metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.63D14, (C2×C28).75D4, C22⋊Q8.2D7, (C2×Q8).25D14, Q8⋊Dic7⋊12C2, C4.Dic14⋊37C2, C14.71(C2×SD16), (C2×C14).16SD16, (C22×C14).89D4, C22.6(Q8⋊D7), C28.187(C4○D4), C4.93(D4⋊2D7), (C2×C28).362C23, C28.55D4.6C2, (C22×C4).124D14, C23.60(C7⋊D4), C7⋊5(C23.47D4), (Q8×C14).43C22, C4⋊Dic7.337C22, C2.14(D4.9D14), C14.116(C8.C22), (C22×C28).166C22, C14.80(C22.D4), C2.14(C23.18D14), C2.8(C2×Q8⋊D7), (C7×C22⋊Q8).1C2, (C2×C14).493(C2×D4), (C2×C4).53(C7⋊D4), (C2×C7⋊C8).112C22, (C2×C4⋊Dic7).37C2, (C7×C4⋊C4).110C22, (C2×C4).462(C22×D7), C22.168(C2×C7⋊D4), SmallGroup(448,577)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊Q8.D7
G = < a,b,c,d,e,f | a2=b2=c4=e7=1, d2=c2, f2=b, dad-1=ab=ba, ac=ca, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fcf-1=c-1, ce=ec, de=ed, fdf-1=c-1d, fef-1=e-1 >
Subgroups: 412 in 104 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, Q8, C23, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, Q8⋊C4, C4.Q8, C2×C4⋊C4, C22⋊Q8, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×C14, C23.47D4, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C22×Dic7, C22×C28, Q8×C14, C4.Dic14, C28.55D4, Q8⋊Dic7, C2×C4⋊Dic7, C7×C22⋊Q8, C22⋊Q8.D7
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, C4○D4, D14, C22.D4, C2×SD16, C8.C22, C7⋊D4, C22×D7, C23.47D4, Q8⋊D7, D4⋊2D7, C2×C7⋊D4, C23.18D14, C2×Q8⋊D7, D4.9D14, C22⋊Q8.D7
►On 224 points
Generators in S224
(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)
(1 41 13 34)(2 42 14 35)(3 36 8 29)(4 37 9 30)(5 38 10 31)(6 39 11 32)(7 40 12 33)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)(57 85 64 92)(58 86 65 93)(59 87 66 94)(60 88 67 95)(61 89 68 96)(62 90 69 97)(63 91 70 98)(71 99 78 106)(72 100 79 107)(73 101 80 108)(74 102 81 109)(75 103 82 110)(76 104 83 111)(77 105 84 112)(113 141 120 148)(114 142 121 149)(115 143 122 150)(116 144 123 151)(117 145 124 152)(118 146 125 153)(119 147 126 154)(127 155 134 162)(128 156 135 163)(129 157 136 164)(130 158 137 165)(131 159 138 166)(132 160 139 167)(133 161 140 168)(169 204 176 197)(170 205 177 198)(171 206 178 199)(172 207 179 200)(173 208 180 201)(174 209 181 202)(175 210 182 203)(183 218 190 211)(184 219 191 212)(185 220 192 213)(186 221 193 214)(187 222 194 215)(188 223 195 216)(189 224 196 217)
(1 69 13 62)(2 70 14 63)(3 64 8 57)(4 65 9 58)(5 66 10 59)(6 67 11 60)(7 68 12 61)(15 78 22 71)(16 79 23 72)(17 80 24 73)(18 81 25 74)(19 82 26 75)(20 83 27 76)(21 84 28 77)(29 92 36 85)(30 93 37 86)(31 94 38 87)(32 95 39 88)(33 96 40 89)(34 97 41 90)(35 98 42 91)(43 106 50 99)(44 107 51 100)(45 108 52 101)(46 109 53 102)(47 110 54 103)(48 111 55 104)(49 112 56 105)(113 204 120 197)(114 205 121 198)(115 206 122 199)(116 207 123 200)(117 208 124 201)(118 209 125 202)(119 210 126 203)(127 218 134 211)(128 219 135 212)(129 220 136 213)(130 221 137 214)(131 222 138 215)(132 223 139 216)(133 224 140 217)(141 169 148 176)(142 170 149 177)(143 171 150 178)(144 172 151 179)(145 173 152 180)(146 174 153 181)(147 175 154 182)(155 183 162 190)(156 184 163 191)(157 185 164 192)(158 186 165 193)(159 187 166 194)(160 188 167 195)(161 189 168 196)
