direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×Q8⋊6D4, C14.1622+ 1+4, Q8⋊6(C7×D4), (C7×Q8)⋊24D4, (D4×C28)⋊47C2, (C4×D4)⋊18C14, C4⋊1D4⋊8C14, (Q8×C28)⋊33C2, (C4×Q8)⋊13C14, C4.44(D4×C14), C28⋊19(C4○D4), C4⋊D4⋊14C14, C28.405(C2×D4), C42.45(C2×C14), (C2×C28).677C23, (C4×C28).286C22, (C2×C14).370C24, C14.198(C22×D4), (D4×C14).220C22, C22.44(C23×C14), C23.17(C22×C14), (Q8×C14).286C22, C2.14(C7×2+ 1+4), (C22×C14).101C23, (C22×C28).456C22, C4⋊3(C7×C4○D4), C2.22(D4×C2×C14), (C2×C4○D4)⋊9C14, (C14×C4○D4)⋊25C2, (C7×C4⋊1D4)⋊19C2, C4⋊C4.72(C2×C14), (C7×C4⋊D4)⋊41C2, C2.23(C14×C4○D4), (C2×D4).68(C2×C14), C14.242(C2×C4○D4), (C2×Q8).74(C2×C14), (C7×C4⋊C4).398C22, C22⋊C4.21(C2×C14), (C22×C4).68(C2×C14), (C2×C4).137(C22×C14), (C7×C22⋊C4).154C22, SmallGroup(448,1333)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×Q8⋊6D4
G = < a,b,c,d,e | a7=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >
Subgroups: 506 in 312 conjugacy classes, 166 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C28, C28, C2×C14, C2×C14, C4×D4, C4×Q8, C4⋊D4, C4⋊1D4, C2×C4○D4, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, Q8⋊6D4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, C7×C4○D4, D4×C28, Q8×C28, C7×C4⋊D4, C7×C4⋊1D4, C14×C4○D4, C7×Q8⋊6D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C24, C2×C14, C22×D4, C2×C4○D4, 2+ 1+4, C7×D4, C22×C14, Q8⋊6D4, D4×C14, C7×C4○D4, C23×C14, D4×C2×C14, C14×C4○D4, C7×2+ 1+4, C7×Q8⋊6D4
►On 224 points
Generators in S224
(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 67 34 63)(2 68 35 57)(3 69 29 58)(4 70 30 59)(5 64 31 60)(6 65 32 61)(7 66 33 62)(8 204 222 200)(9 205 223 201)(10 206 224 202)(11 207 218 203)(12 208 219 197)(13 209 220 198)(14 210 221 199)(15 195 22 212)(16 196 23 213)(17 190 24 214)(18 191 25 215)(19 192 26 216)(20 193 27 217)(21 194 28 211)(36 71 43 54)(37 72 44 55)(38 73 45 56)(39 74 46 50)(40 75 47 51)(41 76 48 52)(42 77 49 53)(78 119 102 123)(79 113 103 124)(80 114 104 125)(81 115 105 126)(82 116 99 120)(83 117 100 121)(84 118 101 122)(85 107 96 131)(86 108 97 132)(87 109 98 133)(88 110 92 127)(89 111 93 128)(90 112 94 129)(91 106 95 130)(134 179 158 175)(135 180 159 169)(136 181 160 170)(137 182 161 171)(138 176 155 172)(139 177 156 173)(140 178 157 174)(141 187 152 163)(142 188 153 164)(143 189 154 165)(144 183 148 166)(145 184 149 167)(146 185 150 168)(147 186 151 162)
(1 151 34 147)(2 152 35 141)(3 153 29 142)(4 154 30 143)(5 148 31 144)(6 149 32 145)(7 150 33 146)(8 116 222 120)(9 117 223 121)(10 118 224 122)(11 119 218 123)(12 113 219 124)(13 114 220 125)(14 115 221 126)(15 128 22 111)(16 129 23 112)(17 130 24 106)(18 131 25 107)(19 132 26 108)(20 133 27 109)(21 127 28 110)(36 155 43 138)(37 156 44 139)(38 157 45 140)(39 158 46 134)(40 159 47 135)(41 160 48 136)(42 161 49 137)(50 175 74 179)(51 169 75 180)(52 170 76 181)(53 171 77 182)(54 172 71 176)(55 173 72 177)(56 174 73 178)(57 163 68 187)(58 164 69 188)(59 165 70 189)(60 166 64 183)(61 167 65 184)(62 168 66 185)(63 162 67 186)(78 203 102 207)(79 197 103 208)(80 198 104 209)(81 199 105 210)(82 200 99 204)(83 201 100 205)(84 202 101 206)(85 191 96 215)(86 192 97 216)(87 193 98 217)(88 194 92 211)(89 195 93 212)(90 196 94 213)(91 190 95 214)
