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G = C7×C42.C22  order 448 = 26·7

Direct product of C7 and C42.C22

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

Aliases: C7×C42.C22, C8⋊C4⋊6C14, C14.24C4≀C2, (C2×D4).1C28, (D4×C14).4C4, (C2×Q8).1C28, (Q8×C14).4C4, (C2×C28).445D4, C42.1(C2×C14), C4.4D4.1C14, (C4×C28).241C22, C14.12(C4.D4), C2.6(C7×C4≀C2), (C2×C4).9(C2×C28), (C7×C8⋊C4)⋊16C2, (C2×C4).97(C7×D4), C2.3(C7×C4.D4), (C2×C28).176(C2×C4), (C7×C4.4D4).10C2, C22.37(C7×C22⋊C4), (C2×C14).124(C22⋊C4), SmallGroup(448,133)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C4 — C7×C42.C22
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×C28 — C7×C8⋊C4 — C7×C42.C22
Lower central
C1 — C22 — C2×C4 — C7×C42.C22
Upper central
C1 — C2×C14 — C4×C28 — C7×C42.C22

Generators and relations for C7×C42.C22
 G = < a,b,c,d,e | a7=b4=c4=e2=1, d2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc2, ebe=b-1, cd=dc, ece=b2c-1, ede=b-1c2d >

Subgroups: 146 in 70 conjugacy classes, 30 normal (18 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C42, C22⋊C4, C2×C8, C2×D4, C2×Q8, C28, C2×C14, C2×C14, C8⋊C4, C4.4D4, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C42.C22, C4×C28, C7×C22⋊C4, C2×C56, D4×C14, Q8×C14, C7×C8⋊C4, C7×C4.4D4, C7×C42.C22
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C14, C22⋊C4, C28, C2×C14, C4.D4, C4≀C2, C2×C28, C7×D4, C42.C22, C7×C22⋊C4, C7×C4.D4, C7×C4≀C2, C7×C42.C22

Smallest permutation representation of C7×C42.C22
►On 224 points
Generators in S224
(1 63 55 47 39 31 23)(2 64 56 48 40 32 24)(3 57 49 41 33 25 17)(4 58 50 42 34 26 18)(5 59 51 43 35 27 19)(6 60 52 44 36 28 20)(7 61 53 45 37 29 21)(8 62 54 46 38 30 22)(9 161 153 145 137 129 121)(10 162 154 146 138 130 122)(11 163 155 147 139 131 123)(12 164 156 148 140 132 124)(13 165 157 149 141 133 125)(14 166 158 150 142 134 126)(15 167 159 151 143 135 127)(16 168 160 152 144 136 128)(65 120 105 97 89 81 73)(66 113 106 98 90 82 74)(67 114 107 99 91 83 75)(68 115 108 100 92 84 76)(69 116 109 101 93 85 77)(70 117 110 102 94 86 78)(71 118 111 103 95 87 79)(72 119 112 104 96 88 80)(169 217 209 201 193 185 177)(170 218 210 202 194 186 178)(171 219 211 203 195 187 179)(172 220 212 204 196 188 180)(173 221 213 205 197 189 181)(174 222 214 206 198 190 182)(175 223 215 207 199 191 183)(176 224 216 208 200 192 184)

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

(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)

(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)

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

133 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F7A···7F8A···8H14A···14R14S···14X28A···28X28Y···28AD28AE···28AJ56A···56AV
order122224444447···78···814···1414···1428···2828···2828···2856···56
size111182222481···14···41···18···82···24···48···84···4

133 irreducible representations

dim1111111111222244
type+++++
imageC1C2C2C4C4C7C14C14C28C28D4C4≀C2C7×D4C7×C4≀C2C4.D4C7×C4.D4
kernelC7×C42.C22C7×C8⋊C4C7×C4.4D4D4×C14Q8×C14C42.C22C8⋊C4C4.4D4C2×D4C2×Q8C2×C28C14C2×C4C2C14C2
# reps121226126121228124816

Matrix representation of C7×C42.C22 ►in GL4(𝔽113) generated by

1000
0100
00300
00030
,
19800
8311200
001111
001112
,
15100
29800
00980
00098
,
1610600
99000
00099
001060
,
1000
8311200
0010
001112
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,30,0,0,0,0,30],[1,83,0,0,98,112,0,0,0,0,1,1,0,0,111,112],[15,2,0,0,1,98,0,0,0,0,98,0,0,0,0,98],[16,99,0,0,106,0,0,0,0,0,0,106,0,0,99,0],[1,83,0,0,0,112,0,0,0,0,1,1,0,0,0,112] >;
 

C7×C42.C22 in GAP, Magma, Sage, TeX 

C_7\times C_4^2.C_2^2
 
% in TeX
 
G:=Group("C7xC4^2.C2^2");
 
// GroupNames label
 
G:=SmallGroup(448,133);
 
// by ID
 
G=gap.SmallGroup(448,133);
 
# by ID
 
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,3923,3538,248,6871,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^7=b^4=c^4=e^2=1,d^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^-1=b*c^2,e*b*e=b^-1,c*d=d*c,e*c*e=b^2*c^-1,e*d*e=b^-1*c^2*d>;
 
// generators/relations
 

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