C7xC4^2.28C2^2 - GroupNames

G = C₇×C₄².₂₈C₂² order 448 = 2⁶·7

Direct product of C₇ and C₄².₂₈C₂²

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

Aliases: C₇×C₄².₂₈C₂², C₄⋊Q₈⋊₆C₁₄, C₈⋊C₄⋊₁₀C₁₄, (C₂×C₂₈).₃₄₀D₄, Q₈⋊C₄⋊₁₉C₁₄, D₄⋊C₄.₇C₁₄, C₄².₂₆(C₂×C₁₄), C₄.₄D₄.₆C₁₄, C₂₈.₂₇₁(C₄○D₄), (C₄×C₂₈).₂₆₈C₂², (C₂×C₅₆).₃₃₅C₂², (C₂×C₂₈).₉₄₅C₂³, C₂².₁₁₀(D₄×C₁₄), C₁₄.₁₄₅(C₈⋊C₂²), C₁₄.₇₄(C₄.₄D₄), (D₄×C₁₄).₂₀₀C₂², (Q₈×C₁₄).₁₇₄C₂², C₁₄.₁₄₅(C₈.C₂²), (C₇×C₄⋊Q₈)⋊₂₇C₂, (C₇×C₈⋊C₄)⋊₂₄C₂, C₄.₁₆(C₇×C₄○D₄), (C₂×C₄).₄₁(C₇×D₄), C₄⋊C₄.₂₀(C₂×C₁₄), (C₂×C₈).₅₆(C₂×C₁₄), C₂.₂₀(C₇×C₈⋊C₂²), (C₇×Q₈⋊C₄)⋊₄₂C₂, (C₂×D₄).₂₃(C₂×C₁₄), (C₂×C₁₄).₆₆₆(C₂×D₄), (C₂×Q₈).₁₈(C₂×C₁₄), C₂.₁₂(C₇×C₄.₄D₄), C₂.₂₀(C₇×C₈.C₂²), (C₇×D₄⋊C₄).₁₆C₂, (C₇×C₄⋊C₄).₂₄₀C₂², (C₇×C₄.₄D₄).₁₅C₂, (C₂×C₄).₁₂₀(C₂²×C₁₄), SmallGroup(448,897)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₄ — C₇×C₄².₂₈C₂²

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₂₈ — D₄×C₁₄ — C₇×D₄⋊C₄ — C₇×C₄².₂₈C₂²

Lower central

C₁ — C₂ — C₂×C₄ — C₇×C₄².₂₈C₂²

Upper central

C₁ — C₂×C₁₄ — C₄×C₂₈ — C₇×C₄².₂₈C₂²

Generators and relations for C₇×C₄².₂₈C₂²
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^-1c², ebe^-1=bc², dcd=c^-1, ce=ec, ede^-1=b²c^-1d >

Subgroups: 194 in 100 conjugacy classes, 50 normal (30 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₁₄, C₁₄, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂₈, C₂₈, C₂×C₁₄, C₂×C₁₄, C₈⋊C₄, D₄⋊C₄, Q₈⋊C₄, C₄.₄D₄, C₄⋊Q₈, C₅₆, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₇×Q₈, C₂²×C₁₄, C₄².₂₈C₂², C₄×C₂₈, C₇×C₂²⋊C₄, C₇×C₄⋊C₄, C₇×C₄⋊C₄, C₂×C₅₆, D₄×C₁₄, Q₈×C₁₄, Q₈×C₁₄, C₇×C₈⋊C₄, C₇×D₄⋊C₄, C₇×Q₈⋊C₄, C₇×C₄.₄D₄, C₇×C₄⋊Q₈, C₇×C₄².₂₈C₂²
Quotients: C₁, C₂, C₂², C₇, D₄, C₂³, C₁₄, C₂×D₄, C₄○D₄, C₂×C₁₄, C₄.₄D₄, C₈⋊C₂², C₈.C₂², C₇×D₄, C₂²×C₁₄, C₄².₂₈C₂², D₄×C₁₄, C₇×C₄○D₄, C₇×C₄.₄D₄, C₇×C₈⋊C₂², C₇×C₈.C₂², C₇×C₄².₂₈C₂²

Smallest permutation representation of C₇×C₄².₂₈C₂²
►On 224 points

Generators in S₂₂₄

(1 157 45 149 37 141 29)(2 158 46 150 38 142 30)(3 159 47 151 39 143 31)(4 160 48 152 40 144 32)(5 153 41 145 33 137 25)(6 154 42 146 34 138 26)(7 155 43 147 35 139 27)(8 156 44 148 36 140 28)(9 59 163 51 129 20 126)(10 60 164 52 130 21 127)(11 61 165 53 131 22 128)(12 62 166 54 132 23 121)(13 63 167 55 133 24 122)(14 64 168 56 134 17 123)(15 57 161 49 135 18 124)(16 58 162 50 136 19 125)(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)(89 117 217 105 209 97 201)(90 118 218 106 210 98 202)(91 119 219 107 211 99 203)(92 120 220 108 212 100 204)(93 113 221 109 213 101 205)(94 114 222 110 214 102 206)(95 115 223 111 215 103 207)(96 116 224 112 216 104 208)

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

(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)

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

(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)| (1,157,45,149,37,141,29)(2,158,46,150,38,142,30)(3,159,47,151,39,143,31)(4,160,48,152,40,144,32)(5,153,41,145,33,137,25)(6,154,42,146,34,138,26)(7,155,43,147,35,139,27)(8,156,44,148,36,140,28)(9,59,163,51,129,20,126)(10,60,164,52,130,21,127)(11,61,165,53,131,22,128)(12,62,166,54,132,23,121)(13,63,167,55,133,24,122)(14,64,168,56,134,17,123)(15,57,161,49,135,18,124)(16,58,162,50,136,19,125)(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)(89,117,217,105,209,97,201)(90,118,218,106,210,98,202)(91,119,219,107,211,99,203)(92,120,220,108,212,100,204)(93,113,221,109,213,101,205)(94,114,222,110,214,102,206)(95,115,223,111,215,103,207)(96,116,224,112,216,104,208), (1,96,169,52)(2,93,170,49)(3,90,171,54)(4,95,172,51)(5,92,173,56)(6,89,174,53)(7,94,175,50)(8,91,176,55)(9,40,215,76)(10,37,216,73)(11,34,209,78)(12,39,210,75)(13,36,211,80)(14,33,212,77)(15,38,213,74)(16,35,214,79)(17,41,220,85)(18,46,221,82)(19,43,222,87)(20,48,223,84)(21,45,224,81)(22,42,217,86)(23,47,218,83)(24,44,219,88)(25,204,69,168)(26,201,70,165)(27,206,71,162)(28,203,72,167)(29,208,65,164)(30,205,66,161)(31,202,67,166)(32,207,68,163)(57,142,101,178)(58,139,102,183)(59,144,103,180)(60,141,104,177)(61,138,97,182)(62,143,98,179)(63,140,99,184)(64,137,100,181)(105,190,128,146)(106,187,121,151)(107,192,122,148)(108,189,123,145)(109,186,124,150)(110,191,125,147)(111,188,126,152)(112,185,127,149)(113,194,135,158)(114,199,136,155)(115,196,129,160)(116,193,130,157)(117,198,131,154)(118,195,132,159)(119,200,133,156)(120,197,134,153), (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), (2,172)(3,7)(4,170)(6,176)(8,174)(9,11)(10,212)(12,210)(13,15)(14,216)(16,214)(17,224)(18,24)(19,222)(20,22)(21,220)(23,218)(26,72)(27,31)(28,70)(30,68)(32,66)(34,80)(35,39)(36,78)(38,76)(40,74)(42,88)(43,47)(44,86)(46,84)(48,82)(49,55)(50,94)(51,53)(52,92)(54,90)(56,96)(57,63)(58,102)(59,61)(60,100)(62,98)(64,104)(67,71)(75,79)(83,87)(89,95)(91,93)(97,103)(99,101)(105,111)(106,121)(107,109)(108,127)(110,125)(112,123)(113,119)(114,136)(115,117)(116,134)(118,132)(120,130)(122,124)(126,128)(129,131)(133,135)(138,184)(139,143)(140,182)(142,180)(144,178)(146,192)(147,151)(148,190)(150,188)(152,186)(154,200)(155,159)(156,198)(158,196)(160,194)(161,167)(162,206)(163,165)(164,204)(166,202)(168,208)(171,175)(179,183)(187,191)(195,199)(201,207)(203,205)(209,215)(211,213)(217,223)(219,221), (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( (1,157,45,149,37,141,29)(2,158,46,150,38,142,30)(3,159,47,151,39,143,31)(4,160,48,152,40,144,32)(5,153,41,145,33,137,25)(6,154,42,146,34,138,26)(7,155,43,147,35,139,27)(8,156,44,148,36,140,28)(9,59,163,51,129,20,126)(10,60,164,52,130,21,127)(11,61,165,53,131,22,128)(12,62,166,54,132,23,121)(13,63,167,55,133,24,122)(14,64,168,56,134,17,123)(15,57,161,49,135,18,124)(16,58,162,50,136,19,125)(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)(89,117,217,105,209,97,201)(90,118,218,106,210,98,202)(91,119,219,107,211,99,203)(92,120,220,108,212,100,204)(93,113,221,109,213,101,205)(94,114,222,110,214,102,206)(95,115,223,111,215,103,207)(96,116,224,112,216,104,208), (1,96,169,52)(2,93,170,49)(3,90,171,54)(4,95,172,51)(5,92,173,56)(6,89,174,53)(7,94,175,50)(8,91,176,55)(9,40,215,76)(10,37,216,73)(11,34,209,78)(12,39,210,75)(13,36,211,80)(14,33,212,77)(15,38,213,74)(16,35,214,79)(17,41,220,85)(18,46,221,82)(19,43,222,87)(20,48,223,84)(21,45,224,81)(22,42,217,86)(23,47,218,83)(24,44,219,88)(25,204,69,168)(26,201,70,165)(27,206,71,162)(28,203,72,167)(29,208,65,164)(30,205,66,161)(31,202,67,166)(32,207,68,163)(57,142,101,178)(58,139,102,183)(59,144,103,180)(60,141,104,177)(61,138,97,182)(62,143,98,179)(63,140,99,184)(64,137,100,181)(105,190,128,146)(106,187,121,151)(107,192,122,148)(108,189,123,145)(109,186,124,150)(110,191,125,147)(111,188,126,152)(112,185,127,149)(113,194,135,158)(114,199,136,155)(115,196,129,160)(116,193,130,157)(117,198,131,154)(118,195,132,159)(119,200,133,156)(120,197,134,153), (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), (2,172)(3,7)(4,170)(6,176)(8,174)(9,11)(10,212)(12,210)(13,15)(14,216)(16,214)(17,224)(18,24)(19,222)(20,22)(21,220)(23,218)(26,72)(27,31)(28,70)(30,68)(32,66)(34,80)(35,39)(36,78)(38,76)(40,74)(42,88)(43,47)(44,86)(46,84)(48,82)(49,55)(50,94)(51,53)(52,92)(54,90)(56,96)(57,63)(58,102)(59,61)(60,100)(62,98)(64,104)(67,71)(75,79)(83,87)(89,95)(91,93)(97,103)(99,101)(105,111)(106,121)(107,109)(108,127)(110,125)(112,123)(113,119)(114,136)(115,117)(116,134)(118,132)(120,130)(122,124)(126,128)(129,131)(133,135)(138,184)(139,143)(140,182)(142,180)(144,178)(146,192)(147,151)(148,190)(150,188)(152,186)(154,200)(155,159)(156,198)(158,196)(160,194)(161,167)(162,206)(163,165)(164,204)(166,202)(168,208)(171,175)(179,183)(187,191)(195,199)(201,207)(203,205)(209,215)(211,213)(217,223)(219,221), (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([[(1,157,45,149,37,141,29),(2,158,46,150,38,142,30),(3,159,47,151,39,143,31),(4,160,48,152,40,144,32),(5,153,41,145,33,137,25),(6,154,42,146,34,138,26),(7,155,43,147,35,139,27),(8,156,44,148,36,140,28),(9,59,163,51,129,20,126),(10,60,164,52,130,21,127),(11,61,165,53,131,22,128),(12,62,166,54,132,23,121),(13,63,167,55,133,24,122),(14,64,168,56,134,17,123),(15,57,161,49,135,18,124),(16,58,162,50,136,19,125),(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),(89,117,217,105,209,97,201),(90,118,218,106,210,98,202),(91,119,219,107,211,99,203),(92,120,220,108,212,100,204),(93,113,221,109,213,101,205),(94,114,222,110,214,102,206),(95,115,223,111,215,103,207),(96,116,224,112,216,104,208)], [(1,96,169,52),(2,93,170,49),(3,90,171,54),(4,95,172,51),(5,92,173,56),(6,89,174,53),(7,94,175,50),(8,91,176,55),(9,40,215,76),(10,37,216,73),(11,34,209,78),(12,39,210,75),(13,36,211,80),(14,33,212,77),(15,38,213,74),(16,35,214,79),(17,41,220,85),(18,46,221,82),(19,43,222,87),(20,48,223,84),(21,45,224,81),(22,42,217,86),(23,47,218,83),(24,44,219,88),(25,204,69,168),(26,201,70,165),(27,206,71,162),(28,203,72,167),(29,208,65,164),(30,205,66,161),(31,202,67,166),(32,207,68,163),(57,142,101,178),(58,139,102,183),(59,144,103,180),(60,141,104,177),(61,138,97,182),(62,143,98,179),(63,140,99,184),(64,137,100,181),(105,190,128,146),(106,187,121,151),(107,192,122,148),(108,189,123,145),(109,186,124,150),(110,191,125,147),(111,188,126,152),(112,185,127,149),(113,194,135,158),(114,199,136,155),(115,196,129,160),(116,193,130,157),(117,198,131,154),(118,195,132,159),(119,200,133,156),(120,197,134,153)], [(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)], [(2,172),(3,7),(4,170),(6,176),(8,174),(9,11),(10,212),(12,210),(13,15),(14,216),(16,214),(17,224),(18,24),(19,222),(20,22),(21,220),(23,218),(26,72),(27,31),(28,70),(30,68),(32,66),(34,80),(35,39),(36,78),(38,76),(40,74),(42,88),(43,47),(44,86),(46,84),(48,82),(49,55),(50,94),(51,53),(52,92),(54,90),(56,96),(57,63),(58,102),(59,61),(60,100),(62,98),(64,104),(67,71),(75,79),(83,87),(89,95),(91,93),(97,103),(99,101),(105,111),(106,121),(107,109),(108,127),(110,125),(112,123),(113,119),(114,136),(115,117),(116,134),(118,132),(120,130),(122,124),(126,128),(129,131),(133,135),(138,184),(139,143),(140,182),(142,180),(144,178),(146,192),(147,151),(148,190),(150,188),(152,186),(154,200),(155,159),(156,198),(158,196),(160,194),(161,167),(162,206),(163,165),(164,204),(166,202),(168,208),(171,175),(179,183),(187,191),(195,199),(201,207),(203,205),(209,215),(211,213),(217,223),(219,221)], [(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	4A	4B	4C	4D	4E	4F	4G	7A	···	7F	8A	8B	8C	8D	14A	···	14R	14S	···	14X	28A	···	28L	28M	···	28X	28Y	···	28AP	56A	···	56X
order	1	2	2	2	2	4	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	2	2	4	4	8	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	1	1	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+							+				+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₇	C₁₄	C₁₄	C₁₄	C₁₄	C₁₄	D₄	C₄○D₄	C₇×D₄	C₇×C₄○D₄	C₈⋊C₂²	C₈.C₂²	C₇×C₈⋊C₂²	C₇×C₈.C₂²
kernel	C₇×C₄².₂₈C₂²	C₇×C₈⋊C₄	C₇×D₄⋊C₄	C₇×Q₈⋊C₄	C₇×C₄.₄D₄	C₇×C₄⋊Q₈	C₄².₂₈C₂²	C₈⋊C₄	D₄⋊C₄	Q₈⋊C₄	C₄.₄D₄	C₄⋊Q₈	C₂×C₂₈	C₂₈	C₂×C₄	C₄	C₁₄	C₁₄	C₂	C₂
# reps	1	1	2	2	1	1	6	6	12	12	6	6	2	4	12	24	1	1	6	6