(1 95 39 102)(2 96 40 103)(3 97 41 104)(4 98 42 105)(5 92 36 99)(6 93 37 100)(7 94 38 101)(8 183 28 176)(9 184 22 177)(10 185 23 178)(11 186 24 179)(12 187 25 180)(13 188 26 181)(14 189 27 182)(15 173 223 167)(16 174 224 168)(17 175 218 162)(18 169 219 163)(19 170 220 164)(20 171 221 165)(21 172 222 166)(29 86 48 80)(30 87 49 81)(31 88 43 82)(32 89 44 83)(33 90 45 84)(34 91 46 78)(35 85 47 79)(50 123 63 130)(51 124 57 131)(52 125 58 132)(53 126 59 133)(54 120 60 127)(55 121 61 128)(56 122 62 129)(64 110 71 116)(65 111 72 117)(66 112 73 118)(67 106 74 119)(68 107 75 113)(69 108 76 114)(70 109 77 115)(134 207 147 214)(135 208 141 215)(136 209 142 216)(137 210 143 217)(138 204 144 211)(139 205 145 212)(140 206 146 213)(148 194 155 200)(149 195 156 201)(150 196 157 202)(151 190 158 203)(152 191 159 197)(153 192 160 198)(154 193 161 199)
(1 158)(2 159)(3 160)(4 161)(5 155)(6 156)(7 157)(8 120)(9 121)(10 122)(11 123)(12 124)(13 125)(14 126)(15 111)(16 112)(17 106)(18 107)(19 108)(20 109)(21 110)(22 128)(23 129)(24 130)(25 131)(26 132)(27 133)(28 127)(29 136)(30 137)(31 138)(32 139)(33 140)(34 134)(35 135)(36 148)(37 149)(38 150)(39 151)(40 152)(41 153)(42 154)(43 144)(44 145)(45 146)(46 147)(47 141)(48 142)(49 143)(50 186)(51 187)(52 188)(53 189)(54 183)(55 184)(56 185)(57 180)(58 181)(59 182)(60 176)(61 177)(62 178)(63 179)(64 172)(65 173)(66 174)(67 175)(68 169)(69 170)(70 171)(71 166)(72 167)(73 168)(74 162)(75 163)(76 164)(77 165)(78 207)(79 208)(80 209)(81 210)(82 204)(83 205)(84 206)(85 215)(86 216)(87 217)(88 211)(89 212)(90 213)(91 214)(92 194)(93 195)(94 196)(95 190)(96 191)(97 192)(98 193)(99 200)(100 201)(101 202)(102 203)(103 197)(104 198)(105 199)(113 219)(114 220)(115 221)(116 222)(117 223)(118 224)(119 218)
G:=sub<Sym(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), (1,67,34,63)(2,68,35,57)(3,69,29,58)(4,70,30,59)(5,64,31,60)(6,65,32,61)(7,66,33,62)(8,204,222,200)(9,205,223,201)(10,206,224,202)(11,207,218,203)(12,208,219,197)(13,209,220,198)(14,210,221,199)(15,195,22,212)(16,196,23,213)(17,190,24,214)(18,191,25,215)(19,192,26,216)(20,193,27,217)(21,194,28,211)(36,71,43,54)(37,72,44,55)(38,73,45,56)(39,74,46,50)(40,75,47,51)(41,76,48,52)(42,77,49,53)(78,119,102,123)(79,113,103,124)(80,114,104,125)(81,115,105,126)(82,116,99,120)(83,117,100,121)(84,118,101,122)(85,107,96,131)(86,108,97,132)(87,109,98,133)(88,110,92,127)(89,111,93,128)(90,112,94,129)(91,106,95,130)(134,179,158,175)(135,180,159,169)(136,181,160,170)(137,182,161,171)(138,176,155,172)(139,177,156,173)(140,178,157,174)(141,187,152,163)(142,188,153,164)(143,189,154,165)(144,183,148,166)(145,184,149,167)(146,185,150,168)(147,186,151,162), (1,151,34,147)(2,152,35,141)(3,153,29,142)(4,154,30,143)(5,148,31,144)(6,149,32,145)(7,150,33,146)(8,116,222,120)(9,117,223,121)(10,118,224,122)(11,119,218,123)(12,113,219,124)(13,114,220,125)(14,115,221,126)(15,128,22,111)(16,129,23,112)(17,130,24,106)(18,131,25,107)(19,132,26,108)(20,133,27,109)(21,127,28,110)(36,155,43,138)(37,156,44,139)(38,157,45,140)(39,158,46,134)(40,159,47,135)(41,160,48,136)(42,161,49,137)(50,175,74,179)(51,169,75,180)(52,170,76,181)(53,171,77,182)(54,172,71,176)(55,173,72,177)(56,174,73,178)(57,163,68,187)(58,164,69,188)(59,165,70,189)(60,166,64,183)(61,167,65,184)(62,168,66,185)(63,162,67,186)(78,203,102,207)(79,197,103,208)(80,198,104,209)(81,199,105,210)(82,200,99,204)(83,201,100,205)(84,202,101,206)(85,191,96,215)(86,192,97,216)(87,193,98,217)(88,194,92,211)(89,195,93,212)(90,196,94,213)(91,190,95,214), (1,95,39,102)(2,96,40,103)(3,97,41,104)(4,98,42,105)(5,92,36,99)(6,93,37,100)(7,94,38,101)(8,183,28,176)(9,184,22,177)(10,185,23,178)(11,186,24,179)(12,187,25,180)(13,188,26,181)(14,189,27,182)(15,173,223,167)(16,174,224,168)(17,175,218,162)(18,169,219,163)(19,170,220,164)(20,171,221,165)(21,172,222,166)(29,86,48,80)(30,87,49,81)(31,88,43,82)(32,89,44,83)(33,90,45,84)(34,91,46,78)(35,85,47,79)(50,123,63,130)(51,124,57,131)(52,125,58,132)(53,126,59,133)(54,120,60,127)(55,121,61,128)(56,122,62,129)(64,110,71,116)(65,111,72,117)(66,112,73,118)(67,106,74,119)(68,107,75,113)(69,108,76,114)(70,109,77,115)(134,207,147,214)(135,208,141,215)(136,209,142,216)(137,210,143,217)(138,204,144,211)(139,205,145,212)(140,206,146,213)(148,194,155,200)(149,195,156,201)(150,196,157,202)(151,190,158,203)(152,191,159,197)(153,192,160,198)(154,193,161,199), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,120)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,111)(16,112)(17,106)(18,107)(19,108)(20,109)(21,110)(22,128)(23,129)(24,130)(25,131)(26,132)(27,133)(28,127)(29,136)(30,137)(31,138)(32,139)(33,140)(34,134)(35,135)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,144)(44,145)(45,146)(46,147)(47,141)(48,142)(49,143)(50,186)(51,187)(52,188)(53,189)(54,183)(55,184)(56,185)(57,180)(58,181)(59,182)(60,176)(61,177)(62,178)(63,179)(64,172)(65,173)(66,174)(67,175)(68,169)(69,170)(70,171)(71,166)(72,167)(73,168)(74,162)(75,163)(76,164)(77,165)(78,207)(79,208)(80,209)(81,210)(82,204)(83,205)(84,206)(85,215)(86,216)(87,217)(88,211)(89,212)(90,213)(91,214)(92,194)(93,195)(94,196)(95,190)(96,191)(97,192)(98,193)(99,200)(100,201)(101,202)(102,203)(103,197)(104,198)(105,199)(113,219)(114,220)(115,221)(116,222)(117,223)(118,224)(119,218)>;
