direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×Q8×D7, C42.230D14, C28⋊9(C2×Q8), (Q8×C28)⋊6C2, Dic7⋊7(C2×Q8), C4⋊C4.322D14, (Q8×Dic7)⋊31C2, D14.14(C2×Q8), (D7×C42).5C2, Dic14⋊14(C2×C4), (C4×Dic14)⋊37C2, C14.24(C23×C4), C28.34(C22×C4), (C2×Q8).199D14, D14.36(C4○D4), Dic7⋊3Q8⋊46C2, C14.28(C22×Q8), (C2×C14).115C24, (C4×C28).167C22, (C2×C28).494C23, D14.24(C22×C4), C22.34(C23×D7), C4⋊Dic7.365C22, (Q8×C14).215C22, Dic7.10(C22×C4), Dic7⋊C4.136C22, (C4×Dic7).294C22, (C2×Dic7).309C23, (C22×D7).255C23, (C2×Dic14).289C22, C7⋊3(C2×C4×Q8), C2.3(C2×Q8×D7), C4.34(C2×C4×D7), (C2×Q8×D7).9C2, C2.5(D7×C4○D4), (D7×C4⋊C4).14C2, (C7×Q8)⋊10(C2×C4), C2.26(D7×C22×C4), (C4×D7).17(C2×C4), C14.144(C2×C4○D4), (C2×C4×D7).294C22, (C7×C4⋊C4).343C22, (C2×C4).820(C22×D7), SmallGroup(448,1024)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×Q8×D7
G = < a,b,c,d,e | a4=b4=d7=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1060 in 298 conjugacy classes, 169 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, D7, C14, C42, C42, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, D14, C2×C14, C2×C42, C2×C4⋊C4, C4×Q8, C4×Q8, C22×Q8, Dic14, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×D7, C2×C4×Q8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×C4×D7, Q8×D7, Q8×C14, C4×Dic14, D7×C42, Dic7⋊3Q8, D7×C4⋊C4, Q8×Dic7, Q8×C28, C2×Q8×D7, C4×Q8×D7
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D7, C22×C4, C2×Q8, C4○D4, C24, D14, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, C4×D7, C22×D7, C2×C4×Q8, C2×C4×D7, Q8×D7, C23×D7, D7×C22×C4, C2×Q8×D7, D7×C4○D4, C4×Q8×D7
►On 224 points
Generators in S224
(1 48 20 34)(2 49 21 35)(3 43 15 29)(4 44 16 30)(5 45 17 31)(6 46 18 32)(7 47 19 33)(8 50 22 36)(9 51 23 37)(10 52 24 38)(11 53 25 39)(12 54 26 40)(13 55 27 41)(14 56 28 42)(57 99 71 85)(58 100 72 86)(59 101 73 87)(60 102 74 88)(61 103 75 89)(62 104 76 90)(63 105 77 91)(64 106 78 92)(65 107 79 93)(66 108 80 94)(67 109 81 95)(68 110 82 96)(69 111 83 97)(70 112 84 98)(113 155 127 141)(114 156 128 142)(115 157 129 143)(116 158 130 144)(117 159 131 145)(118 160 132 146)(119 161 133 147)(120 162 134 148)(121 163 135 149)(122 164 136 150)(123 165 137 151)(124 166 138 152)(125 167 139 153)(126 168 140 154)(169 211 183 197)(170 212 184 198)(171 213 185 199)(172 214 186 200)(173 215 187 201)(174 216 188 202)(175 217 189 203)(176 218 190 204)(177 219 191 205)(178 220 192 206)(179 221 193 207)(180 222 194 208)(181 223 195 209)(182 224 196 210)
(1 83 13 76)(2 84 14 77)(3 78 8 71)(4 79 9 72)(5 80 10 73)(6 81 11 74)(7 82 12 75)(15 64 22 57)(16 65 23 58)(17 66 24 59)(18 67 25 60)(19 68 26 61)(20 69 27 62)(21 70 28 63)(29 106 36 99)(30 107 37 100)(31 108 38 101)(32 109 39 102)(33 110 40 103)(34 111 41 104)(35 112 42 105)(43 92 50 85)(44 93 51 86)(45 94 52 87)(46 95 53 88)(47 96 54 89)(48 97 55 90)(49 98 56 91)(113 183 120 190)(114 184 121 191)(115 185 122 192)(116 186 123 193)(117 187 124 194)(118 188 125 195)(119 189 126 196)(127 169 134 176)(128 170 135 177)(129 171 136 178)(130 172 137 179)(131 173 138 180)(132 174 139 181)(133 175 140 182)(141 211 148 218)(142 212 149 219)(143 213 150 220)(144 214 151 221)(145 215 152 222)(146 216 153 223)(147 217 154 224)(155 197 162 204)(156 198 163 205)(157 199 164 206)(158 200 165 207)(159 201 166 208)(160 202 167 209)(161 203 168 210)
(1 139 13 132)(2 140 14 133)(3 134 8 127)(4 135 9 128)(5 136 10 129)(6 137 11 130)(7 138 12 131)(15 120 22 113)(16 121 23 114)(17 122 24 115)(18 123 25 116)(19 124 26 117)(20 125 27 118)(21 126 28 119)(29 162 36 155)(30 163 37 156)(31 164 38 157)(32 165 39 158)(33 166 40 159)(34 167 41 160)(35 168 42 161)(43 148 50 141)(44 149 51 142)(45 150 52 143)(46 151 53 144)(47 152 54 145)(48 153 55 146)(49 154 56 147)(57 190 64 183)(58 191 65 184)(59 192 66 185)(60 193 67 186)(61 194 68 187)(62 195 69 188)(63 196 70 189)(71 176 78 169)(72 177 79 170)(73 178 80 171)(74 179 81 172)(75 180 82 173)(76 181 83 174)(77 182 84 175)(85 218 92 211)(86 219 93 212)(87 220 94 213)(88 221 95 214)(89 222 96 215)(90 223 97 216)(91 224 98 217)(99 204 106 197)(100 205 107 198)(101 206 108 199)(102 207 109 200)(103 208 110 201)(104 209 111 202)(105 210 112 203)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(43 52)(44 51)(45 50)(46 56)(47 55)(48 54)(49 53)(57 66)(58 65)(59 64)(60 70)(61 69)(62 68)(63 67)(71 80)(72 79)(73 78)(74 84)(75 83)(76 82)(77 81)(85 94)(86 93)(87 92)(88 98)(89 97)(90 96)(91 95)(99 108)(100 107)(101 106)(102 112)(103 111)(104 110)(105 109)(113 122)(114 121)(115 120)(116 126)(117 125)(118 124)(119 123)(127 136)(128 135)(129 134)(130 140)(131 139)(132 138)(133 137)(141 150)(142 149)(143 148)(144 154)(145 153)(146 152)(147 151)(155 164)(156 163)(157 162)(158 168)(159 167)(160 166)(161 165)(169 178)(170 177)(171 176)(172 182)(173 181)(174 180)(175 179)(183 192)(184 191)(185 190)(186 196)(187 195)(188 194)(189 193)(197 206)(198 205)(199 204)(200 210)(201 209)(202 208)(203 207)(211 220)(212 219)(213 218)(214 224)(215 223)(216 222)(217 221)
G:=sub<Sym(224)| (1,48,20,34)(2,49,21,35)(3,43,15,29)(4,44,16,30)(5,45,17,31)(6,46,18,32)(7,47,19,33)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(57,99,71,85)(58,100,72,86)(59,101,73,87)(60,102,74,88)(61,103,75,89)(62,104,76,90)(63,105,77,91)(64,106,78,92)(65,107,79,93)(66,108,80,94)(67,109,81,95)(68,110,82,96)(69,111,83,97)(70,112,84,98)(113,155,127,141)(114,156,128,142)(115,157,129,143)(116,158,130,144)(117,159,131,145)(118,160,132,146)(119,161,133,147)(120,162,134,148)(121,163,135,149)(122,164,136,150)(123,165,137,151)(124,166,138,152)(125,167,139,153)(126,168,140,154)(169,211,183,197)(170,212,184,198)(171,213,185,199)(172,214,186,200)(173,215,187,201)(174,216,188,202)(175,217,189,203)(176,218,190,204)(177,219,191,205)(178,220,192,206)(179,221,193,207)(180,222,194,208)(181,223,195,209)(182,224,196,210), (1,83,13,76)(2,84,14,77)(3,78,8,71)(4,79,9,72)(5,80,10,73)(6,81,11,74)(7,82,12,75)(15,64,22,57)(16,65,23,58)(17,66,24,59)(18,67,25,60)(19,68,26,61)(20,69,27,62)(21,70,28,63)(29,106,36,99)(30,107,37,100)(31,108,38,101)(32,109,39,102)(33,110,40,103)(34,111,41,104)(35,112,42,105)(43,92,50,85)(44,93,51,86)(45,94,52,87)(46,95,53,88)(47,96,54,89)(48,97,55,90)(49,98,56,91)(113,183,120,190)(114,184,121,191)(115,185,122,192)(116,186,123,193)(117,187,124,194)(118,188,125,195)(119,189,126,196)(127,169,134,176)(128,170,135,177)(129,171,136,178)(130,172,137,179)(131,173,138,180)(132,174,139,181)(133,175,140,182)(141,211,148,218)(142,212,149,219)(143,213,150,220)(144,214,151,221)(145,215,152,222)(146,216,153,223)(147,217,154,224)(155,197,162,204)(156,198,163,205)(157,199,164,206)(158,200,165,207)(159,201,166,208)(160,202,167,209)(161,203,168,210), (1,139,13,132)(2,140,14,133)(3,134,8,127)(4,135,9,128)(5,136,10,129)(6,137,11,130)(7,138,12,131)(15,120,22,113)(16,121,23,114)(17,122,24,115)(18,123,25,116)(19,124,26,117)(20,125,27,118)(21,126,28,119)(29,162,36,155)(30,163,37,156)(31,164,38,157)(32,165,39,158)(33,166,40,159)(34,167,41,160)(35,168,42,161)(43,148,50,141)(44,149,51,142)(45,150,52,143)(46,151,53,144)(47,152,54,145)(48,153,55,146)(49,154,56,147)(57,190,64,183)(58,191,65,184)(59,192,66,185)(60,193,67,186)(61,194,68,187)(62,195,69,188)(63,196,70,189)(71,176,78,169)(72,177,79,170)(73,178,80,171)(74,179,81,172)(75,180,82,173)(76,181,83,174)(77,182,84,175)(85,218,92,211)(86,219,93,212)(87,220,94,213)(88,221,95,214)(89,222,96,215)(90,223,97,216)(91,224,98,217)(99,204,106,197)(100,205,107,198)(101,206,108,199)(102,207,109,200)(103,208,110,201)(104,209,111,202)(105,210,112,203), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53)(57,66)(58,65)(59,64)(60,70)(61,69)(62,68)(63,67)(71,80)(72,79)(73,78)(74,84)(75,83)(76,82)(77,81)(85,94)(86,93)(87,92)(88,98)(89,97)(90,96)(91,95)(99,108)(100,107)(101,106)(102,112)(103,111)(104,110)(105,109)(113,122)(114,121)(115,120)(116,126)(117,125)(118,124)(119,123)(127,136)(128,135)(129,134)(130,140)(131,139)(132,138)(133,137)(141,150)(142,149)(143,148)(144,154)(145,153)(146,152)(147,151)(155,164)(156,163)(157,162)(158,168)(159,167)(160,166)(161,165)(169,178)(170,177)(171,176)(172,182)(173,181)(174,180)(175,179)(183,192)(184,191)(185,190)(186,196)(187,195)(188,194)(189,193)(197,206)(198,205)(199,204)(200,210)(201,209)(202,208)(203,207)(211,220)(212,219)(213,218)(214,224)(215,223)(216,222)(217,221)>;
G:=Group( (1,48,20,34)(2,49,21,35)(3,43,15,29)(4,44,16,30)(5,45,17,31)(6,46,18,32)(7,47,19,33)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(57,99,71,85)(58,100,72,86)(59,101,73,87)(60,102,74,88)(61,103,75,89)(62,104,76,90)(63,105,77,91)(64,106,78,92)(65,107,79,93)(66,108,80,94)(67,109,81,95)(68,110,82,96)(69,111,83,97)(70,112,84,98)(113,155,127,141)(114,156,128,142)(115,157,129,143)(116,158,130,144)(117,159,131,145)(118,160,132,146)(119,161,133,147)(120,162,134,148)(121,163,135,149)(122,164,136,150)(123,165,137,151)(124,166,138,152)(125,167,139,153)(126,168,140,154)(169,211,183,197)(170,212,184,198)(171,213,185,199)(172,214,186,200)(173,215,187,201)(174,216,188,202)(175,217,189,203)(176,218,190,204)(177,219,191,205)(178,220,192,206)(179,221,193,207)(180,222,194,208)(181,223,195,209)(182,224,196,210), (1,83,13,76)(2,84,14,77)(3,78,8,71)(4,79,9,72)(5,80,10,73)(6,81,11,74)(7,82,12,75)(15,64,22,57)(16,65,23,58)(17,66,24,59)(18,67,25,60)(19,68,26,61)(20,69,27,62)(21,70,28,63)(29,106,36,99)(30,107,37,100)(31,108,38,101)(32,109,39,102)(33,110,40,103)(34,111,41,104)(35,112,42,105)(43,92,50,85)(44,93,51,86)(45,94,52,87)(46,95,53,88)(47,96,54,89)(48,97,55,90)(49,98,56,91)(113,183,120,190)(114,184,121,191)(115,185,122,192)(116,186,123,193)(117,187,124,194)(118,188,125,195)(119,189,126,196)(127,169,134,176)(128,170,135,177)(129,171,136,178)(130,172,137,179)(131,173,138,180)(132,174,139,181)(133,175,140,182)(141,211,148,218)(142,212,149,219)(143,213,150,220)(144,214,151,221)(145,215,152,222)(146,216,153,223)(147,217,154,224)(155,197,162,204)(156,198,163,205)(157,199,164,206)(158,200,165,207)(159,201,166,208)(160,202,167,209)(161,203,168,210), (1,139,13,132)(2,140,14,133)(3,134,8,127)(4,135,9,128)(5,136,10,129)(6,137,11,130)(7,138,12,131)(15,120,22,113)(16,121,23,114)(17,122,24,115)(18,123,25,116)(19,124,26,117)(20,125,27,118)(21,126,28,119)(29,162,36,155)(30,163,37,156)(31,164,38,157)(32,165,39,158)(33,166,40,159)(34,167,41,160)(35,168,42,161)(43,148,50,141)(44,149,51,142)(45,150,52,143)(46,151,53,144)(47,152,54,145)(48,153,55,146)(49,154,56,147)(57,190,64,183)(58,191,65,184)(59,192,66,185)(60,193,67,186)(61,194,68,187)(62,195,69,188)(63,196,70,189)(71,176,78,169)(72,177,79,170)(73,178,80,171)(74,179,81,172)(75,180,82,173)(76,181,83,174)(77,182,84,175)(85,218,92,211)(86,219,93,212)(87,220,94,213)(88,221,95,214)(89,222,96,215)(90,223,97,216)(91,224,98,217)(99,204,106,197)(100,205,107,198)(101,206,108,199)(102,207,109,200)(103,208,110,201)(104,209,111,202)(105,210,112,203), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53)(57,66)(58,65)(59,64)(60,70)(61,69)(62,68)(63,67)(71,80)(72,79)(73,78)(74,84)(75,83)(76,82)(77,81)(85,94)(86,93)(87,92)(88,98)(89,97)(90,96)(91,95)(99,108)(100,107)(101,106)(102,112)(103,111)(104,110)(105,109)(113,122)(114,121)(115,120)(116,126)(117,125)(118,124)(119,123)(127,136)(128,135)(129,134)(130,140)(131,139)(132,138)(133,137)(141,150)(142,149)(143,148)(144,154)(145,153)(146,152)(147,151)(155,164)(156,163)(157,162)(158,168)(159,167)(160,166)(161,165)(169,178)(170,177)(171,176)(172,182)(173,181)(174,180)(175,179)(183,192)(184,191)(185,190)(186,196)(187,195)(188,194)(189,193)(197,206)(198,205)(199,204)(200,210)(201,209)(202,208)(203,207)(211,220)(212,219)(213,218)(214,224)(215,223)(216,222)(217,221) );
G=PermutationGroup([[(1,48,20,34),(2,49,21,35),(3,43,15,29),(4,44,16,30),(5,45,17,31),(6,46,18,32),(7,47,19,33),(8,50,22,36),(9,51,23,37),(10,52,24,38),(11,53,25,39),(12,54,26,40),(13,55,27,41),(14,56,28,42),(57,99,71,85),(58,100,72,86),(59,101,73,87),(60,102,74,88),(61,103,75,89),(62,104,76,90),(63,105,77,91),(64,106,78,92),(65,107,79,93),(66,108,80,94),(67,109,81,95),(68,110,82,96),(69,111,83,97),(70,112,84,98),(113,155,127,141),(114,156,128,142),(115,157,129,143),(116,158,130,144),(117,159,131,145),(118,160,132,146),(119,161,133,147),(120,162,134,148),(121,163,135,149),(122,164,136,150),(123,165,137,151),(124,166,138,152),(125,167,139,153),(126,168,140,154),(169,211,183,197),(170,212,184,198),(171,213,185,199),(172,214,186,200),(173,215,187,201),(174,216,188,202),(175,217,189,203),(176,218,190,204),(177,219,191,205),(178,220,192,206),(179,221,193,207),(180,222,194,208),(181,223,195,209),(182,224,196,210)], [(1,83,13,76),(2,84,14,77),(3,78,8,71),(4,79,9,72),(5,80,10,73),(6,81,11,74),(7,82,12,75),(15,64,22,57),(16,65,23,58),(17,66,24,59),(18,67,25,60),(19,68,26,61),(20,69,27,62),(21,70,28,63),(29,106,36,99),(30,107,37,100),(31,108,38,101),(32,109,39,102),(33,110,40,103),(34,111,41,104),(35,112,42,105),(43,92,50,85),(44,93,51,86),(45,94,52,87),(46,95,53,88),(47,96,54,89),(48,97,55,90),(49,98,56,91),(113,183,120,190),(114,184,121,191),(115,185,122,192),(116,186,123,193),(117,187,124,194),(118,188,125,195),(119,189,126,196),(127,169,134,176),(128,170,135,177),(129,171,136,178),(130,172,137,179),(131,173,138,180),(132,174,139,181),(133,175,140,182),(141,211,148,218),(142,212,149,219),(143,213,150,220),(144,214,151,221),(145,215,152,222),(146,216,153,223),(147,217,154,224),(155,197,162,204),(156,198,163,205),(157,199,164,206),(158,200,165,207),(159,201,166,208),(160,202,167,209),(161,203,168,210)], [(1,139,13,132),(2,140,14,133),(3,134,8,127),(4,135,9,128),(5,136,10,129),(6,137,11,130),(7,138,12,131),(15,120,22,113),(16,121,23,114),(17,122,24,115),(18,123,25,116),(19,124,26,117),(20,125,27,118),(21,126,28,119),(29,162,36,155),(30,163,37,156),(31,164,38,157),(32,165,39,158),(33,166,40,159),(34,167,41,160),(35,168,42,161),(43,148,50,141),(44,149,51,142),(45,150,52,143),(46,151,53,144),(47,152,54,145),(48,153,55,146),(49,154,56,147),(57,190,64,183),(58,191,65,184),(59,192,66,185),(60,193,67,186),(61,194,68,187),(62,195,69,188),(63,196,70,189),(71,176,78,169),(72,177,79,170),(73,178,80,171),(74,179,81,172),(75,180,82,173),(76,181,83,174),(77,182,84,175),(85,218,92,211),(86,219,93,212),(87,220,94,213),(88,221,95,214),(89,222,96,215),(90,223,97,216),(91,224,98,217),(99,204,106,197),(100,205,107,198),(101,206,108,199),(102,207,109,200),(103,208,110,201),(104,209,111,202),(105,210,112,203)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,14),(7,13),(15,24),(16,23),(17,22),(18,28),(19,27),(20,26),(21,25),(29,38),(30,37),(31,36),(32,42),(33,41),(34,40),(35,39),(43,52),(44,51),(45,50),(46,56),(47,55),(48,54),(49,53),(57,66),(58,65),(59,64),(60,70),(61,69),(62,68),(63,67),(71,80),(72,79),(73,78),(74,84),(75,83),(76,82),(77,81),(85,94),(86,93),(87,92),(88,98),(89,97),(90,96),(91,95),(99,108),(100,107),(101,106),(102,112),(103,111),(104,110),(105,109),(113,122),(114,121),(115,120),(116,126),(117,125),(118,124),(119,123),(127,136),(128,135),(129,134),(130,140),(131,139),(132,138),(133,137),(141,150),(142,149),(143,148),(144,154),(145,153),(146,152),(147,151),(155,164),(156,163),(157,162),(158,168),(159,167),(160,166),(161,165),(169,178),(170,177),(171,176),(172,182),(173,181),(174,180),(175,179),(183,192),(184,191),(185,190),(186,196),(187,195),(188,194),(189,193),(197,206),(198,205),(199,204),(200,210),(201,209),(202,208),(203,207),(211,220),(212,219),(213,218),(214,224),(215,223),(216,222),(217,221)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | 4R | 4S | 4T | 4U | ··· | 4AF | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 7 | 7 | 7 | 7 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D7 | C4○D4 | D14 | D14 | D14 | C4×D7 | Q8×D7 | D7×C4○D4 |
| kernel | C4×Q8×D7 | C4×Dic14 | D7×C42 | Dic7⋊3Q8 | D7×C4⋊C4 | Q8×Dic7 | Q8×C28 | C2×Q8×D7 | Q8×D7 | C4×D7 | C4×Q8 | D14 | C42 | C4⋊C4 | C2×Q8 | Q8 | C4 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 16 | 4 | 3 | 4 | 9 | 9 | 3 | 24 | 6 | 6 |
Matrix representation of C4×Q8×D7 ►in GL4(𝔽29) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 12 |
| 0 | 0 | 24 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 21 |
| 0 | 0 | 26 | 27 |
| 26 | 1 | 0 | 0 |
| 27 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 10 | 28 | 0 | 0 |
| 12 | 19 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
G:=sub<GL(4,GF(29))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[28,0,0,0,0,28,0,0,0,0,28,24,0,0,12,1],[1,0,0,0,0,1,0,0,0,0,2,26,0,0,21,27],[26,27,0,0,1,10,0,0,0,0,1,0,0,0,0,1],[10,12,0,0,28,19,0,0,0,0,28,0,0,0,0,28] >;
C4×Q8×D7 in GAP, Magma, Sage, TeX
C_4\times Q_8\times D_7
% in TeX
G:=Group("C4xQ8xD7"); // GroupNames label
G:=SmallGroup(448,1024);
// by ID
G=gap.SmallGroup(448,1024);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,387,184,80,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^7=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=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations