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G = (C2×M4(2))⋊D7order 448 = 26·7

9th semidirect product of C2×M4(2) and D7 acting via D7/C7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D14⋊C840C2, C4.89(C2×D28), (C2×D28).14C4, (C2×C8).189D14, (C2×C4).152D28, C28.444(C2×D4), (C2×C28).171D4, (C2×M4(2))⋊9D7, C23.29(C4×D7), C4.12(D14⋊C4), C14.34(C8○D4), C28.26(C22⋊C4), (C14×M4(2))⋊17C2, (C2×C28).869C23, C2.19(D28.C4), (C2×C56).319C22, C22.2(D14⋊C4), (C2×Dic14).14C4, (C22×C4).350D14, (C22×C28).186C22, (C22×C7⋊C8)⋊6C2, (C2×C4).84(C4×D7), C2.28(C2×D14⋊C4), (C2×C7⋊D4).12C4, C4.135(C2×C7⋊D4), C22.148(C2×C4×D7), (C2×C28).106(C2×C4), C73((C22×C8)⋊C2), (C2×C4○D28).11C2, C14.56(C2×C22⋊C4), (C2×C7⋊C8).325C22, (C2×C4×D7).186C22, (C2×C4).141(C7⋊D4), (C22×C14).69(C2×C4), (C2×Dic7).35(C2×C4), (C22×D7).26(C2×C4), (C2×C4).811(C22×D7), (C2×C14).20(C22⋊C4), (C2×C14).139(C22×C4), SmallGroup(448,663)

Series: Derived Chief Lower central Upper central

C1C2×C14 — (C2×M4(2))⋊D7
C1C7C14C28C2×C28C2×C4×D7C2×C4○D28 — (C2×M4(2))⋊D7
C7C2×C14 — (C2×M4(2))⋊D7
C1C2×C4C2×M4(2)

Generators and relations for (C2×M4(2))⋊D7
 G = < a,b,c,d,e | a2=b8=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b5, bd=db, ebe=ab5, cd=dc, ece=b4c, ede=d-1 >

Subgroups: 740 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×8], C7, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C23 [×2], D7 [×2], C14, C14 [×2], C14 [×2], C2×C8 [×2], C2×C8 [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic7 [×2], C28 [×2], C28 [×2], D14 [×6], C2×C14, C2×C14 [×2], C2×C14 [×2], C22⋊C8 [×4], C22×C8, C2×M4(2), C2×C4○D4, C7⋊C8 [×2], C56 [×2], Dic14 [×2], C4×D7 [×4], D28 [×2], C2×Dic7 [×2], C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×4], C22×D7 [×2], C22×C14, (C22×C8)⋊C2, C2×C7⋊C8 [×2], C2×C7⋊C8 [×2], C2×C56 [×2], C7×M4(2) [×2], C2×Dic14, C2×C4×D7 [×2], C2×D28, C4○D28 [×4], C2×C7⋊D4 [×2], C22×C28, D14⋊C8 [×4], C22×C7⋊C8, C14×M4(2), C2×C4○D28, (C2×M4(2))⋊D7
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D7, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D14 [×3], C2×C22⋊C4, C8○D4 [×2], C4×D7 [×2], D28 [×2], C7⋊D4 [×2], C22×D7, (C22×C8)⋊C2, D14⋊C4 [×4], C2×C4×D7, C2×D28, C2×C7⋊D4, D28.C4 [×2], C2×D14⋊C4, (C2×M4(2))⋊D7

