C7xC2^2.M4(2) - GroupNames

G = C₇×C₂².M₄(2) order 448 = 2⁶·7

Direct product of C₇ and C₂².M₄(2)

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

Aliases: C₇×C₂².M₄(2), (C₂×C₄)⋊C₅₆, (C₂×C₂₈)⋊₂C₈, (C₂×C₂₈).₄₄₂D₄, C₂²⋊C₈.₁C₁₄, C₂².₃(C₂×C₅₆), (C₂²×C₂₈).₅C₄, (C₂²×C₄).₂C₂₈, C₂³.₂₂(C₂×C₂₈), C₁₄.₂₈(C₂³⋊C₄), C₁₄.₂₁(C₂²⋊C₈), (C₂×C₁₄).₁₅M₄(2), (C₂²×C₂₈).₂C₂², C₂².₃(C₇×M₄(2)), C₁₄.₁₀(C₄.₁₀D₄), (C₂×C₄⋊C₄).₁C₁₄, (C₂×C₄).₉₂(C₇×D₄), C₂.₂(C₇×C₂³⋊C₄), C₂.₄(C₇×C₂²⋊C₈), (C₁₄×C₄⋊C₄).₂₈C₂, (C₂×C₁₄).₂₁(C₂×C₈), (C₇×C₂²⋊C₈).₃C₂, (C₂²×C₄).₂(C₂×C₁₄), C₂.₁(C₇×C₄.₁₀D₄), C₂².₂₄(C₇×C₂²⋊C₄), (C₂²×C₁₄).₁₀₇(C₂×C₄), (C₂×C₁₄).₁₁₉(C₂²⋊C₄), SmallGroup(448,128)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₇×C₂².M₄(2)

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂²×C₂₈ — C₇×C₂²⋊C₈ — C₇×C₂².M₄(2)

Lower central

C₁ — C₂ — C₂² — C₇×C₂².M₄(2)

Upper central

C₁ — C₂×C₁₄ — C₂²×C₂₈ — C₇×C₂².M₄(2)

Generators and relations for C₇×C₂².M₄(2)
G = < a,b,c,d,e | a⁷=b²=c²=d⁸=1, e²=c, ab=ba, ac=ca, ad=da, ae=ea, dbd^-1=bc=cb, be=eb, cd=dc, ce=ec, ede^-1=bcd⁵ >

Subgroups: 138 in 78 conjugacy classes, 38 normal (26 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, C₂³, C₁₄, C₁₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂₈, C₂×C₁₄, C₂×C₁₄, C₂²⋊C₈, C₂×C₄⋊C₄, C₅₆, C₂×C₂₈, C₂×C₂₈, C₂²×C₁₄, C₂².M₄(2), C₇×C₄⋊C₄, C₂×C₅₆, C₂²×C₂₈, C₇×C₂²⋊C₈, C₁₄×C₄⋊C₄, C₇×C₂².M₄(2)
Quotients: C₁, C₂, C₄, C₂², C₇, C₈, C₂×C₄, D₄, C₁₄, C₂²⋊C₄, C₂×C₈, M₄(2), C₂₈, C₂×C₁₄, C₂²⋊C₈, C₂³⋊C₄, C₄.₁₀D₄, C₅₆, C₂×C₂₈, C₇×D₄, C₂².M₄(2), C₇×C₂²⋊C₄, C₂×C₅₆, C₇×M₄(2), C₇×C₂²⋊C₈, C₇×C₂³⋊C₄, C₇×C₄.₁₀D₄, C₇×C₂².M₄(2)

Smallest permutation representation of C₇×C₂².M₄(2)
►On 224 points

Generators in S₂₂₄

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

(2 118)(4 120)(6 114)(8 116)(10 137)(12 139)(14 141)(16 143)(18 145)(20 147)(22 149)(24 151)(26 153)(28 155)(30 157)(32 159)(34 171)(36 173)(38 175)(40 169)(42 124)(44 126)(46 128)(48 122)(50 132)(52 134)(54 136)(56 130)(58 218)(60 220)(62 222)(64 224)(66 202)(68 204)(70 206)(72 208)(74 210)(76 212)(78 214)(80 216)(81 164)(83 166)(85 168)(87 162)(89 180)(91 182)(93 184)(95 178)(97 188)(99 190)(101 192)(103 186)(105 196)(107 198)(109 200)(111 194)

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

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

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

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

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

154 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	7A	···	7F	8A	···	8H	14A	···	14R	14S	···	14AD	28A	···	28X	28Y	···	28AV	56A	···	56AV
order	1	2	2	2	2	2	4	4	4	4	4	4	4	4	7	···	7	8	···	8	14	···	14	14	···	14	28	···	28	28	···	28	56	···	56
size	1	1	1	1	2	2	2	2	2	2	4	4	4	4	1	···	1	4	···	4	1	···	1	2	···	2	2	···	2	4	···	4	4	···	4

154 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4	4	4
type	+	+	+								+				+	-
image	C₁	C₂	C₂	C₄	C₇	C₈	C₁₄	C₁₄	C₂₈	C₅₆	D₄	M₄(2)	C₇×D₄	C₇×M₄(2)	C₂³⋊C₄	C₄.₁₀D₄	C₇×C₂³⋊C₄	C₇×C₄.₁₀D₄
kernel	C₇×C₂².M₄(2)	C₇×C₂²⋊C₈	C₁₄×C₄⋊C₄	C₂²×C₂₈	C₂².M₄(2)	C₂×C₂₈	C₂²⋊C₈	C₂×C₄⋊C₄	C₂²×C₄	C₂×C₄	C₂×C₂₈	C₂×C₁₄	C₂×C₄	C₂²	C₁₄	C₁₄	C₂	C₂
# reps	1	2	1	4	6	8	12	6	24	48	2	2	12	12	1	1	6	6

Matrix representation of C₇×C₂².M₄(2) ►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

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	98	90	112	0
0	0	61	102	0	112

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

102	61	0	0	0	0
102	11	0	0	0	0
0	0	77	52	7	77
0	0	88	93	36	106
0	0	40	96	42	68
0	0	83	10	72	14

1	111	0	0	0	0
0	112	0	0	0	0
0	0	53	18	0	0
0	0	95	60	0	0
0	0	110	53	53	18
0	0	33	3	95	60

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,98,61,0,0,0,1,90,102,0,0,0,0,112,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[102,102,0,0,0,0,61,11,0,0,0,0,0,0,77,88,40,83,0,0,52,93,96,10,0,0,7,36,42,72,0,0,77,106,68,14],[1,0,0,0,0,0,111,112,0,0,0,0,0,0,53,95,110,33,0,0,18,60,53,3,0,0,0,0,53,95,0,0,0,0,18,60] >;

C₇×C₂².M₄(2) in GAP, Magma, Sage, TeX

C_7\times C_2^2.M_4(2)

% in TeX

G:=Group("C7xC2^2.M4(2)");

// GroupNames label

G:=SmallGroup(448,128);

// by ID

G=gap.SmallGroup(448,128);

# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,792,3923,2951,172]);

// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^2=c^2=d^8=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^5>;

// generators/relations

G = C7×C22.M4(2) order 448 = 26·7

Direct product of C7 and C22.M4(2)

G = C₇×C₂².M₄(2) order 448 = 2⁶·7

Direct product of C₇ and C₂².M₄(2)