(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)
(1 133 20 119)(2 132 21 118)(3 131 15 117)(4 130 16 116)(5 129 17 115)(6 128 18 114)(7 127 19 113)(8 138 22 124)(9 137 23 123)(10 136 24 122)(11 135 25 121)(12 134 26 120)(13 140 27 126)(14 139 28 125)(29 159 43 145)(30 158 44 144)(31 157 45 143)(32 156 46 142)(33 155 47 141)(34 161 48 147)(35 160 49 146)(36 166 50 152)(37 165 51 151)(38 164 52 150)(39 163 53 149)(40 162 54 148)(41 168 55 154)(42 167 56 153)(57 187 71 173)(58 186 72 172)(59 185 73 171)(60 184 74 170)(61 183 75 169)(62 189 76 175)(63 188 77 174)(64 194 78 180)(65 193 79 179)(66 192 80 178)(67 191 81 177)(68 190 82 176)(69 196 83 182)(70 195 84 181)(85 215 99 201)(86 214 100 200)(87 213 101 199)(88 212 102 198)(89 211 103 197)(90 217 104 203)(91 216 105 202)(92 222 106 208)(93 221 107 207)(94 220 108 206)(95 219 109 205)(96 218 110 204)(97 224 111 210)(98 223 112 209)
G:=sub<Sym(224)| (57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (1,41,13,34)(2,42,14,35)(3,36,8,29)(4,37,9,30)(5,38,10,31)(6,39,11,32)(7,40,12,33)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49)(57,85,64,92)(58,86,65,93)(59,87,66,94)(60,88,67,95)(61,89,68,96)(62,90,69,97)(63,91,70,98)(71,99,78,106)(72,100,79,107)(73,101,80,108)(74,102,81,109)(75,103,82,110)(76,104,83,111)(77,105,84,112)(113,141,120,148)(114,142,121,149)(115,143,122,150)(116,144,123,151)(117,145,124,152)(118,146,125,153)(119,147,126,154)(127,155,134,162)(128,156,135,163)(129,157,136,164)(130,158,137,165)(131,159,138,166)(132,160,139,167)(133,161,140,168)(169,204,176,197)(170,205,177,198)(171,206,178,199)(172,207,179,200)(173,208,180,201)(174,209,181,202)(175,210,182,203)(183,218,190,211)(184,219,191,212)(185,220,192,213)(186,221,193,214)(187,222,194,215)(188,223,195,216)(189,224,196,217), (1,69,13,62)(2,70,14,63)(3,64,8,57)(4,65,9,58)(5,66,10,59)(6,67,11,60)(7,68,12,61)(15,78,22,71)(16,79,23,72)(17,80,24,73)(18,81,25,74)(19,82,26,75)(20,83,27,76)(21,84,28,77)(29,92,36,85)(30,93,37,86)(31,94,38,87)(32,95,39,88)(33,96,40,89)(34,97,41,90)(35,98,42,91)(43,106,50,99)(44,107,51,100)(45,108,52,101)(46,109,53,102)(47,110,54,103)(48,111,55,104)(49,112,56,105)(113,204,120,197)(114,205,121,198)(115,206,122,199)(116,207,123,200)(117,208,124,201)(118,209,125,202)(119,210,126,203)(127,218,134,211)(128,219,135,212)(129,220,136,213)(130,221,137,214)(131,222,138,215)(132,223,139,216)(133,224,140,217)(141,169,148,176)(142,170,149,177)(143,171,150,178)(144,172,151,179)(145,173,152,180)(146,174,153,181)(147,175,154,182)(155,183,162,190)(156,184,163,191)(157,185,164,192)(158,186,165,193)(159,187,166,194)(160,188,167,195)(161,189,168,196), (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), (1,133,20,119)(2,132,21,118)(3,131,15,117)(4,130,16,116)(5,129,17,115)(6,128,18,114)(7,127,19,113)(8,138,22,124)(9,137,23,123)(10,136,24,122)(11,135,25,121)(12,134,26,120)(13,140,27,126)(14,139,28,125)(29,159,43,145)(30,158,44,144)(31,157,45,143)(32,156,46,142)(33,155,47,141)(34,161,48,147)(35,160,49,146)(36,166,50,152)(37,165,51,151)(38,164,52,150)(39,163,53,149)(40,162,54,148)(41,168,55,154)(42,167,56,153)(57,187,71,173)(58,186,72,172)(59,185,73,171)(60,184,74,170)(61,183,75,169)(62,189,76,175)(63,188,77,174)(64,194,78,180)(65,193,79,179)(66,192,80,178)(67,191,81,177)(68,190,82,176)(69,196,83,182)(70,195,84,181)(85,215,99,201)(86,214,100,200)(87,213,101,199)(88,212,102,198)(89,211,103,197)(90,217,104,203)(91,216,105,202)(92,222,106,208)(93,221,107,207)(94,220,108,206)(95,219,109,205)(96,218,110,204)(97,224,111,210)(98,223,112,209)>;
G:=Group( (57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (1,41,13,34)(2,42,14,35)(3,36,8,29)(4,37,9,30)(5,38,10,31)(6,39,11,32)(7,40,12,33)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49)(57,85,64,92)(58,86,65,93)(59,87,66,94)(60,88,67,95)(61,89,68,96)(62,90,69,97)(63,91,70,98)(71,99,78,106)(72,100,79,107)(73,101,80,108)(74,102,81,109)(75,103,82,110)(76,104,83,111)(77,105,84,112)(113,141,120,148)(114,142,121,149)(115,143,122,150)(116,144,123,151)(117,145,124,152)(118,146,125,153)(119,147,126,154)(127,155,134,162)(128,156,135,163)(129,157,136,164)(130,158,137,165)(131,159,138,166)(132,160,139,167)(133,161,140,168)(169,204,176,197)(170,205,177,198)(171,206,178,199)(172,207,179,200)(173,208,180,201)(174,209,181,202)(175,210,182,203)(183,218,190,211)(184,219,191,212)(185,220,192,213)(186,221,193,214)(187,222,194,215)(188,223,195,216)(189,224,196,217), (1,69,13,62)(2,70,14,63)(3,64,8,57)(4,65,9,58)(5,66,10,59)(6,67,11,60)(7,68,12,61)(15,78,22,71)(16,79,23,72)(17,80,24,73)(18,81,25,74)(19,82,26,75)(20,83,27,76)(21,84,28,77)(29,92,36,85)(30,93,37,86)(31,94,38,87)(32,95,39,88)(33,96,40,89)(34,97,41,90)(35,98,42,91)(43,106,50,99)(44,107,51,100)(45,108,52,101)(46,109,53,102)(47,110,54,103)(48,111,55,104)(49,112,56,105)(113,204,120,197)(114,205,121,198)(115,206,122,199)(116,207,123,200)(117,208,124,201)(118,209,125,202)(119,210,126,203)(127,218,134,211)(128,219,135,212)(129,220,136,213)(130,221,137,214)(131,222,138,215)(132,223,139,216)(133,224,140,217)(141,169,148,176)(142,170,149,177)(143,171,150,178)(144,172,151,179)(145,173,152,180)(146,174,153,181)(147,175,154,182)(155,183,162,190)(156,184,163,191)(157,185,164,192)(158,186,165,193)(159,187,166,194)(160,188,167,195)(161,189,168,196), (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), (1,133,20,119)(2,132,21,118)(3,131,15,117)(4,130,16,116)(5,129,17,115)(6,128,18,114)(7,127,19,113)(8,138,22,124)(9,137,23,123)(10,136,24,122)(11,135,25,121)(12,134,26,120)(13,140,27,126)(14,139,28,125)(29,159,43,145)(30,158,44,144)(31,157,45,143)(32,156,46,142)(33,155,47,141)(34,161,48,147)(35,160,49,146)(36,166,50,152)(37,165,51,151)(38,164,52,150)(39,163,53,149)(40,162,54,148)(41,168,55,154)(42,167,56,153)(57,187,71,173)(58,186,72,172)(59,185,73,171)(60,184,74,170)(61,183,75,169)(62,189,76,175)(63,188,77,174)(64,194,78,180)(65,193,79,179)(66,192,80,178)(67,191,81,177)(68,190,82,176)(69,196,83,182)(70,195,84,181)(85,215,99,201)(86,214,100,200)(87,213,101,199)(88,212,102,198)(89,211,103,197)(90,217,104,203)(91,216,105,202)(92,222,106,208)(93,221,107,207)(94,220,108,206)(95,219,109,205)(96,218,110,204)(97,224,111,210)(98,223,112,209) );
