direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C14×C22⋊Q8, C23⋊4(C7×Q8), C4.63(D4×C14), (C22×C14)⋊6Q8, C22⋊1(Q8×C14), C28.470(C2×D4), (C2×C28).524D4, (C22×Q8)⋊3C14, (C23×C4).12C14, (C23×C28).25C2, C24.34(C2×C14), (Q8×C14)⋊48C22, C22.60(D4×C14), C14.57(C22×Q8), (C2×C14).343C24, (C2×C28).656C23, C14.182(C22×D4), C23.70(C22×C14), C22.17(C23×C14), (C23×C14).91C22, (C22×C28).444C22, (C22×C14).258C23, C2.6(D4×C2×C14), C2.3(Q8×C2×C14), (C2×C4⋊C4)⋊15C14, (C14×C4⋊C4)⋊42C2, (Q8×C2×C14)⋊15C2, (C2×C14)⋊5(C2×Q8), C4⋊C4⋊10(C2×C14), (C2×Q8)⋊8(C2×C14), C2.6(C14×C4○D4), (C7×C4⋊C4)⋊66C22, (C2×C4).135(C7×D4), C14.225(C2×C4○D4), (C2×C14).682(C2×D4), C22.30(C7×C4○D4), C22⋊C4.10(C2×C14), (C2×C22⋊C4).11C14, (C14×C22⋊C4).31C2, (C2×C4).12(C22×C14), (C22×C4).53(C2×C14), (C2×C14).230(C4○D4), (C7×C22⋊C4).144C22, SmallGroup(448,1306)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×C22⋊Q8
G = < a,b,c,d,e | a14=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 450 in 322 conjugacy classes, 194 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C23, C23, C14, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C28, C28, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, C2×C28, C2×C28, C7×Q8, C22×C14, C22×C14, C22×C14, C2×C22⋊Q8, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C22×C28, C22×C28, Q8×C14, Q8×C14, C23×C14, C14×C22⋊C4, C14×C4⋊C4, C14×C4⋊C4, C7×C22⋊Q8, C23×C28, Q8×C2×C14, C14×C22⋊Q8
Quotients: C1, C2, C22, C7, D4, Q8, C23, C14, C2×D4, C2×Q8, C4○D4, C24, C2×C14, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C7×D4, C7×Q8, C22×C14, C2×C22⋊Q8, D4×C14, Q8×C14, C7×C4○D4, C23×C14, C7×C22⋊Q8, D4×C2×C14, Q8×C2×C14, C14×C4○D4, C14×C22⋊Q8
►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)
(1 122)(2 123)(3 124)(4 125)(5 126)(6 113)(7 114)(8 115)(9 116)(10 117)(11 118)(12 119)(13 120)(14 121)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 111)(30 112)(31 99)(32 100)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(41 109)(42 110)(43 180)(44 181)(45 182)(46 169)(47 170)(48 171)(49 172)(50 173)(51 174)(52 175)(53 176)(54 177)(55 178)(56 179)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(127 134)(128 135)(129 136)(130 137)(131 138)(132 139)(133 140)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(183 190)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)(211 218)(212 219)(213 220)(214 221)(215 222)(216 223)(217 224)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 113)(14 114)(15 222)(16 223)(17 224)(18 211)(19 212)(20 213)(21 214)(22 215)(23 216)(24 217)(25 218)(26 219)(27 220)(28 221)(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 99)(39 100)(40 101)(41 102)(42 103)(43 173)(44 174)(45 175)(46 176)(47 177)(48 178)(49 179)(50 180)(51 181)(52 182)(53 169)(54 170)(55 171)(56 172)(57 76)(58 77)(59 78)(60 79)(61 80)(62 81)(63 82)(64 83)(65 84)(66 71)(67 72)(68 73)(69 74)(70 75)(85 210)(86 197)(87 198)(88 199)(89 200)(90 201)(91 202)(92 203)(93 204)(94 205)(95 206)(96 207)(97 208)(98 209)(127 185)(128 186)(129 187)(130 188)(131 189)(132 190)(133 191)(134 192)(135 193)(136 194)(137 195)(138 196)(139 183)(140 184)(141 162)(142 163)(143 164)(144 165)(145 166)(146 167)(147 168)(148 155)(149 156)(150 157)(151 158)(152 159)(153 160)(154 161)
