direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×Q8⋊D7, C28.30C24, D28.27C23, C7⋊C8⋊10C23, (C2×Q8)⋊26D14, C14⋊4(C2×SD16), Q8⋊4(C22×D7), (C7×Q8)⋊4C23, (C22×Q8)⋊3D7, C7⋊4(C22×SD16), (C2×C14)⋊15SD16, C28.254(C2×D4), (C2×C28).210D4, C4.30(C23×D7), (Q8×C14)⋊33C22, (C2×C28).547C23, (C22×D28).18C2, (C22×C14).209D4, (C22×C4).381D14, C14.149(C22×D4), (C2×D28).276C22, C23.106(C7⋊D4), (C22×C28).279C22, (Q8×C2×C14)⋊2C2, (C2×C7⋊C8)⋊40C22, (C22×C7⋊C8)⋊13C2, C4.24(C2×C7⋊D4), (C2×C14).584(C2×D4), C2.22(C22×C7⋊D4), (C2×C4).154(C7⋊D4), (C2×C4).629(C22×D7), C22.112(C2×C7⋊D4), SmallGroup(448,1260)
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=f2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fcf=c-1, ce=ec, de=ed, fdf=c-1d, fef=e-1 >
Subgroups: 1524 in 298 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, D7, C14, C14, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C24, C28, C28, C28, D14, C2×C14, C22×C8, C2×SD16, C22×D4, C22×Q8, C7⋊C8, D28, D28, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×D7, C22×C14, C22×SD16, C2×C7⋊C8, Q8⋊D7, C2×D28, C2×D28, C22×C28, C22×C28, Q8×C14, Q8×C14, C23×D7, C22×C7⋊C8, C2×Q8⋊D7, C22×D28, Q8×C2×C14, C22×Q8⋊D7
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, C24, D14, C2×SD16, C22×D4, C7⋊D4, C22×D7, C22×SD16, Q8⋊D7, C2×C7⋊D4, C23×D7, C2×Q8⋊D7, C22×C7⋊D4, C22×Q8⋊D7
►On 224 points
Generators in S224
(1 120)(2 121)(3 122)(4 123)(5 124)(6 125)(7 126)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 134)(16 135)(17 136)(18 137)(19 138)(20 139)(21 140)(22 127)(23 128)(24 129)(25 130)(26 131)(27 132)(28 133)(29 148)(30 149)(31 150)(32 151)(33 152)(34 153)(35 154)(36 141)(37 142)(38 143)(39 144)(40 145)(41 146)(42 147)(43 162)(44 163)(45 164)(46 165)(47 166)(48 167)(49 168)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 161)(57 176)(58 177)(59 178)(60 179)(61 180)(62 181)(63 182)(64 169)(65 170)(66 171)(67 172)(68 173)(69 174)(70 175)(71 190)(72 191)(73 192)(74 193)(75 194)(76 195)(77 196)(78 183)(79 184)(80 185)(81 186)(82 187)(83 188)(84 189)(85 204)(86 205)(87 206)(88 207)(89 208)(90 209)(91 210)(92 197)(93 198)(94 199)(95 200)(96 201)(97 202)(98 203)(99 218)(100 219)(101 220)(102 221)(103 222)(104 223)(105 224)(106 211)(107 212)(108 213)(109 214)(110 215)(111 216)(112 217)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(113 169)(114 170)(115 171)(116 172)(117 173)(118 174)(119 175)(120 176)(121 177)(122 178)(123 179)(124 180)(125 181)(126 182)(127 183)(128 184)(129 185)(130 186)(131 187)(132 188)(133 189)(134 190)(135 191)(136 192)(137 193)(138 194)(139 195)(140 196)(141 197)(142 198)(143 199)(144 200)(145 201)(146 202)(147 203)(148 204)(149 205)(150 206)(151 207)(152 208)(153 209)(154 210)(155 211)(156 212)(157 213)(158 214)(159 215)(160 216)(161 217)(162 218)(163 219)(164 220)(165 221)(166 222)(167 223)(168 224)
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(35 49 42 56)(57 78 64 71)(58 79 65 72)(59 80 66 73)(60 81 67 74)(61 82 68 75)(62 83 69 76)(63 84 70 77)(85 99 92 106)(86 100 93 107)(87 101 94 108)(88 102 95 109)(89 103 96 110)(90 104 97 111)(91 105 98 112)(113 134 120 127)(114 135 121 128)(115 136 122 129)(116 137 123 130)(117 138 124 131)(118 139 125 132)(119 140 126 133)(141 155 148 162)(142 156 149 163)(143 157 150 164)(144 158 151 165)(145 159 152 166)(146 160 153 167)(147 161 154 168)(169 190 176 183)(170 191 177 184)(171 192 178 185)(172 193 179 186)(173 194 180 187)(174 195 181 188)(175 196 182 189)(197 211 204 218)(198 212 205 219)(199 213 206 220)(200 214 207 221)(201 215 208 222)(202 216 209 223)(203 217 210 224)
(1 204 8 197)(2 205 9 198)(3 206 10 199)(4 207 11 200)(5 208 12 201)(6 209 13 202)(7 210 14 203)(15 218 22 211)(16 219 23 212)(17 220 24 213)(18 221 25 214)(19 222 26 215)(20 223 27 216)(21 224 28 217)(29 169 36 176)(30 170 37 177)(31 171 38 178)(32 172 39 179)(33 173 40 180)(34 174 41 181)(35 175 42 182)(43 183 50 190)(44 184 51 191)(45 185 52 192)(46 186 53 193)(47 187 54 194)(48 188 55 195)(49 189 56 196)(57 148 64 141)(58 149 65 142)(59 150 66 143)(60 151 67 144)(61 152 68 145)(62 153 69 146)(63 154 70 147)(71 162 78 155)(72 163 79 156)(73 164 80 157)(74 165 81 158)(75 166 82 159)(76 167 83 160)(77 168 84 161)(85 113 92 120)(86 114 93 121)(87 115 94 122)(88 116 95 123)(89 117 96 124)(90 118 97 125)(91 119 98 126)(99 127 106 134)(100 128 107 135)(101 129 108 136)(102 130 109 137)(103 131 110 138)(104 132 111 139)(105 133 112 140)
(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 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 84)(16 83)(17 82)(18 81)(19 80)(20 79)(21 78)(22 77)(23 76)(24 75)(25 74)(26 73)(27 72)(28 71)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 99)(36 112)(37 111)(38 110)(39 109)(40 108)(41 107)(42 106)(43 91)(44 90)(45 89)(46 88)(47 87)(48 86)(49 85)(50 98)(51 97)(52 96)(53 95)(54 94)(55 93)(56 92)(113 175)(114 174)(115 173)(116 172)(117 171)(118 170)(119 169)(120 182)(121 181)(122 180)(123 179)(124 178)(125 177)(126 176)(127 196)(128 195)(129 194)(130 193)(131 192)(132 191)(133 190)(134 189)(135 188)(136 187)(137 186)(138 185)(139 184)(140 183)(141 217)(142 216)(143 215)(144 214)(145 213)(146 212)(147 211)(148 224)(149 223)(150 222)(151 221)(152 220)(153 219)(154 218)(155 203)(156 202)(157 201)(158 200)(159 199)(160 198)(161 197)(162 210)(163 209)(164 208)(165 207)(166 206)(167 205)(168 204)
G:=sub<Sym(224)| (1,120)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,140)(22,127)(23,128)(24,129)(25,130)(26,131)(27,132)(28,133)(29,148)(30,149)(31,150)(32,151)(33,152)(34,153)(35,154)(36,141)(37,142)(38,143)(39,144)(40,145)(41,146)(42,147)(43,162)(44,163)(45,164)(46,165)(47,166)(48,167)(49,168)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,161)(57,176)(58,177)(59,178)(60,179)(61,180)(62,181)(63,182)(64,169)(65,170)(66,171)(67,172)(68,173)(69,174)(70,175)(71,190)(72,191)(73,192)(74,193)(75,194)(76,195)(77,196)(78,183)(79,184)(80,185)(81,186)(82,187)(83,188)(84,189)(85,204)(86,205)(87,206)(88,207)(89,208)(90,209)(91,210)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(99,218)(100,219)(101,220)(102,221)(103,222)(104,223)(105,224)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(113,169)(114,170)(115,171)(116,172)(117,173)(118,174)(119,175)(120,176)(121,177)(122,178)(123,179)(124,180)(125,181)(126,182)(127,183)(128,184)(129,185)(130,186)(131,187)(132,188)(133,189)(134,190)(135,191)(136,192)(137,193)(138,194)(139,195)(140,196)(141,197)(142,198)(143,199)(144,200)(145,201)(146,202)(147,203)(148,204)(149,205)(150,206)(151,207)(152,208)(153,209)(154,210)(155,211)(156,212)(157,213)(158,214)(159,215)(160,216)(161,217)(162,218)(163,219)(164,220)(165,221)(166,222)(167,223)(168,224), (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56)(57,78,64,71)(58,79,65,72)(59,80,66,73)(60,81,67,74)(61,82,68,75)(62,83,69,76)(63,84,70,77)(85,99,92,106)(86,100,93,107)(87,101,94,108)(88,102,95,109)(89,103,96,110)(90,104,97,111)(91,105,98,112)(113,134,120,127)(114,135,121,128)(115,136,122,129)(116,137,123,130)(117,138,124,131)(118,139,125,132)(119,140,126,133)(141,155,148,162)(142,156,149,163)(143,157,150,164)(144,158,151,165)(145,159,152,166)(146,160,153,167)(147,161,154,168)(169,190,176,183)(170,191,177,184)(171,192,178,185)(172,193,179,186)(173,194,180,187)(174,195,181,188)(175,196,182,189)(197,211,204,218)(198,212,205,219)(199,213,206,220)(200,214,207,221)(201,215,208,222)(202,216,209,223)(203,217,210,224), (1,204,8,197)(2,205,9,198)(3,206,10,199)(4,207,11,200)(5,208,12,201)(6,209,13,202)(7,210,14,203)(15,218,22,211)(16,219,23,212)(17,220,24,213)(18,221,25,214)(19,222,26,215)(20,223,27,216)(21,224,28,217)(29,169,36,176)(30,170,37,177)(31,171,38,178)(32,172,39,179)(33,173,40,180)(34,174,41,181)(35,175,42,182)(43,183,50,190)(44,184,51,191)(45,185,52,192)(46,186,53,193)(47,187,54,194)(48,188,55,195)(49,189,56,196)(57,148,64,141)(58,149,65,142)(59,150,66,143)(60,151,67,144)(61,152,68,145)(62,153,69,146)(63,154,70,147)(71,162,78,155)(72,163,79,156)(73,164,80,157)(74,165,81,158)(75,166,82,159)(76,167,83,160)(77,168,84,161)(85,113,92,120)(86,114,93,121)(87,115,94,122)(88,116,95,123)(89,117,96,124)(90,118,97,125)(91,119,98,126)(99,127,106,134)(100,128,107,135)(101,129,108,136)(102,130,109,137)(103,131,110,138)(104,132,111,139)(105,133,112,140), (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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,84)(16,83)(17,82)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,112)(37,111)(38,110)(39,109)(40,108)(41,107)(42,106)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)(49,85)(50,98)(51,97)(52,96)(53,95)(54,94)(55,93)(56,92)(113,175)(114,174)(115,173)(116,172)(117,171)(118,170)(119,169)(120,182)(121,181)(122,180)(123,179)(124,178)(125,177)(126,176)(127,196)(128,195)(129,194)(130,193)(131,192)(132,191)(133,190)(134,189)(135,188)(136,187)(137,186)(138,185)(139,184)(140,183)(141,217)(142,216)(143,215)(144,214)(145,213)(146,212)(147,211)(148,224)(149,223)(150,222)(151,221)(152,220)(153,219)(154,218)(155,203)(156,202)(157,201)(158,200)(159,199)(160,198)(161,197)(162,210)(163,209)(164,208)(165,207)(166,206)(167,205)(168,204)>;
G:=Group( (1,120)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,140)(22,127)(23,128)(24,129)(25,130)(26,131)(27,132)(28,133)(29,148)(30,149)(31,150)(32,151)(33,152)(34,153)(35,154)(36,141)(37,142)(38,143)(39,144)(40,145)(41,146)(42,147)(43,162)(44,163)(45,164)(46,165)(47,166)(48,167)(49,168)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,161)(57,176)(58,177)(59,178)(60,179)(61,180)(62,181)(63,182)(64,169)(65,170)(66,171)(67,172)(68,173)(69,174)(70,175)(71,190)(72,191)(73,192)(74,193)(75,194)(76,195)(77,196)(78,183)(79,184)(80,185)(81,186)(82,187)(83,188)(84,189)(85,204)(86,205)(87,206)(88,207)(89,208)(90,209)(91,210)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(99,218)(100,219)(101,220)(102,221)(103,222)(104,223)(105,224)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(113,169)(114,170)(115,171)(116,172)(117,173)(118,174)(119,175)(120,176)(121,177)(122,178)(123,179)(124,180)(125,181)(126,182)(127,183)(128,184)(129,185)(130,186)(131,187)(132,188)(133,189)(134,190)(135,191)(136,192)(137,193)(138,194)(139,195)(140,196)(141,197)(142,198)(143,199)(144,200)(145,201)(146,202)(147,203)(148,204)(149,205)(150,206)(151,207)(152,208)(153,209)(154,210)(155,211)(156,212)(157,213)(158,214)(159,215)(160,216)(161,217)(162,218)(163,219)(164,220)(165,221)(166,222)(167,223)(168,224), (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56)(57,78,64,71)(58,79,65,72)(59,80,66,73)(60,81,67,74)(61,82,68,75)(62,83,69,76)(63,84,70,77)(85,99,92,106)(86,100,93,107)(87,101,94,108)(88,102,95,109)(89,103,96,110)(90,104,97,111)(91,105,98,112)(113,134,120,127)(114,135,121,128)(115,136,122,129)(116,137,123,130)(117,138,124,131)(118,139,125,132)(119,140,126,133)(141,155,148,162)(142,156,149,163)(143,157,150,164)(144,158,151,165)(145,159,152,166)(146,160,153,167)(147,161,154,168)(169,190,176,183)(170,191,177,184)(171,192,178,185)(172,193,179,186)(173,194,180,187)(174,195,181,188)(175,196,182,189)(197,211,204,218)(198,212,205,219)(199,213,206,220)(200,214,207,221)(201,215,208,222)(202,216,209,223)(203,217,210,224), (1,204,8,197)(2,205,9,198)(3,206,10,199)(4,207,11,200)(5,208,12,201)(6,209,13,202)(7,210,14,203)(15,218,22,211)(16,219,23,212)(17,220,24,213)(18,221,25,214)(19,222,26,215)(20,223,27,216)(21,224,28,217)(29,169,36,176)(30,170,37,177)(31,171,38,178)(32,172,39,179)(33,173,40,180)(34,174,41,181)(35,175,42,182)(43,183,50,190)(44,184,51,191)(45,185,52,192)(46,186,53,193)(47,187,54,194)(48,188,55,195)(49,189,56,196)(57,148,64,141)(58,149,65,142)(59,150,66,143)(60,151,67,144)(61,152,68,145)(62,153,69,146)(63,154,70,147)(71,162,78,155)(72,163,79,156)(73,164,80,157)(74,165,81,158)(75,166,82,159)(76,167,83,160)(77,168,84,161)(85,113,92,120)(86,114,93,121)(87,115,94,122)(88,116,95,123)(89,117,96,124)(90,118,97,125)(91,119,98,126)(99,127,106,134)(100,128,107,135)(101,129,108,136)(102,130,109,137)(103,131,110,138)(104,132,111,139)(105,133,112,140), (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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,84)(16,83)(17,82)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,112)(37,111)(38,110)(39,109)(40,108)(41,107)(42,106)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)(49,85)(50,98)(51,97)(52,96)(53,95)(54,94)(55,93)(56,92)(113,175)(114,174)(115,173)(116,172)(117,171)(118,170)(119,169)(120,182)(121,181)(122,180)(123,179)(124,178)(125,177)(126,176)(127,196)(128,195)(129,194)(130,193)(131,192)(132,191)(133,190)(134,189)(135,188)(136,187)(137,186)(138,185)(139,184)(140,183)(141,217)(142,216)(143,215)(144,214)(145,213)(146,212)(147,211)(148,224)(149,223)(150,222)(151,221)(152,220)(153,219)(154,218)(155,203)(156,202)(157,201)(158,200)(159,199)(160,198)(161,197)(162,210)(163,209)(164,208)(165,207)(166,206)(167,205)(168,204) );
G=PermutationGroup([[(1,120),(2,121),(3,122),(4,123),(5,124),(6,125),(7,126),(8,113),(9,114),(10,115),(11,116),(12,117),(13,118),(14,119),(15,134),(16,135),(17,136),(18,137),(19,138),(20,139),(21,140),(22,127),(23,128),(24,129),(25,130),(26,131),(27,132),(28,133),(29,148),(30,149),(31,150),(32,151),(33,152),(34,153),(35,154),(36,141),(37,142),(38,143),(39,144),(40,145),(41,146),(42,147),(43,162),(44,163),(45,164),(46,165),(47,166),(48,167),(49,168),(50,155),(51,156),(52,157),(53,158),(54,159),(55,160),(56,161),(57,176),(58,177),(59,178),(60,179),(61,180),(62,181),(63,182),(64,169),(65,170),(66,171),(67,172),(68,173),(69,174),(70,175),(71,190),(72,191),(73,192),(74,193),(75,194),(76,195),(77,196),(78,183),(79,184),(80,185),(81,186),(82,187),(83,188),(84,189),(85,204),(86,205),(87,206),(88,207),(89,208),(90,209),(91,210),(92,197),(93,198),(94,199),(95,200),(96,201),(97,202),(98,203),(99,218),(100,219),(101,220),(102,221),(103,222),(104,223),(105,224),(106,211),(107,212),(108,213),(109,214),(110,215),(111,216),(112,217)], [(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,73),(18,74),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112),(113,169),(114,170),(115,171),(116,172),(117,173),(118,174),(119,175),(120,176),(121,177),(122,178),(123,179),(124,180),(125,181),(126,182),(127,183),(128,184),(129,185),(130,186),(131,187),(132,188),(133,189),(134,190),(135,191),(136,192),(137,193),(138,194),(139,195),(140,196),(141,197),(142,198),(143,199),(144,200),(145,201),(146,202),(147,203),(148,204),(149,205),(150,206),(151,207),(152,208),(153,209),(154,210),(155,211),(156,212),(157,213),(158,214),(159,215),(160,216),(161,217),(162,218),(163,219),(164,220),(165,221),(166,222),(167,223),(168,224)], [(1,22,8,15),(2,23,9,16),(3,24,10,17),(4,25,11,18),(5,26,12,19),(6,27,13,20),(7,28,14,21),(29,43,36,50),(30,44,37,51),(31,45,38,52),(32,46,39,53),(33,47,40,54),(34,48,41,55),(35,49,42,56),(57,78,64,71),(58,79,65,72),(59,80,66,73),(60,81,67,74),(61,82,68,75),(62,83,69,76),(63,84,70,77),(85,99,92,106),(86,100,93,107),(87,101,94,108),(88,102,95,109),(89,103,96,110),(90,104,97,111),(91,105,98,112),(113,134,120,127),(114,135,121,128),(115,136,122,129),(116,137,123,130),(117,138,124,131),(118,139,125,132),(119,140,126,133),(141,155,148,162),(142,156,149,163),(143,157,150,164),(144,158,151,165),(145,159,152,166),(146,160,153,167),(147,161,154,168),(169,190,176,183),(170,191,177,184),(171,192,178,185),(172,193,179,186),(173,194,180,187),(174,195,181,188),(175,196,182,189),(197,211,204,218),(198,212,205,219),(199,213,206,220),(200,214,207,221),(201,215,208,222),(202,216,209,223),(203,217,210,224)], [(1,204,8,197),(2,205,9,198),(3,206,10,199),(4,207,11,200),(5,208,12,201),(6,209,13,202),(7,210,14,203),(15,218,22,211),(16,219,23,212),(17,220,24,213),(18,221,25,214),(19,222,26,215),(20,223,27,216),(21,224,28,217),(29,169,36,176),(30,170,37,177),(31,171,38,178),(32,172,39,179),(33,173,40,180),(34,174,41,181),(35,175,42,182),(43,183,50,190),(44,184,51,191),(45,185,52,192),(46,186,53,193),(47,187,54,194),(48,188,55,195),(49,189,56,196),(57,148,64,141),(58,149,65,142),(59,150,66,143),(60,151,67,144),(61,152,68,145),(62,153,69,146),(63,154,70,147),(71,162,78,155),(72,163,79,156),(73,164,80,157),(74,165,81,158),(75,166,82,159),(76,167,83,160),(77,168,84,161),(85,113,92,120),(86,114,93,121),(87,115,94,122),(88,116,95,123),(89,117,96,124),(90,118,97,125),(91,119,98,126),(99,127,106,134),(100,128,107,135),(101,129,108,136),(102,130,109,137),(103,131,110,138),(104,132,111,139),(105,133,112,140)], [(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)], [(1,63),(2,62),(3,61),(4,60),(5,59),(6,58),(7,57),(8,70),(9,69),(10,68),(11,67),(12,66),(13,65),(14,64),(15,84),(16,83),(17,82),(18,81),(19,80),(20,79),(21,78),(22,77),(23,76),(24,75),(25,74),(26,73),(27,72),(28,71),(29,105),(30,104),(31,103),(32,102),(33,101),(34,100),(35,99),(36,112),(37,111),(38,110),(39,109),(40,108),(41,107),(42,106),(43,91),(44,90),(45,89),(46,88),(47,87),(48,86),(49,85),(50,98),(51,97),(52,96),(53,95),(54,94),(55,93),(56,92),(113,175),(114,174),(115,173),(116,172),(117,171),(118,170),(119,169),(120,182),(121,181),(122,180),(123,179),(124,178),(125,177),(126,176),(127,196),(128,195),(129,194),(130,193),(131,192),(132,191),(133,190),(134,189),(135,188),(136,187),(137,186),(138,185),(139,184),(140,183),(141,217),(142,216),(143,215),(144,214),(145,213),(146,212),(147,211),(148,224),(149,223),(150,222),(151,221),(152,220),(153,219),(154,218),(155,203),(156,202),(157,201),(158,200),(159,199),(160,198),(161,197),(162,210),(163,209),(164,208),(165,207),(166,206),(167,205),(168,204)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | SD16 | D14 | D14 | C7⋊D4 | C7⋊D4 | Q8⋊D7 |
| kernel | C22×Q8⋊D7 | C22×C7⋊C8 | C2×Q8⋊D7 | C22×D28 | Q8×C2×C14 | C2×C28 | C22×C14 | C22×Q8 | C2×C14 | C22×C4 | C2×Q8 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 12 | 1 | 1 | 3 | 1 | 3 | 8 | 3 | 18 | 18 | 6 | 12 |
Matrix representation of C22×Q8⋊D7 ►in GL5(𝔽113)
| 112 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 112 | 112 | 0 | 0 |
| 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 26 | 13 | 0 | 0 |
| 0 | 87 | 87 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 25 | 112 |
| 0 | 0 | 0 | 26 | 112 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 34 | 89 |
| 0 | 0 | 0 | 34 | 79 |
G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,112,0,0,0,0,0,112],[1,0,0,0,0,0,112,2,0,0,0,112,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,26,87,0,0,0,13,87,0,0,0,0,0,112,0,0,0,0,0,112],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,25,26,0,0,0,112,112],[1,0,0,0,0,0,1,0,0,0,0,1,112,0,0,0,0,0,34,34,0,0,0,89,79] >;
C22×Q8⋊D7 in GAP, Magma, Sage, TeX
C_2^2\times Q_8\rtimes D_7
% in TeX
G:=Group("C2^2xQ8:D7"); // GroupNames label
G:=SmallGroup(448,1260);
// by ID
G=gap.SmallGroup(448,1260);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,675,136,1684,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=e^7=f^2=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*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=c^-1,c*e=e*c,d*e=e*d,f*d*f=c^-1*d,f*e*f=e^-1>;
// generators/relations