G:=Group( (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), (1,67,34,63)(2,68,35,57)(3,69,29,58)(4,70,30,59)(5,64,31,60)(6,65,32,61)(7,66,33,62)(8,204,222,200)(9,205,223,201)(10,206,224,202)(11,207,218,203)(12,208,219,197)(13,209,220,198)(14,210,221,199)(15,195,22,212)(16,196,23,213)(17,190,24,214)(18,191,25,215)(19,192,26,216)(20,193,27,217)(21,194,28,211)(36,71,43,54)(37,72,44,55)(38,73,45,56)(39,74,46,50)(40,75,47,51)(41,76,48,52)(42,77,49,53)(78,119,102,123)(79,113,103,124)(80,114,104,125)(81,115,105,126)(82,116,99,120)(83,117,100,121)(84,118,101,122)(85,107,96,131)(86,108,97,132)(87,109,98,133)(88,110,92,127)(89,111,93,128)(90,112,94,129)(91,106,95,130)(134,179,158,175)(135,180,159,169)(136,181,160,170)(137,182,161,171)(138,176,155,172)(139,177,156,173)(140,178,157,174)(141,187,152,163)(142,188,153,164)(143,189,154,165)(144,183,148,166)(145,184,149,167)(146,185,150,168)(147,186,151,162), (1,151,34,147)(2,152,35,141)(3,153,29,142)(4,154,30,143)(5,148,31,144)(6,149,32,145)(7,150,33,146)(8,116,222,120)(9,117,223,121)(10,118,224,122)(11,119,218,123)(12,113,219,124)(13,114,220,125)(14,115,221,126)(15,128,22,111)(16,129,23,112)(17,130,24,106)(18,131,25,107)(19,132,26,108)(20,133,27,109)(21,127,28,110)(36,155,43,138)(37,156,44,139)(38,157,45,140)(39,158,46,134)(40,159,47,135)(41,160,48,136)(42,161,49,137)(50,175,74,179)(51,169,75,180)(52,170,76,181)(53,171,77,182)(54,172,71,176)(55,173,72,177)(56,174,73,178)(57,163,68,187)(58,164,69,188)(59,165,70,189)(60,166,64,183)(61,167,65,184)(62,168,66,185)(63,162,67,186)(78,203,102,207)(79,197,103,208)(80,198,104,209)(81,199,105,210)(82,200,99,204)(83,201,100,205)(84,202,101,206)(85,191,96,215)(86,192,97,216)(87,193,98,217)(88,194,92,211)(89,195,93,212)(90,196,94,213)(91,190,95,214), (1,95,39,102)(2,96,40,103)(3,97,41,104)(4,98,42,105)(5,92,36,99)(6,93,37,100)(7,94,38,101)(8,183,28,176)(9,184,22,177)(10,185,23,178)(11,186,24,179)(12,187,25,180)(13,188,26,181)(14,189,27,182)(15,173,223,167)(16,174,224,168)(17,175,218,162)(18,169,219,163)(19,170,220,164)(20,171,221,165)(21,172,222,166)(29,86,48,80)(30,87,49,81)(31,88,43,82)(32,89,44,83)(33,90,45,84)(34,91,46,78)(35,85,47,79)(50,123,63,130)(51,124,57,131)(52,125,58,132)(53,126,59,133)(54,120,60,127)(55,121,61,128)(56,122,62,129)(64,110,71,116)(65,111,72,117)(66,112,73,118)(67,106,74,119)(68,107,75,113)(69,108,76,114)(70,109,77,115)(134,207,147,214)(135,208,141,215)(136,209,142,216)(137,210,143,217)(138,204,144,211)(139,205,145,212)(140,206,146,213)(148,194,155,200)(149,195,156,201)(150,196,157,202)(151,190,158,203)(152,191,159,197)(153,192,160,198)(154,193,161,199), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,120)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,111)(16,112)(17,106)(18,107)(19,108)(20,109)(21,110)(22,128)(23,129)(24,130)(25,131)(26,132)(27,133)(28,127)(29,136)(30,137)(31,138)(32,139)(33,140)(34,134)(35,135)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,144)(44,145)(45,146)(46,147)(47,141)(48,142)(49,143)(50,186)(51,187)(52,188)(53,189)(54,183)(55,184)(56,185)(57,180)(58,181)(59,182)(60,176)(61,177)(62,178)(63,179)(64,172)(65,173)(66,174)(67,175)(68,169)(69,170)(70,171)(71,166)(72,167)(73,168)(74,162)(75,163)(76,164)(77,165)(78,207)(79,208)(80,209)(81,210)(82,204)(83,205)(84,206)(85,215)(86,216)(87,217)(88,211)(89,212)(90,213)(91,214)(92,194)(93,195)(94,196)(95,190)(96,191)(97,192)(98,193)(99,200)(100,201)(101,202)(102,203)(103,197)(104,198)(105,199)(113,219)(114,220)(115,221)(116,222)(117,223)(118,224)(119,218) );