G=PermutationGroup([[(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(169,183),(170,184),(171,185),(172,186),(173,187),(174,188),(175,189),(176,190),(177,191),(178,192),(179,193),(180,194),(181,195),(182,196),(197,211),(198,212),(199,213),(200,214),(201,215),(202,216),(203,217),(204,218),(205,219),(206,220),(207,221),(208,222),(209,223),(210,224)], [(1,20),(2,21),(3,15),(4,16),(5,17),(6,18),(7,19),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(141,155),(142,156),(143,157),(144,158),(145,159),(146,160),(147,161),(148,162),(149,163),(150,164),(151,165),(152,166),(153,167),(154,168),(169,183),(170,184),(171,185),(172,186),(173,187),(174,188),(175,189),(176,190),(177,191),(178,192),(179,193),(180,194),(181,195),(182,196),(197,211),(198,212),(199,213),(200,214),(201,215),(202,216),(203,217),(204,218),(205,219),(206,220),(207,221),(208,222),(209,223),(210,224)], [(1,41,13,34),(2,42,14,35),(3,36,8,29),(4,37,9,30),(5,38,10,31),(6,39,11,32),(7,40,12,33),(15,50,22,43),(16,51,23,44),(17,52,24,45),(18,53,25,46),(19,54,26,47),(20,55,27,48),(21,56,28,49),(57,85,64,92),(58,86,65,93),(59,87,66,94),(60,88,67,95),(61,89,68,96),(62,90,69,97),(63,91,70,98),(71,99,78,106),(72,100,79,107),(73,101,80,108),(74,102,81,109),(75,103,82,110),(76,104,83,111),(77,105,84,112),(113,141,120,148),(114,142,121,149),(115,143,122,150),(116,144,123,151),(117,145,124,152),(118,146,125,153),(119,147,126,154),(127,155,134,162),(128,156,135,163),(129,157,136,164),(130,158,137,165),(131,159,138,166),(132,160,139,167),(133,161,140,168),(169,204,176,197),(170,205,177,198),(171,206,178,199),(172,207,179,200),(173,208,180,201),(174,209,181,202),(175,210,182,203),(183,218,190,211),(184,219,191,212),(185,220,192,213),(186,221,193,214),(187,222,194,215),(188,223,195,216),(189,224,196,217)], [(1,69,13,62),(2,70,14,63),(3,64,8,57),(4,65,9,58),(5,66,10,59),(6,67,11,60),(7,68,12,61),(15,78,22,71),(16,79,23,72),(17,80,24,73),(18,81,25,74),(19,82,26,75),(20,83,27,76),(21,84,28,77),(29,92,36,85),(30,93,37,86),(31,94,38,87),(32,95,39,88),(33,96,40,89),(34,97,41,90),(35,98,42,91),(43,106,50,99),(44,107,51,100),(45,108,52,101),(46,109,53,102),(47,110,54,103),(48,111,55,104),(49,112,56,105),(113,204,120,197),(114,205,121,198),(115,206,122,199),(116,207,123,200),(117,208,124,201),(118,209,125,202),(119,210,126,203),(127,218,134,211),(128,219,135,212),(129,220,136,213),(130,221,137,214),(131,222,138,215),(132,223,139,216),(133,224,140,217),(141,169,148,176),(142,170,149,177),(143,171,150,178),(144,172,151,179),(145,173,152,180),(146,174,153,181),(147,175,154,182),(155,183,162,190),(156,184,163,191),(157,185,164,192),(158,186,165,193),(159,187,166,194),(160,188,167,195),(161,189,168,196)], [(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)], [(1,133,20,119),(2,132,21,118),(3,131,15,117),(4,130,16,116),(5,129,17,115),(6,128,18,114),(7,127,19,113),(8,138,22,124),(9,137,23,123),(10,136,24,122),(11,135,25,121),(12,134,26,120),(13,140,27,126),(14,139,28,125),(29,159,43,145),(30,158,44,144),(31,157,45,143),(32,156,46,142),(33,155,47,141),(34,161,48,147),(35,160,49,146),(36,166,50,152),(37,165,51,151),(38,164,52,150),(39,163,53,149),(40,162,54,148),(41,168,55,154),(42,167,56,153),(57,187,71,173),(58,186,72,172),(59,185,73,171),(60,184,74,170),(61,183,75,169),(62,189,76,175),(63,188,77,174),(64,194,78,180),(65,193,79,179),(66,192,80,178),(67,191,81,177),(68,190,82,176),(69,196,83,182),(70,195,84,181),(85,215,99,201),(86,214,100,200),(87,213,101,199),(88,212,102,198),(89,211,103,197),(90,217,104,203),(91,216,105,202),(92,222,106,208),(93,221,107,207),(94,220,108,206),(95,219,109,205),(96,218,110,204),(97,224,111,210),(98,223,112,209)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | C4○D4 | SD16 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | C8.C22 | D4⋊2D7 | Q8⋊D7 | D4.9D14 |
| kernel | C22⋊Q8.D7 | C4.Dic14 | C28.55D4 | Q8⋊Dic7 | C2×C4⋊Dic7 | C7×C22⋊Q8 | C2×C28 | C22×C14 | C22⋊Q8 | C28 | C2×C14 | C4⋊C4 | C22×C4 | C2×Q8 | C2×C4 | C23 | C14 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 3 | 3 | 3 | 6 | 6 | 1 | 6 | 6 | 6 |
Matrix representation of C22⋊Q8.D7 ►in GL6(𝔽113)
| 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 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 44 | 112 |
| 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 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 37 | 36 | 0 | 0 | 0 | 0 |
| 31 | 76 | 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 | 0 | 0 | 1 |
| 106 | 2 | 0 | 0 | 0 | 0 |
| 88 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 77 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 1 | 0 | 0 |
| 0 | 0 | 32 | 80 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 30 | 0 | 0 | 0 | 0 |
| 48 | 97 | 0 | 0 | 0 | 0 |
| 0 | 0 | 66 | 16 | 0 | 0 |
| 0 | 0 | 88 | 47 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,44,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[37,31,0,0,0,0,36,76,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[106,88,0,0,0,0,2,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,77,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,32,0,0,0,0,1,80,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,48,0,0,0,0,30,97,0,0,0,0,0,0,66,88,0,0,0,0,16,47,0,0,0,0,0,0,15,0,0,0,0,0,0,15] >;
C22⋊Q8.D7 in GAP, Magma, Sage, TeX
C_2^2\rtimes Q_8.D_7
% in TeX
G:=Group("C2^2:Q8.D7"); // GroupNames label
G:=SmallGroup(448,577);
// by ID
G=gap.SmallGroup(448,577);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,254,219,184,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=e^7=1,d^2=c^2,f^2=b,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=f*c*f^-1=c^-1,c*e=e*c,d*e=e*d,f*d*f^-1=c^-1*d,f*e*f^-1=e^-1>;
// generators/relations