direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×Q8⋊6D4, C14.1622+ (1+4), Q8⋊6(C7×D4), (C7×Q8)⋊24D4, (C4×D4)⋊18C14, (D4×C28)⋊47C2, 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×C14).370C24, (C4×C28).286C22, (C2×C28).677C23, 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, (C7×C4⋊D4)⋊41C2, C4⋊C4.72(C2×C14), 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
Subgroups: 506 in 312 conjugacy classes, 166 normal (20 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×5], C22, C22 [×18], C7, C2×C4, C2×C4 [×8], C2×C4 [×12], D4 [×24], Q8 [×4], C23 [×6], C14 [×3], C14 [×6], C42 [×3], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×3], C22×C4 [×6], C2×D4 [×15], C2×Q8, C4○D4 [×8], C28 [×8], C28 [×5], C2×C14, C2×C14 [×18], C4×D4 [×3], C4×Q8, C4⋊D4 [×6], C4⋊1D4 [×3], C2×C4○D4 [×2], C2×C28, C2×C28 [×8], C2×C28 [×12], C7×D4 [×24], C7×Q8 [×4], C22×C14 [×6], Q8⋊6D4, C4×C28 [×3], C7×C22⋊C4 [×6], C7×C4⋊C4, C7×C4⋊C4 [×3], C22×C28 [×6], D4×C14 [×15], Q8×C14, C7×C4○D4 [×8], D4×C28 [×3], Q8×C28, C7×C4⋊D4 [×6], C7×C4⋊1D4 [×3], C14×C4○D4 [×2], C7×Q8⋊6D4
Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×4], C23 [×15], C14 [×15], C2×D4 [×6], C4○D4 [×2], C24, C2×C14 [×35], C22×D4, C2×C4○D4, 2+ (1+4), C7×D4 [×4], C22×C14 [×15], Q8⋊6D4, D4×C14 [×6], C7×C4○D4 [×2], C23×C14, D4×C2×C14, C14×C4○D4, C7×2+ (1+4), C7×Q8⋊6D4
Generators and relations
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 >
►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 71 39 63)(2 72 40 57)(3 73 41 58)(4 74 42 59)(5 75 36 60)(6 76 37 61)(7 77 38 62)(8 196 23 205)(9 190 24 206)(10 191 25 207)(11 192 26 208)(12 193 27 209)(13 194 28 210)(14 195 22 204)(15 211 218 203)(16 212 219 197)(17 213 220 198)(18 214 221 199)(19 215 222 200)(20 216 223 201)(21 217 224 202)(29 51 44 67)(30 52 45 68)(31 53 46 69)(32 54 47 70)(33 55 48 64)(34 56 49 65)(35 50 43 66)(78 119 94 127)(79 113 95 128)(80 114 96 129)(81 115 97 130)(82 116 98 131)(83 117 92 132)(84 118 93 133)(85 107 100 123)(86 108 101 124)(87 109 102 125)(88 110 103 126)(89 111 104 120)(90 112 105 121)(91 106 99 122)(134 183 150 175)(135 184 151 169)(136 185 152 170)(137 186 153 171)(138 187 154 172)(139 188 148 173)(140 189 149 174)(141 179 156 163)(142 180 157 164)(143 181 158 165)(144 182 159 166)(145 176 160 167)(146 177 161 168)(147 178 155 162)
(1 155 39 147)(2 156 40 141)(3 157 41 142)(4 158 42 143)(5 159 36 144)(6 160 37 145)(7 161 38 146)(8 121 23 112)(9 122 24 106)(10 123 25 107)(11 124 26 108)(12 125 27 109)(13 126 28 110)(14 120 22 111)(15 119 218 127)(16 113 219 128)(17 114 220 129)(18 115 221 130)(19 116 222 131)(20 117 223 132)(21 118 224 133)(29 135 44 151)(30 136 45 152)(31 137 46 153)(32 138 47 154)(33 139 48 148)(34 140 49 149)(35 134 43 150)(50 175 66 183)(51 169 67 184)(52 170 68 185)(53 171 69 186)(54 172 70 187)(55 173 64 188)(56 174 65 189)(57 163 72 179)(58 164 73 180)(59 165 74 181)(60 166 75 182)(61 167 76 176)(62 168 77 177)(63 162 71 178)(78 203 94 211)(79 197 95 212)(80 198 96 213)(81 199 97 214)(82 200 98 215)(83 201 92 216)(84 202 93 217)(85 191 100 207)(86 192 101 208)(87 193 102 209)(88 194 103 210)(89 195 104 204)(90 196 105 205)(91 190 99 206)