G=PermutationGroup([[(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)], [(1,67,34,63),(2,68,35,57),(3,69,29,58),(4,70,30,59),(5,64,31,60),(6,65,32,61),(7,66,33,62),(8,204,222,200),(9,205,223,201),(10,206,224,202),(11,207,218,203),(12,208,219,197),(13,209,220,198),(14,210,221,199),(15,195,22,212),(16,196,23,213),(17,190,24,214),(18,191,25,215),(19,192,26,216),(20,193,27,217),(21,194,28,211),(36,71,43,54),(37,72,44,55),(38,73,45,56),(39,74,46,50),(40,75,47,51),(41,76,48,52),(42,77,49,53),(78,119,102,123),(79,113,103,124),(80,114,104,125),(81,115,105,126),(82,116,99,120),(83,117,100,121),(84,118,101,122),(85,107,96,131),(86,108,97,132),(87,109,98,133),(88,110,92,127),(89,111,93,128),(90,112,94,129),(91,106,95,130),(134,179,158,175),(135,180,159,169),(136,181,160,170),(137,182,161,171),(138,176,155,172),(139,177,156,173),(140,178,157,174),(141,187,152,163),(142,188,153,164),(143,189,154,165),(144,183,148,166),(145,184,149,167),(146,185,150,168),(147,186,151,162)], [(1,151,34,147),(2,152,35,141),(3,153,29,142),(4,154,30,143),(5,148,31,144),(6,149,32,145),(7,150,33,146),(8,116,222,120),(9,117,223,121),(10,118,224,122),(11,119,218,123),(12,113,219,124),(13,114,220,125),(14,115,221,126),(15,128,22,111),(16,129,23,112),(17,130,24,106),(18,131,25,107),(19,132,26,108),(20,133,27,109),(21,127,28,110),(36,155,43,138),(37,156,44,139),(38,157,45,140),(39,158,46,134),(40,159,47,135),(41,160,48,136),(42,161,49,137),(50,175,74,179),(51,169,75,180),(52,170,76,181),(53,171,77,182),(54,172,71,176),(55,173,72,177),(56,174,73,178),(57,163,68,187),(58,164,69,188),(59,165,70,189),(60,166,64,183),(61,167,65,184),(62,168,66,185),(63,162,67,186),(78,203,102,207),(79,197,103,208),(80,198,104,209),(81,199,105,210),(82,200,99,204),(83,201,100,205),(84,202,101,206),(85,191,96,215),(86,192,97,216),(87,193,98,217),(88,194,92,211),(89,195,93,212),(90,196,94,213),(91,190,95,214)], [(1,95,39,102),(2,96,40,103),(3,97,41,104),(4,98,42,105),(5,92,36,99),(6,93,37,100),(7,94,38,101),(8,183,28,176),(9,184,22,177),(10,185,23,178),(11,186,24,179),(12,187,25,180),(13,188,26,181),(14,189,27,182),(15,173,223,167),(16,174,224,168),(17,175,218,162),(18,169,219,163),(19,170,220,164),(20,171,221,165),(21,172,222,166),(29,86,48,80),(30,87,49,81),(31,88,43,82),(32,89,44,83),(33,90,45,84),(34,91,46,78),(35,85,47,79),(50,123,63,130),(51,124,57,131),(52,125,58,132),(53,126,59,133),(54,120,60,127),(55,121,61,128),(56,122,62,129),(64,110,71,116),(65,111,72,117),(66,112,73,118),(67,106,74,119),(68,107,75,113),(69,108,76,114