Matrix representation of C₇×C₄².₂₈C₂² ►in GL₆(𝔽₁₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	16	0	0	0
0	0	0	16	0	0
0	0	0	0	16	0
0	0	0	0	0	16

15	30	0	0	0	0
0	98	0	0	0	0
0	0	112	0	1	1
0	0	0	112	112	1
0	0	112	1	1	0
0	0	112	112	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	112	0	0	0
0	0	0	0	0	1
0	0	0	0	112	0

1	0	0	0	0	0
112	112	0	0	0	0
0	0	1	0	0	0
0	0	0	112	0	0
0	0	0	0	0	1
0	0	0	0	1	0

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

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[15,0,0,0,0,0,30,98,0,0,0,0,0,0,112,0,112,112,0,0,0,112,1,112,0,0,1,112,1,0,0,0,1,1,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[1,112,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[112,0,0,0,0,0,111,1,0,0,0,0,0,0,112,1,112,0,0,0,112,112,0,112,0,0,0,112,1,112,0,0,1,0,1,1] >;

C₇×C₄².₂₈C₂² in GAP, Magma, Sage, TeX

C_7\times C_4^2._{28}C_2^2

% in TeX

G:=Group("C7xC4^2.28C2^2");

// GroupNames label

G:=SmallGroup(448,897);

// by ID

G=gap.SmallGroup(448,897);

# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,792,2438,2403,310,9804,172,14117,124]);

// 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*c^2,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×C42.28C22 order 448 = 26·7

Direct product of C7 and C42.28C22

G = C₇×C₄².₂₈C₂² order 448 = 2⁶·7

Direct product of C₇ and C₄².₂₈C₂²