Smallest permutation representation of (C2×M4(2))⋊D7
On 224 points
Generators in S224
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 97)(8 98)(9 165)(10 166)(11 167)(12 168)(13 161)(14 162)(15 163)(16 164)(17 129)(18 130)(19 131)(20 132)(21 133)(22 134)(23 135)(24 136)(25 118)(26 119)(27 120)(28 113)(29 114)(30 115)(31 116)(32 117)(33 126)(34 127)(35 128)(36 121)(37 122)(38 123)(39 124)(40 125)(41 183)(42 184)(43 177)(44 178)(45 179)(46 180)(47 181)(48 182)(49 143)(50 144)(51 137)(52 138)(53 139)(54 140)(55 141)(56 142)(57 151)(58 152)(59 145)(60 146)(61 147)(62 148)(63 149)(64 150)(65 171)(66 172)(67 173)(68 174)(69 175)(70 176)(71 169)(72 170)(73 108)(74 109)(75 110)(76 111)(77 112)(78 105)(79 106)(80 107)(81 188)(82 189)(83 190)(84 191)(85 192)(86 185)(87 186)(88 187)(89 196)(90 197)(91 198)(92 199)(93 200)(94 193)(95 194)(96 195)(153 202)(154 203)(155 204)(156 205)(157 206)(158 207)(159 208)(160 201)(209 222)(210 223)(211 224)(212 217)(213 218)(214 219)(215 220)(216 221)
(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)
(1 103)(2 100)(3 97)(4 102)(5 99)(6 104)(7 101)(8 98)(9 161)(10 166)(11 163)(12 168)(13 165)(14 162)(15 167)(16 164)(17 129)(18 134)(19 131)(20 136)(21 133)(22 130)(23 135)(24 132)(25 114)(26 119)(27 116)(28 113)(29 118)(30 115)(31 120)(32 117)(33 126)(34 123)(35 128)(36 125)(37 122)(38 127)(39 124)(40 121)(41 179)(42 184)(43 181)(44 178)(45 183)(46 180)(47 177)(48 182)(49 139)(50 144)(51 141)(52 138)(53 143)(54 140)(55 137)(56 142)(57 151)(58 148)(59 145)(60 150)(61 147)(62 152)(63 149)(64 146)(65 171)(66 176)(67 173)(68 170)(69 175)(70 172)(71 169)(72 174)(73 108)(74 105)(75 110)(76 107)(77 112)(78 109)(79 106)(80 111)(81 192)(82 189)(83 186)(84 191)(85 188)(86 185)(87 190)(88 187)(89 196)(90 193)(91 198)(92 195)(93 200)(94 197)(95 194)(96 199)(153 202)(154 207)(155 204)(156 201)(157 206)(158 203)(159 208)(160 205)(209 218)(210 223)(211 220)(212 217)(213 222)(214 219)(215 224)(216 221)
(1 13 45 137 176 20 74)(2 14 46 138 169 21 75)(3 15 47 139 170 22 76)(4 16 48 140 171 23 77)(5 9 41 141 172 24 78)(6 10 42 142 173 17 79)(7 11 43 143 174 18 80)(8 12 44 144 175 19 73)(25 190 152 38 90 203 224)(26 191 145 39 91 204 217)(27 192 146 40 92 205 218)(28 185 147 33 93 206 219)(29 186 148 34 94 207 220)(30 187 149 35 95 208 221)(31 188 150 36 96 201 222)(32 189 151 37 89 202 223)(49 68 130 107 97 167 177)(50 69 131 108 98 168 178)(51 70 132 109 99 161 179)(52 71 133 110 100 162 180)(53 72 134 111 101 163 181)(54 65 135 112 102 164 182)(55 66 136 105 103 165 183)(56 67 129 106 104 166 184)(57 122 196 153 210 117 82)(58 123 197 154 211 118 83)(59 124 198 155 212 119 84)(60 125 199 156 213 120 85)(61 126 200 157 214 113 86)(62 127 193 158 215 114 87)(63 128 194 159 216 115 88)(64 121 195 160 209 116 81)
(1 91)(2 195)(3 93)(4 197)(5 95)(6 199)(7 89)(8 193)(9 35)(10 125)(11 37)(12 127)(13 39)(14 121)(15 33)(16 123)(17 213)(18 223)(19 215)(20 217)(21 209)(22 219)(23 211)(24 221)(25 65)(26 176)(27 67)(28 170)(29 69)(30 172)(31 71)(32 174)(34 168)(36 162)(38 164)(40 166)(41 149)(42 60)(43 151)(44 62)(45 145)(46 64)(47 147)(48 58)(49 82)(50 186)(51 84)(52 188)(53 86)(54 190)(55 88)(56 192)(57 177)(59 179)(61 181)(63 183)(66 115)(68 117)(70 119)(72 113)(73 158)(74 204)(75 160)(76 206)(77 154)(78 208)(79 156)(80 202)(81 138)(83 140)(85 142)(87 144)(90 102)(92 104)(94 98)(96 100)(97 196)(99 198)(101 200)(103 194)(105 159)(106 205)(107 153)(108 207)(109 155)(110 201)(111 157)(112 203)(114 175)(116 169)(118 171)(120 173)(122 167)(124 161)(126 163)(128 165)(129 218)(130 210)(131 220)(132 212)(133 222)(134 214)(135 224)(136 216)(137 191)(139 185)(141 187)(143 189)(146 184)(148 178)(150 180)(152 182)

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

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

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

88 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H7A7B7C8A8B8C8D8E···8L14A···14I14J···14O28A···28L28M···28R56A···56X
order122222224444444477788888···814···1414···1428···2828···2856···56
size11112228281111222828222444414···142···24···42···24···44···4

88 irreducible representations

dim111111112222222224
type++++++++++
imageC1C2C2C2C2C4C4C4D4D7D14D14C8○D4C4×D7D28C7⋊D4C4×D7D28.C4
kernel(C2×M4(2))⋊D7D14⋊C8C22×C7⋊C8C14×M4(2)C2×C4○D28C2×Dic14C2×D28C2×C7⋊D4C2×C28C2×M4(2)C2×C8C22×C4C14C2×C4C2×C4C2×C4C23C2
# reps141112244363861212612

Matrix representation of (C2×M4(2))⋊D7 in GL4(𝔽113) generated by

112000
011200
0010
0001
,
587500
385500
00982
0010615
,
112000
011200
0010
0015112
,
0100
112900
0010
0001
,
104100
33900
006936
003144
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[58,38,0,0,75,55,0,0,0,0,98,106,0,0,2,15],[112,0,0,0,0,112,0,0,0,0,1,15,0,0,0,112],[0,112,0,0,1,9,0,0,0,0,1,0,0,0,0,1],[104,33,0,0,1,9,0,0,0,0,69,31,0,0,36,44] >;

(C2×M4(2))⋊D7 in GAP, Magma, Sage, TeX

(C_2\times M_{4(2}))\rtimes D_7
% in TeX

G:=Group("(C2xM4(2)):D7");
// GroupNames label

G:=SmallGroup(448,663);
// by ID

G=gap.SmallGroup(448,663);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,422,387,58,136,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^5,b*d=d*b,e*b*e=a*b^5,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