₂×C₂₈).₉₄₆C₂³, C₂².₁₁₁(D₄×C₁₄), C₁₄.₁₄₆(C₈⋊C₂²), C₁₄.₇₅(C₄.₄D₄), (D₄×C₁₄).₂₀₁C₂², (C₇×C₈⋊C₄)⋊₂₅C₂, C₄.₁₇(C₇×C₄○D₄), (C₂×C₄).₄₂(C₇×D₄), C₄⋊C₄.₂₁(C₂×C₁₄), (C₂×C₈).₅₇(C₂×C₁₄), C₂.₂₁(C₇×C₈⋊C₂²), (C₇×D₄⋊C₄)⋊₄₂C₂, (C₂×D₄).₂₄(C₂×C₁₄), (C₂×C₁₄).₆₆₇(C₂×D₄), (C₇×C₄⋊₁D₄).₁₂C₂, (C₇×C₄².C₂)⋊₂₀C₂, C₂.₁₃(C₇×C₄.₄D₄), (C₇×C₄⋊C₄).₂₄₁C₂², (C₂×C₄).₁₂₁(C₂²×C₁₄), SmallGroup(448,898)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₄ — C₇×C₄².₂₉C₂²

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₂₈ — D₄×C₁₄ — C₇×D₄⋊C₄ — C₇×C₄².₂₉C₂²

Lower central

C₁ — C₂ — C₂×C₄ — C₇×C₄².₂₉C₂²

Upper central

C₁ — C₂×C₁₄ — C₄×C₂₈ — C₇×C₄².₂₉C₂²

Generators and relations for C₇×C₄².₂₉C₂²
G = < a,b,c,d,e | a⁷=b⁴=c⁴=d²=1, e²=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b^-1, ebe^-1=bc², dcd=c^-1, ce=ec, ede^-1=b²c^-1d >

Subgroups: 242 in 110 conjugacy classes, 50 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, C₂³, C₁₄, C₁₄, C₁₄, C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×D₄, C₂×D₄, C₂₈, C₂₈, C₂×C₁₄, C₂×C₁₄, C₈⋊C₄, D₄⋊C₄, C₄².C₂, C₄⋊₁D₄, C₅₆, C₂×C₂₈, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₂²×C₁₄, C₄².₂₉C₂², C₄×C₂₈, C₇×C₄⋊C₄, C₇×C₄⋊C₄, C₂×C₅₆, D₄×C₁₄, D₄×C₁₄, C₇×C₈⋊C₄, C₇×D₄⋊C₄, C₇×C₄².C₂, C₇×C₄⋊₁D₄, C₇×C₄².₂₉C₂²
Quotients: C₁, C₂, C₂², C₇, D₄, C₂³, C₁₄, C₂×D₄, C₄○D₄, C₂×C₁₄, C₄.₄D₄, C₈⋊C₂², C₇×D₄, C₂²×C₁₄, C₄².₂₉C₂², D₄×C₁₄, C₇×C₄○D₄, C₇×C₄.₄D₄, C₇×C₈⋊C₂², C₇×C₄².₂₉C₂²

Smallest permutation representation of C₇×C₄².₂₉C₂²
►On 224 points

Generators in S₂₂₄

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

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

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

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

(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)

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

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

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

112 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	7A	···	7F	8A	8B	8C	8D	14A	···	14R	14S	···	14AD	28A	···	28L	28M	···	28X	28Y	···	28AJ	56A	···	56X
order	1	2	2	2	2	2	4	4	4	4	4	4	7	···	7	8	8	8	8	14	···	14	14	···	14	28	···	28	28	···	28	28	···	28	56	···	56
size	1	1	1	1	8	8	2	2	4	4	8	8	1	···	1	4	4	4	4	1	···	1	8	···	8	2	···	2	4	···	4	8	···	8	4	···	4

112 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+	+	+						+				+
image	C₁	C₂	C₂	C₂	C₂	C₇	C₁₄	C₁₄	C₁₄	C₁₄	D₄	C₄○D₄	C₇×D₄	C₇×C₄○D₄	C₈⋊C₂²	C₇×C₈⋊C₂²
kernel	C₇×C₄².₂₉C₂²	C₇×C₈⋊C₄	C₇×D₄⋊C₄	C₇×C₄².C₂	C₇×C₄⋊₁D₄	C₄².₂₉C₂²	C₈⋊C₄	D₄⋊C₄	C₄².C₂	C₄⋊₁D₄	C₂×C₂₈	C₂₈	C₂×C₄	C₄	C₁₄	C₂
# reps	1	1	4	1	1	6	6	24	6	6	2	4	12	24	2	12

Matrix representation of C₇×C₄².₂₉C₂² ►in GL₆(𝔽₁₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	106	0	0	0
0	0	0	106	0	0
0	0	0	0	106	0
0	0	0	0	0	106

0	98	0	0	0	0
98	0	0	0	0	0
0	0	61	0	0	78
0	0	0	61	74	35
0	0	78	78	52	0
0	0	74	0	0	52

112	0	0	0	0	0
0	112	0	0	0	0
0	0	1	2	0	0
0	0	112	112	0	0
0	0	0	0	1	2
0	0	0	0	112	112

1	0	0	0	0	0
0	112	0	0	0	0
0	0	1	0	0	0
0	0	112	112	0	0
0	0	0	0	112	111
0	0	0	0	0	1

98	0	0	0	0	0
0	98	0	0	0	0
0	0	0	35	61	9
0	0	39	78	52	52
0	0	61	0	0	78
0	0	0	61	74	35

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,106,0,0,0,0,0,0,106,0,0,0,0,0,0,106,0,0,0,0,0,0,106],[0,98,0,0,0,0,98,0,0,0,0,0,0,0,61,0,78,74,0,0,0,61,78,0,0,0,0,74,52,0,0,0,78,35,0,52],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,112,0,0,0,0,2,112,0,0,0,0,0,0,1,112,0,0,0,0,2,112],[1,0,0,0,0,0,0,112,0,0,0,0,0,0,1,112,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,111,1],[98,0,0,0,0,0,0,98,0,0,0,0,0,0,0,39,61,0,0,0,35,78,0,61,0,0,61,52,0,74,0,0,9,52,78,35] >;

C₇×C₄².₂₉C₂² in GAP, Magma, Sage, TeX

C_7\times C_4^2._{29}C_2^2

% in TeX

G:=Group("C7xC4^2.29C2^2");

// GroupNames label

G:=SmallGroup(448,898);

// by ID

G=gap.SmallGroup(448,898);

# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,2360,2438,2403,310,9804,172,14117,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^4=c^4=d^2=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d=b^-1,e*b*e^-1=b*c^2,d*c*d=c^-1,c*e=e*c,e*d*e^-1=b^2*c^-1*d>;

// generators/relations

G = C7×C42.29C22 order 448 = 26·7

Direct product of C7 and C42.29C22

G = C₇×C₄².₂₉C₂² order 448 = 2⁶·7

Direct product of C₇ and C₄².₂₉C₂²