(1 99 43 94)(2 100 44 95)(3 101 45 96)(4 102 46 97)(5 103 47 98)(6 104 48 92)(7 105 49 93)(8 174 224 168)(9 175 218 162)(10 169 219 163)(11 170 220 164)(12 171 221 165)(13 172 222 166)(14 173 223 167)(15 178 24 183)(16 179 25 184)(17 180 26 185)(18 181 27 186)(19 182 28 187)(20 176 22 188)(21 177 23 189)(29 79 40 85)(30 80 41 86)(31 81 42 87)(32 82 36 88)(33 83 37 89)(34 84 38 90)(35 78 39 91)(50 127 63 122)(51 128 57 123)(52 129 58 124)(53 130 59 125)(54 131 60 126)(55 132 61 120)(56 133 62 121)(64 117 76 111)(65 118 77 112)(66 119 71 106)(67 113 72 107)(68 114 73 108)(69 115 74 109)(70 116 75 110)(134 211 147 206)(135 212 141 207)(136 213 142 208)(137 214 143 209)(138 215 144 210)(139 216 145 204)(140 217 146 205)(148 201 160 195)(149 202 161 196)(150 203 155 190)(151 197 156 191)(152 198 157 192)(153 199 158 193)(154 200 159 194)
(1 150)(2 151)(3 152)(4 153)(5 154)(6 148)(7 149)(8 112)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 127)(16 128)(17 129)(18 130)(19 131)(20 132)(21 133)(22 120)(23 121)(24 122)(25 123)(26 124)(27 125)(28 126)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 138)(37 139)(38 140)(39 134)(40 135)(41 136)(42 137)(43 155)(44 156)(45 157)(46 158)(47 159)(48 160)(49 161)(50 178)(51 179)(52 180)(53 181)(54 182)(55 176)(56 177)(57 184)(58 185)(59 186)(60 187)(61 188)(62 189)(63 183)(64 167)(65 168)(66 162)(67 163)(68 164)(69 165)(70 166)(71 175)(72 169)(73 170)(74 171)(75 172)(76 173)(77 174)(78 211)(79 212)(80 213)(81 214)(82 215)(83 216)(84 217)(85 207)(86 208)(87 209)(88 210)(89 204)(90 205)(91 206)(92 201)(93 202)(94 203)(95 197)(96 198)(97 199)(98 200)(99 190)(100 191)(101 192)(102 193)(103 194)(104 195)(105 196)(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), (1,71,39,63)(2,72,40,57)(3,73,41,58)(4,74,42,59)(5,75,36,60)(6,76,37,61)(7,77,38,62)(8,196,23,205)(9,190,24,206)(10,191,25,207)(11,192,26,208)(12,193,27,209)(13,194,28,210)(14,195,22,204)(15,211,218,203)(16,212,219,197)(17,213,220,198)(18,214,221,199)(19,215,222,200)(20,216,223,201)(21,217,224,202)(29,51,44,67)(30,52,45,68)(31,53,46,69)(32,54,47,70)(33,55,48,64)(34,56,49,65)(35,50,43,66)(78,119,94,127)(79,113,95,128)(80,114,96,129)(81,115,97,130)(82,116,98,131)(83,117,92,132)(84,118,93,133)(85,107,100,123)(86,108,101,124)(87,109,102,125)(88,110,103,126)(89,111,104,120)(90,112,105,121)(91,106,99,122)(134,183,150,175)(135,184,151,169)(136,185,152,170)(137,186,153,171)(138,187,154,172)(139,188,148,173)(140,189,149,174)(141,179,156,163)(142,180,157,164)(143,181,158,165)(144,182,159,166)(145,176,160,167)(146,177,161,168)(147,178,155,162), (1,155,39,147)(2,156,40,141)(3,157,41,142)(4,158,42,143)(5,159,36,144)(6,160,37,145)(7,161,38,146)(8,121,23,112)(9,122,24,106)(10,123,25,107)(11,124,26,108)(12,125,27,109)(13,126,28,110)(14,120,22,111)(15,119,218,127)(16,113,219,128)(17,114,220,129)(18,115,221,130)(19,116,222,131)(20,117,223,132)(21,118,224,133)(29,135,44,151)(30,136,45,152)(31,137,46,153)(32,138,47,154)(33,139,48,148)(34,140,49,149)(35,134,43,150)(50,175,66,183)(51,169,67,184)(52,170,68,185)(53,171,69,186)(54,172,70,187)(55,173,64,188)(56,174,65,189)(57,163,72,179)(58,164,73,180)(59,165,74,181)(60,166,75,182)(61,167,76,176)(62,168,77,177)(63,162,71,178)(78,203,94,211)(79,197,95,212)(80,198,96,213)(81,199,97,214)(82,200,98,215)(83,201,92,216)(84,202,93,217)(85,191,100,207)(86,192,101,208)(87,193,102,209)(88,194,103,210)(89,195,104,204)(90,196,105,205)(91,190,99,206), (1,99,43,94)(2,100,44,95)(3,101,45,96)(4,102,46,97)(5,103,47,98)(6,104,48,92)(7,105,49,93)(8,174,224,168)(9,175,218,162)(10,169,219,163)(11,170,220,164)(12,171,221,165)(13,172,222,166)(14,173,223,167)(15,178,24,183)(16,179,25,184)(17,180,26,185)(18,181,27,186)(19,182,28,187)(20,176,22,188)(21,177,23,189)(29,79,40,85)(30,80,41,86)(31,81,42,87)(32,82,36,88)(33,83,37,89)(34,84,38,90)(35,78,39,91)(50,127,63,122)(51,128,57,123)(52,129,58,124)(53,130,59,125)(54,131,60,126)(55,132,61,120)(56,133,62,121)(64,117,76,111)(65,118,77,112)(66,119,71,106)(67,113,72,107)(68,114,73,108)(69,115,74,109)(70,116,75,110)(134,211,147,206)(135,212,141,207)(136,213,142,208)(137,214,143,209)(138,215,144,210)(139,216,145,204)(140,217,146,205)(148,201,160,195)(149,202,161,196)(150,203,155,190)(151,197,156,191)(152,198,157,192)(153,199,158,193)(154,200,159,194), (1,150)(2,151)(3,152)(4,153)(5,154)(6,148)(7,149)(8,112)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,127)(16,128)(17,129)(18,130)(19,131)(20,132)(21,133)(22,120)(23,121)(24,122)(25,123)(26,124)(27,125)(28,126)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,138)(37,139)(38,140)(39,134)(40,135)(41,136)(42,137)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(50,178)(51,179)(52,180)(53,181)(54,182)(55,176)(56,177)(57,184)(58,185)(59,186)(60,187)(61,188)(62,189)(63,183)(64,167)(65,168)(66,162)(67,163)(68,164)(69,165)(70,166)(71,175)(72,169)(73,170)(74,171)(75,172)(76,173)(77,174)(78,211)(79,212)(80,213)(81,214)(82,215)(83,216)(84,217)(85,207)(86,208)(87,209)(88,210)(89,204)(90,205)(91,206)(92,201)(93,202)(94,203)(95,197)(96,198)(97,199)(98,200)(99,190)(100,191)(101,192)(102,193)(103,194)(104,195)(105,196)(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), (1,71,39,63)(2,72,40,57)(3,73,41,58)(4,74,42,59)(5,75,36,60)(6,76,37,61)(7,77,38,62)(8,196,23,205)(9,190,24,206)(10,191,25,207)(11,192,26,208)(12,193,27,209)(13,194,28,210)(14,195,22,204)(15,211,218,203)(16,212,219,197)(17,213,220,198)(18,214,221,199)(19,215,222,200)(20,216,223,201)(21,217,224,202)(29,51,44,67)(30,52,45,68)(31,53,46,69)(32,54,47,70)(33,55,48,64)(34,56,49,65)(35,50,43,66)(78,119,94,127)(79,113,95,128)(80,114,96,129)(81,115,97,130)(82,116,98,131)(83,117,92,132)(84,118,93,133)(85,107,100,123)(86,108,101,124)(87,109,102,125)(88,110,103,126)(89,111,104,120)(90,112,105,121)(91,106,99,122)(134,183,150,175)(135,184,151,169)(136,185,152,170)(137,186,153,171)(138,187,154,172)(139,188,148,173)(140,189,149,174)(141,179,156,163)(142,180,157,164)(143,181,158,165)(144,182,159,166)(145,176,160,167)(146,177,161,168)(147,178,155,162), (1,155,39,147)(2,156,40,141)(3,157,41,142)(4,158,42,143)(5,159,36,144)(6,160,37,145)(7,161,38,146)(8,121,23,112)(9,122,24,106)(10,123,25,107)(11,124,26,108)(12,125,27,109)(13,126,28,110)(14,120,22,111)(15,119,218,127)(16,113,219,128)(17,114,220,129)(18,115,221,130)(19,116,222,131)(20,117,223,132)(21,118,224,133)(29,135,44,151)(30,136,45,152)(31,137,46,153)(32,138,47,154)(33,139,48,148)(34,140,49,149)(35,134,43,150)(50,175,66,183)(51,169,67,184)(52,170,68,185)(53,171,69,186)(54,172,70,187)(55,173,64,188)(56,174,65,189)(57,163,72,179)(58,164,73,180)(59,165,74,181)(60,166,75,182)(61,167,76,176)(62,168,77,177)(63,162,71,178)(78,203,94,211)(79,197,95,212)(80,198,96,213)(81,199,97,214)(82,200,98,215)(83,201,92,216)(84,202,93,217)(85,191,100,207)(86,192,101,208)(87,193,102,209)(88,194,103,210)(89,195,104,204)(90,196,105,205)(91,190,99,206), (1,99,43,94)(2,100,44,95)(3,101,45,96)(4,102,46,97)(5,103,47,98)(6,104,48,92)(7,105,49,93)(8,174,224,168)(9,175,218,162)(10,169,219,163)(11,170,220,164)(12,171,221,165)(13,172,222,166)(14,173,223,167)(15,178,24,183)(16,179,25,184)(17,180,26,185)(18,181,27,186)(19,182,28,187)(20,176,22,188)(21,177,23,189)(29,79,40,85)(30,80,41,86)(31,81,42,87)(32,82,36,88)(33,83,37,89)(34,84,38,90)(35,78,39,91)(50,127,63,122)(51,128,57,123)(52,129,58,124)(53,130,59,125)(54,131,60,126)(55,132,61,120)(56,133,62,121)(64,117,76,111)(65,118,77,112)(66,119,71,106)(67,113,72,107)(68,114,73,108)(69,115,74,109)(70,116,75,110)(134,211,147,206)(135,212,141,207)(136,213,142,208)(137,214,143,209)(138,215,144,210)(139,216,145,204)(140,217,146,205)(148,201,160,195)(149,202,161,196)(150,203,155,190)(151,197,156,191)(152,198,157,192)(153,199,158,193)(154,200,159,194), (1,150)(2,151)(3,152)(4,153)(5,154)(6,148)(7,149)(8,112)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,127)(16,128)(17,129)(18,130)(19,131)(20,132)(21,133)(22,120)(23,121)(24,122)(25,123)(26,124)(27,125)(28,126)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,138)(37,139)(38,140)(39,134)(40,135)(41,136)(42,137)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(50,178)(51,179)(52,180)(53,181)(54,182)(55,176)(56,177)(57,184)(58,185)(59,186)(60,187)(61,188)(62,189)(63,183)(64,167)(65,168)(66,162)(67,163)(68,164)(69,165)(70,166)(71,175)(72,169)(73,170)(74,171)(75,172)(76,173)(77,174)(78,211)(79,212)(80,213)(81,214)(82,215)(83,216)(84,217)(85,207)(86,208)(87,209)(88,210)(89,204)(90,205)(91,206)(92,201)(93,202)(94,203)(95,197)(96,198)(97,199)(98,200)(99,190)(100,191)(101,192)(102,193)(103,194)(104,195)(105,196)(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)], [(1,71,39,63),(2,72,40,57),(3,73,41,58),(4,74,42,59),(5,75,36,60),(6,76,37,61),(7,77,38,62),(8,196,23,205),(9,190,24,206),(10,191,25,207),(11,192,26,208),(12,193,27,209),(13,194,28,210),(14,195,22,204),(15,211,218,203),(16,212,219,197),(17,213,220,198),(18,214,221,199),(19,215,222,200),(20,216,223,201),(21,217,224,202),(29,51,44,67),(30,52,45,68),(31,53,46,69),(32,54,47,70),(33,55,48,64),(34,56,49,65),(35,50,43,66),(78,119,94,127),(79,113,95,128),(80,114,96,129),(81,115,97,130),(82,116,98,131),(83,117,92,132),(84,118,93,133),(85,107,100,123),(86,108,101,124),(87,109,102,125),(88,110,103,126),(89,111,104,120),(90,112,105,121),(91,106,99,122),(134,183,150,175),(135,184,151,169),(136,185,152,170),(137,186,153,171),(138,187,154,172),(139,188,148,173),(140,189,149,174),(141,179,156,163),(142,180,157,164),(143,181,158,165),(144,182,159,166),(145,176,160,167),(146,177,161,168),(147,178,155,162)], [(1,155,39,147),(2,156,40,141),(3,157,41,142),(4,158,42,143),(5,159,36,144),(6,160,37,145),(7,161,38,146),(8,121,23,112),(9,122,24,106),(10,123,25,107),(11,124,26,108),(12,125,27,109),(13,126,28,110),(14,120,22,111),(15,119,218,127),(16,113,219,128),(17,114,220,129),(18,115,221,130),(19,116,222,131),(20,117,223,132),(21,118,224,133),(29,135,44,151),(30,136,45,152),(31,137,46,153),(32,138,47,154),(33,139,48,148),(34,140,49,149),(35,134,43,150),(50,175,66,183),(51,169,67,184),(52,170,68,185),(53,171,69,186),(54,172,70,187),(55,173,64,188),(56,174,65,189),(57,163,72,179),(58,164,73,180),(59,165,74,181),(60,166,75,182),(61,167,76,176),(62,168,77,177),(63,162,71,178),(78,203,94,211),(79,197,95,212),(80,198,96,213),(81,199,97,214),(82,200,98,215),(83,201,92,216),(84,202,93,217),(85,191,100,207),(86,192,101,208),(87,193,102,209),(88,194,103,210),(89,195,104,204),(90,196,105,205),(91,190,99,206)], [(1,99,43,94),(2,100,44,95),(3,101,45,96),(4,102,46,97),(5,103,47,98),(6,104,48,92),(7,105,49,93),(8,174,224,168),(9,175,218,162),(10,169,219,163),(11,170,220,164),(12,171,221,165),(13,172,222,166),(14,173,223,167),(15,178,24,183),(16,179,25,184),(17,180,26,185),(18,181,27,186),(19,182,28,187),(20,176,22,188),(21,177,23,189),(29,79,40,85),(30,80,41,86),(31,81,42,87),(32,82,36,88),(33,83,37,89),(34,84,38,90),(35,78,39,91),(50,127,63,122),(51,128,57,123),(52,129,58,124),(53,130,59,125),(54,131,60,126),(55,132,61,120),(56,133,62,121),(64,117,76,111),(65,118,77,112),(66,119,71,106),(67,113,72,107),(68,114,73,108),(69,115,74,109),(70,116,75,110),(134,211,147,206),(135,212,141,207),(136,213,142,208),(137,214,143,209),(138,215,144,210),(139,216,145,204),(140,217,146,205),(148,201,160,195),(149,202,161,196),(150,203,155,190),(151,197,156,191),(152,198,157,192),(153,199,158,193),(154,200,159,194)], [(1,150),(2,151),(3,152),(4,153),(5,154),(6,148),(7,149),(8,112),(9,106),(10,107),(11,108),(12,109),(13,110),(14,111),(15,127),(16,128),(17,129),(18,130),(19,131),(20,132),(21,133),(22,120),(23,121),(24,122),(25,123),(26,124),(27,125),(28,126),(29,141),(30,142),(31,143),(32,144),(33,145),(34,146),(35,147),(36,138),(37,139),(38,140),(39,134),(40,135),(41,136),(42,137),(43,155),(44,156),(45,157),(46,158),(47,159),(48,160),(49,161),(50,178),(51,179),(52,180),(53,181),(54,182),(55,176),(56,177),(57,184),(58,185),(59,186),(60,187),(61,188),(62,189),(63,183),(64,167),(65,168),(66,162),(67,163),(68,164),(69,165),(70,166),(71,175),(72,169),(73,170),(74,171),(75,172),(76,173),(77,174),(78,211),(79,212),(80,213),(81,214),(82,215),(83,216),(84,217),(85,207),(86,208),(87,209),(88,210),(89,204),(90,205),(91,206),(92,201),(93,202),(94,203),(95,197),(96,198),(97,199),(98,200),(99,190),(100,191),(101,192),(102,193),(103,194),(104,195),(105,196),(113,219),(114,220),(115,221),(116,222),(117,223),(118,224),(119,218)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