),(70,109,77,115),(134,207,147,214),(135,208,141,215),(136,209,142,216),(137,210,143,217),(138,204,144,211),(139,205,145,212),(140,206,146,213),(148,194,155,200),(149,195,156,201),(150,196,157,202),(151,190,158,203),(152,191,159,197),(153,192,160,198),(154,193,161,199)], [(1,158),(2,159),(3,160),(4,161),(5,155),(6,156),(7,157),(8,120),(9,121),(10,122),(11,123),(12,124),(13,125),(14,126),(15,111),(16,112),(17,106),(18,107),(19,108),(20,109),(21,110),(22,128),(23,129),(24,130),(25,131),(26,132),(27,133),(28,127),(29,136),(30,137),(31,138),(32,139),(33,140),(34,134),(35,135),(36,148),(37,149),(38,150),(39,151),(40,152),(41,153),(42,154),(43,144),(44,145),(45,146),(46,147),(47,141),(48,142),(49,143),(50,186),(51,187),(52,188),(53,189),(54,183),(55,184),(56,185),(57,180),(58,181),(59,182),(60,176),(61,177),(62,178),(63,179),(64,172),(65,173),(66,174),(67,175),(68,169),(69,170),(70,171),(71,166),(72,167),(73,168),(74,162),(75,163),(76,164),(77,165),(78,207),(79,208),(80,209),(81,210),(82,204),(83,205),(84,206),(85,215),(86,216),(87,217),(88,211),(89,212),(90,213),(91,214),(92,194),(93,195),(94,196),(95,190),(96,191),(97,192),(98,193),(99,200),(100,201),(101,202),(102,203),(103,197),(104,198),(105,199),(113,219),(114,220),(115,221),(116,222),(117,223),(118,224),(119,218)]])
175 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4L | 4M | 4N | 4O | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14BB | 28A | ··· | 28BT | 28BU | ··· | 28CL |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
175 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | C4○D4 | C7×D4 | C7×C4○D4 | 2+ 1+4 | C7×2+ 1+4 |
| kernel | C7×Q8⋊6D4 | D4×C28 | Q8×C28 | C7×C4⋊D4 | C7×C4⋊1D4 | C14×C4○D4 | Q8⋊6D4 | C4×D4 | C4×Q8 | C4⋊D4 | C4⋊1D4 | C2×C4○D4 | C7×Q8 | C28 | Q8 | C4 | C14 | C2 |
| # reps | 1 | 3 | 1 | 6 | 3 | 2 | 6 | 18 | 6 | 36 | 18 | 12 | 4 | 4 | 24 | 24 | 1 | 6 |
Matrix representation of C7×Q8⋊6D4 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 24 | 0 |
| 0 | 0 | 0 | 24 |
| 0 | 1 | 0 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 17 | 0 | 0 |
| 17 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 17 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 27 | 12 |
| 0 | 0 | 2 | 2 |
| 0 | 17 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 28 | 27 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,24,0,0,0,0,24],[0,28,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,17,0,0,17,0,0,0,0,0,1,0,0,0,0,1],[17,0,0,0,0,12,0,0,0,0,27,2,0,0,12,2],[0,12,0,0,17,0,0,0,0,0,28,0,0,0,27,1] >;
C7×Q8⋊6D4 in GAP, Magma, Sage, TeX
C_7\times Q_8\rtimes_6D_4
% in TeX
G:=Group("C7xQ8:6D4"); // GroupNames label
G:=SmallGroup(448,1333);
// by ID
G=gap.SmallGroup(448,1333);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,792,4790,604,1690,416]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=d^4=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations