direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D14⋊Q8, C4⋊C4⋊39D14, D14⋊1(C2×Q8), (C22×D7)⋊4Q8, C14⋊2(C22⋊Q8), C22.33(Q8×D7), (C2×C14).52C24, Dic7.42(C2×D4), C14.44(C22×D4), C22.134(D4×D7), C14.24(C22×Q8), (C2×C28).580C23, Dic7⋊C4⋊61C22, D14⋊C4.92C22, (C2×Dic7).190D4, (C22×Dic14)⋊7C2, (C22×C4).178D14, C22.86(C23×D7), (C2×Dic14)⋊50C22, C22.77(C4○D28), C23.329(C22×D7), (C22×C14).401C23, (C22×C28).434C22, (C2×Dic7).190C23, (C22×D7).158C23, (C23×D7).101C22, (C22×Dic7).81C22, C2.7(C2×Q8×D7), C2.17(C2×D4×D7), (C2×C4⋊C4)⋊17D7, C7⋊2(C2×C22⋊Q8), (C14×C4⋊C4)⋊14C2, (C7×C4⋊C4)⋊47C22, C14.21(C2×C4○D4), C2.23(C2×C4○D28), (C2×C14).93(C2×Q8), (D7×C22×C4).22C2, (C2×Dic7⋊C4)⋊44C2, (C2×D14⋊C4).18C2, (C2×C14).390(C2×D4), (C2×C4×D7).288C22, (C2×C4).142(C22×D7), (C2×C14).107(C4○D4), SmallGroup(448,961)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D14⋊Q8
G = < a,b,c,d,e | a2=b14=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=b5c, ece-1=b12c, ede-1=d-1 >
Subgroups: 1476 in 322 conjugacy classes, 127 normal (31 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C24, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, Dic14, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C2×C22⋊Q8, Dic7⋊C4, D14⋊C4, C7×C4⋊C4, C2×Dic14, C2×Dic14, C2×C4×D7, C2×C4×D7, C22×Dic7, C22×C28, C23×D7, D14⋊Q8, C2×Dic7⋊C4, C2×D14⋊C4, C14×C4⋊C4, C22×Dic14, D7×C22×C4, C2×D14⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, C24, D14, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C22×D7, C2×C22⋊Q8, C4○D28, D4×D7, Q8×D7, C23×D7, D14⋊Q8, C2×C4○D28, C2×D4×D7, C2×Q8×D7, C2×D14⋊Q8
►On 224 points
Generators in S224
(1 197)(2 198)(3 199)(4 200)(5 201)(6 202)(7 203)(8 204)(9 205)(10 206)(11 207)(12 208)(13 209)(14 210)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 29)(24 30)(25 31)(26 32)(27 33)(28 34)(43 220)(44 221)(45 222)(46 223)(47 224)(48 211)(49 212)(50 213)(51 214)(52 215)(53 216)(54 217)(55 218)(56 219)(57 196)(58 183)(59 184)(60 185)(61 186)(62 187)(63 188)(64 189)(65 190)(66 191)(67 192)(68 193)(69 194)(70 195)(71 158)(72 159)(73 160)(74 161)(75 162)(76 163)(77 164)(78 165)(79 166)(80 167)(81 168)(82 155)(83 156)(84 157)(85 176)(86 177)(87 178)(88 179)(89 180)(90 181)(91 182)(92 169)(93 170)(94 171)(95 172)(96 173)(97 174)(98 175)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)(106 121)(107 122)(108 123)(109 124)(110 125)(111 126)(112 113)(127 151)(128 152)(129 153)(130 154)(131 141)(132 142)(133 143)(134 144)(135 145)(136 146)(137 147)(138 148)(139 149)(140 150)
(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)
(1 211)(2 224)(3 223)(4 222)(5 221)(6 220)(7 219)(8 218)(9 217)(10 216)(11 215)(12 214)(13 213)(14 212)(15 102)(16 101)(17 100)(18 99)(19 112)(20 111)(21 110)(22 109)(23 108)(24 107)(25 106)(26 105)(27 104)(28 103)(29 123)(30 122)(31 121)(32 120)(33 119)(34 118)(35 117)(36 116)(37 115)(38 114)(39 113)(40 126)(41 125)(42 124)(43 202)(44 201)(45 200)(46 199)(47 198)(48 197)(49 210)(50 209)(51 208)(52 207)(53 206)(54 205)(55 204)(56 203)(57 146)(58 145)(59 144)(60 143)(61 142)(62 141)(63 154)(64 153)(65 152)(66 151)(67 150)(68 149)(69 148)(70 147)(71 171)(72 170)(73 169)(74 182)(75 181)(76 180)(77 179)(78 178)(79 177)(80 176)(81 175)(82 174)(83 173)(84 172)(85 167)(86 166)(87 165)(88 164)(89 163)(90 162)(91 161)(92 160)(93 159)(94 158)(95 157)(96 156)(97 155)(98 168)(127 191)(128 190)(129 189)(130 188)(131 187)(132 186)(133 185)(134 184)(135 183)(136 196)(137 195)(138 194)(139 193)(140 192)
(1 116 56 17)(2 115 43 16)(3 114 44 15)(4 113 45 28)(5 126 46 27)(6 125 47 26)(7 124 48 25)(8 123 49 24)(9 122 50 23)(10 121 51 22)(11 120 52 21)(12 119 53 20)(13 118 54 19)(14 117 55 18)(29 205 107 213)(30 204 108 212)(31 203 109 211)(32 202 110 224)(33 201 111 223)(34 200 112 222)(35 199 99 221)(36 198 100 220)(37 197 101 219)(38 210 102 218)(39 209 103 217)(40 208 104 216)(41 207 105 215)(42 206 106 214)(57 94 140 76)(58 93 127 75)(59 92 128 74)(60 91 129 73)(61 90 130 72)(62 89 131 71)(63 88 132 84)(64 87 133 83)(65 86 134 82)(66 85 135 81)(67 98 136 80)(68 97 137 79)(69 96 138 78)(70 95 139 77)(141 158 187 180)(142 157 188 179)(143 156 189 178)(144 155 190 177)(145 168 191 176)(146 167 192 175)(147 166 193 174)(148 165 194 173)(149 164 195 172)(150 163 196 171)(151 162 183 170)(152 161 184 169)(153 160 185 182)(154 159 186 181)
(1 145 56 191)(2 144 43 190)(3 143 44 189)(4 142 45 188)(5 141 46 187)(6 154 47 186)(7 153 48 185)(8 152 49 184)(9 151 50 183)(10 150 51 196)(11 149 52 195)(12 148 53 194)(13 147 54 193)(14 146 55 192)(15 156 114 178)(16 155 115 177)(17 168 116 176)(18 167 117 175)(19 166 118 174)(20 165 119 173)(21 164 120 172)(22 163 121 171)(23 162 122 170)(24 161 123 169)(25 160 124 182)(26 159 125 181)(27 158 126 180)(28 157 113 179)(29 75 107 93)(30 74 108 92)(31 73 109 91)(32 72 110 90)(33 71 111 89)(34 84 112 88)(35 83 99 87)(36 82 100 86)(37 81 101 85)(38 80 102 98)(39 79 103 97)(40 78 104 96)(41 77 105 95)(42 76 106 94)(57 206 140 214)(58 205 127 213)(59 204 128 212)(60 203 129 211)(61 202 130 224)(62 201 131 223)(63 200 132 222)(64 199 133 221)(65 198 134 220)(66 197 135 219)(67 210 136 218)(68 209 137 217)(69 208 138 216)(70 207 139 215)
G:=sub<Sym(224)| (1,197)(2,198)(3,199)(4,200)(5,201)(6,202)(7,203)(8,204)(9,205)(10,206)(11,207)(12,208)(13,209)(14,210)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,29)(24,30)(25,31)(26,32)(27,33)(28,34)(43,220)(44,221)(45,222)(46,223)(47,224)(48,211)(49,212)(50,213)(51,214)(52,215)(53,216)(54,217)(55,218)(56,219)(57,196)(58,183)(59,184)(60,185)(61,186)(62,187)(63,188)(64,189)(65,190)(66,191)(67,192)(68,193)(69,194)(70,195)(71,158)(72,159)(73,160)(74,161)(75,162)(76,163)(77,164)(78,165)(79,166)(80,167)(81,168)(82,155)(83,156)(84,157)(85,176)(86,177)(87,178)(88,179)(89,180)(90,181)(91,182)(92,169)(93,170)(94,171)(95,172)(96,173)(97,174)(98,175)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(106,121)(107,122)(108,123)(109,124)(110,125)(111,126)(112,113)(127,151)(128,152)(129,153)(130,154)(131,141)(132,142)(133,143)(134,144)(135,145)(136,146)(137,147)(138,148)(139,149)(140,150), (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), (1,211)(2,224)(3,223)(4,222)(5,221)(6,220)(7,219)(8,218)(9,217)(10,216)(11,215)(12,214)(13,213)(14,212)(15,102)(16,101)(17,100)(18,99)(19,112)(20,111)(21,110)(22,109)(23,108)(24,107)(25,106)(26,105)(27,104)(28,103)(29,123)(30,122)(31,121)(32,120)(33,119)(34,118)(35,117)(36,116)(37,115)(38,114)(39,113)(40,126)(41,125)(42,124)(43,202)(44,201)(45,200)(46,199)(47,198)(48,197)(49,210)(50,209)(51,208)(52,207)(53,206)(54,205)(55,204)(56,203)(57,146)(58,145)(59,144)(60,143)(61,142)(62,141)(63,154)(64,153)(65,152)(66,151)(67,150)(68,149)(69,148)(70,147)(71,171)(72,170)(73,169)(74,182)(75,181)(76,180)(77,179)(78,178)(79,177)(80,176)(81,175)(82,174)(83,173)(84,172)(85,167)(86,166)(87,165)(88,164)(89,163)(90,162)(91,161)(92,160)(93,159)(94,158)(95,157)(96,156)(97,155)(98,168)(127,191)(128,190)(129,189)(130,188)(131,187)(132,186)(133,185)(134,184)(135,183)(136,196)(137,195)(138,194)(139,193)(140,192), (1,116,56,17)(2,115,43,16)(3,114,44,15)(4,113,45,28)(5,126,46,27)(6,125,47,26)(7,124,48,25)(8,123,49,24)(9,122,50,23)(10,121,51,22)(11,120,52,21)(12,119,53,20)(13,118,54,19)(14,117,55,18)(29,205,107,213)(30,204,108,212)(31,203,109,211)(32,202,110,224)(33,201,111,223)(34,200,112,222)(35,199,99,221)(36,198,100,220)(37,197,101,219)(38,210,102,218)(39,209,103,217)(40,208,104,216)(41,207,105,215)(42,206,106,214)(57,94,140,76)(58,93,127,75)(59,92,128,74)(60,91,129,73)(61,90,130,72)(62,89,131,71)(63,88,132,84)(64,87,133,83)(65,86,134,82)(66,85,135,81)(67,98,136,80)(68,97,137,79)(69,96,138,78)(70,95,139,77)(141,158,187,180)(142,157,188,179)(143,156,189,178)(144,155,190,177)(145,168,191,176)(146,167,192,175)(147,166,193,174)(148,165,194,173)(149,164,195,172)(150,163,196,171)(151,162,183,170)(152,161,184,169)(153,160,185,182)(154,159,186,181), (1,145,56,191)(2,144,43,190)(3,143,44,189)(4,142,45,188)(5,141,46,187)(6,154,47,186)(7,153,48,185)(8,152,49,184)(9,151,50,183)(10,150,51,196)(11,149,52,195)(12,148,53,194)(13,147,54,193)(14,146,55,192)(15,156,114,178)(16,155,115,177)(17,168,116,176)(18,167,117,175)(19,166,118,174)(20,165,119,173)(21,164,120,172)(22,163,121,171)(23,162,122,170)(24,161,123,169)(25,160,124,182)(26,159,125,181)(27,158,126,180)(28,157,113,179)(29,75,107,93)(30,74,108,92)(31,73,109,91)(32,72,110,90)(33,71,111,89)(34,84,112,88)(35,83,99,87)(36,82,100,86)(37,81,101,85)(38,80,102,98)(39,79,103,97)(40,78,104,96)(41,77,105,95)(42,76,106,94)(57,206,140,214)(58,205,127,213)(59,204,128,212)(60,203,129,211)(61,202,130,224)(62,201,131,223)(63,200,132,222)(64,199,133,221)(65,198,134,220)(66,197,135,219)(67,210,136,218)(68,209,137,217)(69,208,138,216)(70,207,139,215)>;
G:=Group( (1,197)(2,198)(3,199)(4,200)(5,201)(6,202)(7,203)(8,204)(9,205)(10,206)(11,207)(12,208)(13,209)(14,210)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,29)(24,30)(25,31)(26,32)(27,33)(28,34)(43,220)(44,221)(45,222)(46,223)(47,224)(48,211)(49,212)(50,213)(51,214)(52,215)(53,216)(54,217)(55,218)(56,219)(57,196)(58,183)(59,184)(60,185)(61,186)(62,187)(63,188)(64,189)(65,190)(66,191)(67,192)(68,193)(69,194)(70,195)(71,158)(72,159)(73,160)(74,161)(75,162)(76,163)(77,164)(78,165)(79,166)(80,167)(81,168)(82,155)(83,156)(84,157)(85,176)(86,177)(87,178)(88,179)(89,180)(90,181)(91,182)(92,169)(93,170)(94,171)(95,172)(96,173)(97,174)(98,175)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(106,121)(107,122)(108,123)(109,124)(110,125)(111,126)(112,113)(127,151)(128,152)(129,153)(130,154)(131,141)(132,142)(133,143)(134,144)(135,145)(136,146)(137,147)(138,148)(139,149)(140,150), (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), (1,211)(2,224)(3,223)(4,222)(5,221)(6,220)(7,219)(8,218)(9,217)(10,216)(11,215)(12,214)(13,213)(14,212)(15,102)(16,101)(17,100)(18,99)(19,112)(20,111)(21,110)(22,109)(23,108)(24,107)(25,106)(26,105)(27,104)(28,103)(29,123)(30,122)(31,121)(32,120)(33,119)(34,118)(35,117)(36,116)(37,115)(38,114)(39,113)(40,126)(41,125)(42,124)(43,202)(44,201)(45,200)(46,199)(47,198)(48,197)(49,210)(50,209)(51,208)(52,207)(53,206)(54,205)(55,204)(56,203)(57,146)(58,145)(59,144)(60,143)(61,142)(62,141)(63,154)(64,153)(65,152)(66,151)(67,150)(68,149)(69,148)(70,147)(71,171)(72,170)(73,169)(74,182)(75,181)(76,180)(77,179)(78,178)(79,177)(80,176)(81,175)(82,174)(83,173)(84,172)(85,167)(86,166)(87,165)(88,164)(89,163)(90,162)(91,161)(92,160)(93,159)(94,158)(95,157)(96,156)(97,155)(98,168)(127,191)(128,190)(129,189)(130,188)(131,187)(132,186)(133,185)(134,184)(135,183)(136,196)(137,195)(138,194)(139,193)(140,192), (1,116,56,17)(2,115,43,16)(3,114,44,15)(4,113,45,28)(5,126,46,27)(6,125,47,26)(7,124,48,25)(8,123,49,24)(9,122,50,23)(10,121,51,22)(11,120,52,21)(12,119,53,20)(13,118,54,19)(14,117,55,18)(29,205,107,213)(30,204,108,212)(31,203,109,211)(32,202,110,224)(33,201,111,223)(34,200,112,222)(35,199,99,221)(36,198,100,220)(37,197,101,219)(38,210,102,218)(39,209,103,217)(40,208,104,216)(41,207,105,215)(42,206,106,214)(57,94,140,76)(58,93,127,75)(59,92,128,74)(60,91,129,73)(61,90,130,72)(62,89,131,71)(63,88,132,84)(64,87,133,83)(65,86,134,82)(66,85,135,81)(67,98,136,80)(68,97,137,79)(69,96,138,78)(70,95,139,77)(141,158,187,180)(142,157,188,179)(143,156,189,178)(144,155,190,177)(145,168,191,176)(146,167,192,175)(147,166,193,174)(148,165,194,173)(149,164,195,172)(150,163,196,171)(151,162,183,170)(152,161,184,169)(153,160,185,182)(154,159,186,181), (1,145,56,191)(2,144,43,190)(3,143,44,189)(4,142,45,188)(5,141,46,187)(6,154,47,186)(7,153,48,185)(8,152,49,184)(9,151,50,183)(10,150,51,196)(11,149,52,195)(12,148,53,194)(13,147,54,193)(14,146,55,192)(15,156,114,178)(16,155,115,177)(17,168,116,176)(18,167,117,175)(19,166,118,174)(20,165,119,173)(21,164,120,172)(22,163,121,171)(23,162,122,170)(24,161,123,169)(25,160,124,182)(26,159,125,181)(27,158,126,180)(28,157,113,179)(29,75,107,93)(30,74,108,92)(31,73,109,91)(32,72,110,90)(33,71,111,89)(34,84,112,88)(35,83,99,87)(36,82,100,86)(37,81,101,85)(38,80,102,98)(39,79,103,97)(40,78,104,96)(41,77,105,95)(42,76,106,94)(57,206,140,214)(58,205,127,213)(59,204,128,212)(60,203,129,211)(61,202,130,224)(62,201,131,223)(63,200,132,222)(64,199,133,221)(65,198,134,220)(66,197,135,219)(67,210,136,218)(68,209,137,217)(69,208,138,216)(70,207,139,215) );
G=PermutationGroup([[(1,197),(2,198),(3,199),(4,200),(5,201),(6,202),(7,203),(8,204),(9,205),(10,206),(11,207),(12,208),(13,209),(14,210),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,29),(24,30),(25,31),(26,32),(27,33),(28,34),(43,220),(44,221),(45,222),(46,223),(47,224),(48,211),(49,212),(50,213),(51,214),(52,215),(53,216),(54,217),(55,218),(56,219),(57,196),(58,183),(59,184),(60,185),(61,186),(62,187),(63,188),(64,189),(65,190),(66,191),(67,192),(68,193),(69,194),(70,195),(71,158),(72,159),(73,160),(74,161),(75,162),(76,163),(77,164),(78,165),(79,166),(80,167),(81,168),(82,155),(83,156),(84,157),(85,176),(86,177),(87,178),(88,179),(89,180),(90,181),(91,182),(92,169),(93,170),(94,171),(95,172),(96,173),(97,174),(98,175),(99,114),(100,115),(101,116),(102,117),(103,118),(104,119),(105,120),(106,121),(107,122),(108,123),(109,124),(110,125),(111,126),(112,113),(127,151),(128,152),(129,153),(130,154),(131,141),(132,142),(133,143),(134,144),(135,145),(136,146),(137,147),(138,148),(139,149),(140,150)], [(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)], [(1,211),(2,224),(3,223),(4,222),(5,221),(6,220),(7,219),(8,218),(9,217),(10,216),(11,215),(12,214),(13,213),(14,212),(15,102),(16,101),(17,100),(18,99),(19,112),(20,111),(21,110),(22,109),(23,108),(24,107),(25,106),(26,105),(27,104),(28,103),(29,123),(30,122),(31,121),(32,120),(33,119),(34,118),(35,117),(36,116),(37,115),(38,114),(39,113),(40,126),(41,125),(42,124),(43,202),(44,201),(45,200),(46,199),(47,198),(48,197),(49,210),(50,209),(51,208),(52,207),(53,206),(54,205),(55,204),(56,203),(57,146),(58,145),(59,144),(60,143),(61,142),(62,141),(63,154),(64,153),(65,152),(66,151),(67,150),(68,149),(69,148),(70,147),(71,171),(72,170),(73,169),(74,182),(75,181),(76,180),(77,179),(78,178),(79,177),(80,176),(81,175),(82,174),(83,173),(84,172),(85,167),(86,166),(87,165),(88,164),(89,163),(90,162),(91,161),(92,160),(93,159),(94,158),(95,157),(96,156),(97,155),(98,168),(127,191),(128,190),(129,189),(130,188),(131,187),(132,186),(133,185),(134,184),(135,183),(136,196),(137,195),(138,194),(139,193),(140,192)], [(1,116,56,17),(2,115,43,16),(3,114,44,15),(4,113,45,28),(5,126,46,27),(6,125,47,26),(7,124,48,25),(8,123,49,24),(9,122,50,23),(10,121,51,22),(11,120,52,21),(12,119,53,20),(13,118,54,19),(14,117,55,18),(29,205,107,213),(30,204,108,212),(31,203,109,211),(32,202,110,224),(33,201,111,223),(34,200,112,222),(35,199,99,221),(36,198,100,220),(37,197,101,219),(38,210,102,218),(39,209,103,217),(40,208,104,216),(41,207,105,215),(42,206,106,214),(57,94,140,76),(58,93,127,75),(59,92,128,74),(60,91,129,73),(61,90,130,72),(62,89,131,71),(63,88,132,84),(64,87,133,83),(65,86,134,82),(66,85,135,81),(67,98,136,80),(68,97,137,79),(69,96,138,78),(70,95,139,77),(141,158,187,180),(142,157,188,179),(143,156,189,178),(144,155,190,177),(145,168,191,176),(146,167,192,175),(147,166,193,174),(148,165,194,173),(149,164,195,172),(150,163,196,171),(151,162,183,170),(152,161,184,169),(153,160,185,182),(154,159,186,181)], [(1,145,56,191),(2,144,43,190),(3,143,44,189),(4,142,45,188),(5,141,46,187),(6,154,47,186),(7,153,48,185),(8,152,49,184),(9,151,50,183),(10,150,51,196),(11,149,52,195),(12,148,53,194),(13,147,54,193),(14,146,55,192),(15,156,114,178),(16,155,115,177),(17,168,116,176),(18,167,117,175),(19,166,118,174),(20,165,119,173),(21,164,120,172),(22,163,121,171),(23,162,122,170),(24,161,123,169),(25,160,124,182),(26,159,125,181),(27,158,126,180),(28,157,113,179),(29,75,107,93),(30,74,108,92),(31,73,109,91),(32,72,110,90),(33,71,111,89),(34,84,112,88),(35,83,99,87),(36,82,100,86),(37,81,101,85),(38,80,102,98),(39,79,103,97),(40,78,104,96),(41,77,105,95),(42,76,106,94),(57,206,140,214),(58,205,127,213),(59,204,128,212),(60,203,129,211),(61,202,130,224),(62,201,131,223),(63,200,132,222),(64,199,133,221),(65,198,134,220),(66,197,135,219),(67,210,136,218),(68,209,137,217),(69,208,138,216),(70,207,139,215)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 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 | D4 | Q8 | D7 | C4○D4 | D14 | D14 | C4○D28 | D4×D7 | Q8×D7 |
| kernel | C2×D14⋊Q8 | D14⋊Q8 | C2×Dic7⋊C4 | C2×D14⋊C4 | C14×C4⋊C4 | C22×Dic14 | D7×C22×C4 | C2×Dic7 | C22×D7 | C2×C4⋊C4 | C2×C14 | C4⋊C4 | C22×C4 | C22 | C22 | C22 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 3 | 4 | 12 | 9 | 24 | 6 | 6 |
Matrix representation of C2×D14⋊Q8 ►in GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 11 | 21 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 25 | 3 | 0 | 0 |
| 0 | 24 | 4 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 7 | 24 | 0 | 0 |
| 0 | 10 | 22 | 0 | 0 |
| 0 | 0 | 0 | 1 | 27 |
| 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 10 | 12 | 0 | 0 |
| 0 | 23 | 19 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,11,11,0,0,0,21,0,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,25,24,0,0,0,3,4,0,0,0,0,0,28,28,0,0,0,0,1],[1,0,0,0,0,0,7,10,0,0,0,24,22,0,0,0,0,0,1,0,0,0,0,27,28],[28,0,0,0,0,0,10,23,0,0,0,12,19,0,0,0,0,0,28,0,0,0,0,0,28] >;
C2×D14⋊Q8 in GAP, Magma, Sage, TeX
C_2\times D_{14}\rtimes Q_8 % in TeX
G:=Group("C2xD14:Q8"); // GroupNames label
G:=SmallGroup(448,961);
// by ID
G=gap.SmallGroup(448,961);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,100,1571,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^14=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b^5*c,e*c*e^-1=b^12*c,e*d*e^-1=d^-1>;
// generators/relations