(1 167 29 49)(2 168 30 50)(3 155 31 51)(4 156 32 52)(5 157 33 53)(6 158 34 54)(7 159 35 55)(8 160 36 56)(9 161 37 43)(10 162 38 44)(11 163 39 45)(12 164 40 46)(13 165 41 47)(14 166 42 48)(15 67 88 133)(16 68 89 134)(17 69 90 135)(18 70 91 136)(19 57 92 137)(20 58 93 138)(21 59 94 139)(22 60 95 140)(23 61 96 127)(24 62 97 128)(25 63 98 129)(26 64 85 130)(27 65 86 131)(28 66 87 132)(71 198 190 221)(72 199 191 222)(73 200 192 223)(74 201 193 224)(75 202 194 211)(76 203 195 212)(77 204 196 213)(78 205 183 214)(79 206 184 215)(80 207 185 216)(81 208 186 217)(82 209 187 218)(83 210 188 219)(84 197 189 220)(99 174 124 141)(100 175 125 142)(101 176 126 143)(102 177 113 144)(103 178 114 145)(104 179 115 146)(105 180 116 147)(106 181 117 148)(107 182 118 149)(108 169 119 150)(109 170 120 151)(110 171 121 152)(111 172 122 153)(112 173 123 154)
(1 63 29 129)(2 64 30 130)(3 65 31 131)(4 66 32 132)(5 67 33 133)(6 68 34 134)(7 69 35 135)(8 70 36 136)(9 57 37 137)(10 58 38 138)(11 59 39 139)(12 60 40 140)(13 61 41 127)(14 62 42 128)(15 53 88 157)(16 54 89 158)(17 55 90 159)(18 56 91 160)(19 43 92 161)(20 44 93 162)(21 45 94 163)(22 46 95 164)(23 47 96 165)(24 48 97 166)(25 49 98 167)(26 50 85 168)(27 51 86 155)(28 52 87 156)(71 107 190 118)(72 108 191 119)(73 109 192 120)(74 110 193 121)(75 111 194 122)(76 112 195 123)(77 99 196 124)(78 100 183 125)(79 101 184 126)(80 102 185 113)(81 103 186 114)(82 104 187 115)(83 105 188 116)(84 106 189 117)(141 213 174 204)(142 214 175 205)(143 215 176 206)(144 216 177 207)(145 217 178 208)(146 218 179 209)(147 219 180 210)(148 220 181 197)(149 221 182 198)(150 222 169 199)(151 223 170 200)(152 224 171 201)(153 211 172 202)(154 212 173 203)
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), (1,122)(2,123)(3,124)(4,125)(5,126)(6,113)(7,114)(8,115)(9,116)(10,117)(11,118)(12,119)(13,120)(14,121)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,111)(30,112)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,180)(44,181)(45,182)(46,169)(47,170)(48,171)(49,172)(50,173)(51,174)(52,175)(53,176)(54,177)(55,178)(56,179)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224), (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,113)(14,114)(15,222)(16,223)(17,224)(18,211)(19,212)(20,213)(21,214)(22,215)(23,216)(24,217)(25,218)(26,219)(27,220)(28,221)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,99)(39,100)(40,101)(41,102)(42,103)(43,173)(44,174)(45,175)(46,176)(47,177)(48,178)(49,179)(50,180)(51,181)(52,182)(53,169)(54,170)(55,171)(56,172)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,71)(67,72)(68,73)(69,74)(70,75)(85,210)(86,197)(87,198)(88,199)(89,200)(90,201)(91,202)(92,203)(93,204)(94,205)(95,206)(96,207)(97,208)(98,209)(127,185)(128,186)(129,187)(130,188)(131,189)(132,190)(133,191)(134,192)(135,193)(136,194)(137,195)(138,196)(139,183)(140,184)(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)(147,168)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161), (1,167,29,49)(2,168,30,50)(3,155,31,51)(4,156,32,52)(5,157,33,53)(6,158,34,54)(7,159,35,55)(8,160,36,56)(9,161,37,43)(10,162,38,44)(11,163,39,45)(12,164,40,46)(13,165,41,47)(14,166,42,48)(15,67,88,133)(16,68,89,134)(17,69,90,135)(18,70,91,136)(19,57,92,137)(20,58,93,138)(21,59,94,139)(22,60,95,140)(23,61,96,127)(24,62,97,128)(25,63,98,129)(26,64,85,130)(27,65,86,131)(28,66,87,132)(71,198,190,221)(72,199,191,222)(73,200,192,223)(74,201,193,224)(75,202,194,211)(76,203,195,212)(77,204,196,213)(78,205,183,214)(79,206,184,215)(80,207,185,216)(81,208,186,217)(82,209,187,218)(83,210,188,219)(84,197,189,220)(99,174,124,141)(100,175,125,142)(101,176,126,143)(102,177,113,144)(103,178,114,145)(104,179,115,146)(105,180,116,147)(106,181,117,148)(107,182,118,149)(108,169,119,150)(109,170,120,151)(110,171,121,152)(111,172,122,153)(112,173,123,154), (1,63,29,129)(2,64,30,130)(3,65,31,131)(4,66,32,132)(5,67,33,133)(6,68,34,134)(7,69,35,135)(8,70,36,136)(9,57,37,137)(10,58,38,138)(11,59,39,139)(12,60,40,140)(13,61,41,127)(14,62,42,128)(15,53,88,157)(16,54,89,158)(17,55,90,159)(18,56,91,160)(19,43,92,161)(20,44,93,162)(21,45,94,163)(22,46,95,164)(23,47,96,165)(24,48,97,166)(25,49,98,167)(26,50,85,168)(27,51,86,155)(28,52,87,156)(71,107,190,118)(72,108,191,119)(73,109,192,120)(74,110,193,121)(75,111,194,122)(76,112,195,123)(77,99,196,124)(78,100,183,125)(79,101,184,126)(80,102,185,113)(81,103,186,114)(82,104,187,115)(83,105,188,116)(84,106,189,117)(141,213,174,204)(142,214,175,205)(143,215,176,206)(144,216,177,207)(145,217,178,208)(146,218,179,209)(147,219,180,210)(148,220,181,197)(149,221,182,198)(150,222,169,199)(151,223,170,200)(152,224,171,201)(153,211,172,202)(154,212,173,203)>;
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), (1,122)(2,123)(3,124)(4,125)(5,126)(6,113)(7,114)(8,115)(9,116)(10,117)(11,118)(12,119)(13,120)(14,121)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,111)(30,112)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,180)(44,181)(45,182)(46,169)(47,170)(48,171)(49,172)(50,173)(51,174)(52,175)(53,176)(54,177)(55,178)(56,179)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224), (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,113)(14,114)(15,222)(16,223)(17,224)(18,211)(19,212)(20,213)(21,214)(22,215)(23,216)(24,217)(25,218)(26,219)(27,220)(28,221)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,99)(39,100)(40,101)(41,102)(42,103)(43,173)(44,174)(45,175)(46,176)(47,177)(48,178)(49,179)(50,180)(51,181)(52,182)(53,169)(54,170)(55,171)(56,172)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,71)(67,72)(68,73)(69,74)(70,75)(85,210)(86,197)(87,198)(88,199)(89,200)(90,201)(91,202)(92,203)(93,204)(94,205)(95,206)(96,207)(97,208)(98,209)(127,185)(128,186)(129,187)(130,188)(131,189)(132,190)(133,191)(134,192)(135,193)(136,194)(137,195)(138,196)(139,183)(140,184)(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)(147,168)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161), (1,167,29,49)(2,168,30,50)(3,155,31,51)(4,156,32,52)(5,157,33,53)(6,158,34,54)(7,159,35,55)(8,160,36,56)(9,161,37,43)(10,162,38,44)(11,163,39,45)(12,164,40,46)(13,165,41,47)(14,166,42,48)(15,67,88,133)(16,68,89,134)(17,69,90,135)(18,70,91,136)(19,57,92,137)(20,58,93,138)(21,59,94,139)(22,60,95,140)(23,61,96,127)(24,62,97,128)(25,63,98,129)(26,64,85,130)(27,65,86,131)(28,66,87,132)(71,198,190,221)(72,199,191,222)(73,200,192,223)(74,201,193,224)(75,202,194,211)(76,203,195,212)(77,204,196,213)(78,205,183,214)(79,206,184,215)(80,207,185,216)(81,208,186,217)(82,209,187,218)(83,210,188,219)(84,197,189,220)(99,174,124,141)(100,175,125,142)(101,176,126,143)(102,177,113,144)(103,178,114,145)(104,179,115,146)(105,180,116,147)(106,181,117,148)(107,182,118,149)(108,169,119,150)(109,170,120,151)(110,171,121,152)(111,172,122,153)(112,173,123,154), (1,63,29,129)(2,64,30,130)(3,65,31,131)(4,66,32,132)(5,67,33,133)(6,68,34,134)(7,69,35,135)(8,70,36,136)(9,57,37,137)(10,58,38,138)(11,59,39,139)(12,60,40,140)(13,61,41,127)(14,62,42,128)(15,53,88,157)(16,54,89,158)(17,55,90,159)(18,56,91,160)(19,43,92,161)(20,44,93,162)(21,45,94,163)(22,46,95,164)(23,47,96,165)(24,48,97,166)(25,49,98,167)(26,50,85,168)(27,51,86,155)(28,52,87,156)(71,107,190,118)(72,108,191,119)(73,109,192,120)(74,110,193,121)(75,111,194,122)(76,112,195,123)(77,99,196,124)(78,100,183,125)(79,101,184,126)(80,102,185,113)(81,103,186,114)(82,104,187,115)(83,105,188,116)(84,106,189,117)(141,213,174,204)(142,214,175,205)(143,215,176,206)(144,216,177,207)(145,217,178,208)(146,218,179,209)(147,219,180,210)(148,220,181,197)(149,221,182,198)(150,222,169,199)(151,223,170,200)(152,224,171,201)(153,211,172,202)(154,212,173,203) );
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)], [(1,122),(2,123),(3,124),(4,125),(5,126),(6,113),(7,114),(8,115),(9,116),(10,117),(11,118),(12,119),(13,120),(14,121),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,111),(30,112),(31,99),(32,100),(33,101),(34,102),(35,103),(36,104),(37,105),(38,106),(39,107),(40,108),(41,109),(42,110),(43,180),(44,181),(45,182),(46,169),(47,170),(48,171),(49,172),(50,173),(51,174),(52,175),(53,176),(54,177),(55,178),(56,179),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(127,134),(128,135),(129,136),(130,137),(131,138),(132,139),(133,140),(141,155),(142,156),(143,157),(144,158),(145,159),(146,160),(147,161),(148,162),(149,163),(150,164),(151,165),(152,166),(153,167),(154,168),(183,190),(184,191),(185,192),(186,193),(187,194),(188,195),(189,196),(197,204),(198,205),(199,206),(200,207),(201,208),(202,209),(203,210),(211,218),(212,219),(213,220),(214,221),(215,222),(216,223),(217,224)], [(1,115),(2,116),(3,117),(4,118),(5,119),(6,120),(7,121),(8,122),(9,123),(10,124),(11,125),(12,126),(13,113),(14,114),(15,222),(16,223),(17,224),(18,211),(19,212),(20,213),(21,214),(22,215),(23,216),(24,217),(25,218),(26,219),(27,220),(28,221),(29,104),(30,105),(31,106),(32,107),(33,108),(34,109),(35,110),(36,111),(37,112),(38,99),(39,100),(40,101),(41,102),(42,103),(43,173),(44,174),(45,175),(46,176),(47,177),(48,178),(49,179),(50,180),(51,181),(52,182),(53,169),(54,170),(55,171),(56,172),(57,76),(58,77),(59,78),(60,79),(61,80),(62,81),(63,82),(64,83),(65,84),(66,71),(67,72),(68,73),(69,74),(70,75),(85,210),(86,197),(87,198),(88,199),(89,200),(90,201),(91,202),(92,203),(93,204),(94,205),(95,206),(96,207),(97,208),(98,209),(127,185),(128,186),(129,187),(130,188),(131,189),(132,190),(133,191),(134,192),(135,193),(136,194),(137,195),(138,196),(139,183),(140,184),(141,162),(142,163),(143,164),(144,165),(145,166),(146,167),(147,168),(148,155),(149,156),(150,157),(151,158),(152,159),(153,160),(154,161)], [(1,167,29,49),(2,168,30,50),(3,155,31,51),(4,156,32,52),(5,157,33,53),(6,158,34,54),(7,159,35,55),(8,160,36,56),(9,161,37,43),(10,162,38,44),(11,163,39,45),(12,164,40,46),(13,165,41,47),(14,166,42,48),(15,67,88,133),(16,68,89,134),(17,69,90,135),(18,70,91,136),(19,57,92,137),(20,58,93,138),(21,59,94,139),(22,60,95,140),(23,61,96,127),(24,62,97,128),(25,63,98,129),(26,64,85,130),(27,65,86,131),(28,66,87,132),(71,198,190,221),(72,199,191,222),(73,200,192,223),(74,201,193,224),(75,202,194,211),(76,203,195,212),(77,204,196,213),(78,205,183,214),(79,206,184,215),(80,207,185,216),(81,208,186,217),(82,209,187,218),(83,210,188,219),(84,197,189,220),(99,174,124,141),(100,175,125,142),(101,176,126,143),(102,177,113,144),(103,178,114,145),(104,179,115,146),(105,180,116,147),(106,181,117,148),(107,182,118,149),(108,169,119,150),(109,170,120,151),(110,171,121,152),(111,172,122,153),(112,173,123,154)], [(1,63,29,129),(2,64,30,130),(3,65,31,131),(4,66,32,132),(5,67,33,133),(6,68,34,134),(7,69,35,135),(8,70,36,136),(9,57,37,137),(10,58,38,138),(11,59,39,139),(12,60,40,140),(13,61,41,127),(14,62,42,128),(15,53,88,157),(16,54,89,158),(17,55,90,159),(18,56,91,160),(19,43,92,161),(20,44,93,162),(21,45,94,163),(22,46,95,164),(23,47,96,165),(24,48,97,166),(25,49,98,167),(26,50,85,168),(27,51,86,155),(28,52,87,156),(71,107,190,118),(72,108,191,119),(73,109,192,120),(74,110,193,121),(75,111,194,122),(76,112,195,123),(77,99,196,124),(78,100,183,125),(79,101,184,126),(80,102,185,113),(81,103,186,114),(82,104,187,115),(83,105,188,116),(84,106,189,117),(141,213,174,204),(142,214,175,205),(143,215,176,206),(144,216,177,207),(145,217,178,208),(146,218,179,209),(147,219,180,210),(148,220,181,197),(149,221,182,198),(150,222,169,199),(151,223,170,200),(152,224,171,201),(153,211,172,202),(154,212,173,203)]])
196 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 7A | ··· | 7F | 14A | ··· | 14AP | 14AQ | ··· | 14BN | 28A | ··· | 28AV | 28AW | ··· | 28CR |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | Q8 | C4○D4 | C7×D4 | C7×Q8 | C7×C4○D4 |
| kernel | C14×C22⋊Q8 | C14×C22⋊C4 | C14×C4⋊C4 | C7×C22⋊Q8 | C23×C28 | Q8×C2×C14 | C2×C22⋊Q8 | C2×C22⋊C4 | C2×C4⋊C4 | C22⋊Q8 | C23×C4 | C22×Q8 | C2×C28 | C22×C14 | C2×C14 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 3 | 8 | 1 | 1 | 6 | 12 | 18 | 48 | 6 | 6 | 4 | 4 | 4 | 24 | 24 | 24 |
Matrix representation of C14×C22⋊Q8 ►in GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 | 0 |
| 0 | 0 | 20 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 |
| 0 | 7 | 12 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 4 | 12 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 16 | 1 | 0 | 0 |
| 0 | 4 | 13 | 0 | 0 |
| 0 | 0 | 0 | 10 | 2 |
| 0 | 0 | 0 | 22 | 19 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,20,0,0,0,0,0,20,0,0,0,0,0,13,0,0,0,0,0,13],[1,0,0,0,0,0,28,3,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,17,7,0,0,0,0,12,0,0,0,0,0,17,4,0,0,0,0,12],[28,0,0,0,0,0,16,4,0,0,0,1,13,0,0,0,0,0,10,22,0,0,0,2,19] >;
C14×C22⋊Q8 in GAP, Magma, Sage, TeX
C_{14}\times C_2^2\rtimes Q_8 % in TeX
G:=Group("C14xC2^2:Q8"); // GroupNames label
G:=SmallGroup(448,1306);
// by ID
G=gap.SmallGroup(448,1306);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,792,4790]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^2=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,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations