direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C14×C4○D8, C28.81C24, C56.79C23, D8⋊6(C2×C14), (C2×D8)⋊13C14, (C14×D8)⋊27C2, (C22×C8)⋊8C14, Q16⋊6(C2×C14), C4.84(D4×C14), (C22×C56)⋊22C2, (C2×C56)⋊51C22, (C14×Q16)⋊27C2, (C2×Q16)⋊13C14, SD16⋊5(C2×C14), (C2×C28).539D4, C28.328(C2×D4), (C7×D8)⋊23C22, C4.4(C23×C14), C22.4(D4×C14), C23.28(C7×D4), (C2×SD16)⋊16C14, (C14×SD16)⋊33C2, C8.12(C22×C14), (C7×Q16)⋊20C22, D4.2(C22×C14), (C7×D4).35C23, Q8.2(C22×C14), (C7×Q8).36C23, (C2×C28).974C23, (C7×SD16)⋊22C22, (C22×C14).132D4, C14.202(C22×D4), (D4×C14).328C22, (Q8×C14).281C22, (C22×C28).604C22, C2.26(D4×C2×C14), (C2×C8)⋊13(C2×C14), C4○D4⋊3(C2×C14), (C14×C4○D4)⋊26C2, (C2×C4○D4)⋊10C14, (C2×C4).148(C7×D4), (C2×D4).74(C2×C14), (C2×C14).689(C2×D4), (C7×C4○D4)⋊23C22, (C2×Q8).69(C2×C14), (C22×C4).131(C2×C14), (C2×C4).144(C22×C14), SmallGroup(448,1355)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 402 in 266 conjugacy classes, 162 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], C7, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×4], D4 [×10], Q8 [×4], Q8 [×2], C23, C23 [×2], C14, C14 [×2], C14 [×6], C2×C8 [×2], C2×C8 [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×4], C28 [×2], C28 [×2], C28 [×4], C2×C14, C2×C14 [×2], C2×C14 [×10], C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C56 [×4], C2×C28 [×2], C2×C28 [×4], C2×C28 [×10], C7×D4 [×4], C7×D4 [×10], C7×Q8 [×4], C7×Q8 [×2], C22×C14, C22×C14 [×2], C2×C4○D8, C2×C56 [×2], C2×C56 [×4], C7×D8 [×4], C7×SD16 [×8], C7×Q16 [×4], C22×C28, C22×C28 [×2], D4×C14 [×2], D4×C14 [×2], Q8×C14 [×2], C7×C4○D4 [×8], C7×C4○D4 [×4], C22×C56, C14×D8, C14×SD16 [×2], C14×Q16, C7×C4○D8 [×8], C14×C4○D4 [×2], C14×C4○D8
Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×4], C23 [×15], C14 [×15], C2×D4 [×6], C24, C2×C14 [×35], C4○D8 [×2], C22×D4, C7×D4 [×4], C22×C14 [×15], C2×C4○D8, D4×C14 [×6], C23×C14, C7×C4○D8 [×2], D4×C2×C14, C14×C4○D8
Generators and relations
G = < a,b,c,d | a14=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >
►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)
(1 141 204 30)(2 142 205 31)(3 143 206 32)(4 144 207 33)(5 145 208 34)(6 146 209 35)(7 147 210 36)(8 148 197 37)(9 149 198 38)(10 150 199 39)(11 151 200 40)(12 152 201 41)(13 153 202 42)(14 154 203 29)(15 181 70 133)(16 182 57 134)(17 169 58 135)(18 170 59 136)(19 171 60 137)(20 172 61 138)(21 173 62 139)(22 174 63 140)(23 175 64 127)(24 176 65 128)(25 177 66 129)(26 178 67 130)(27 179 68 131)(28 180 69 132)(43 121 217 112)(44 122 218 99)(45 123 219 100)(46 124 220 101)(47 125 221 102)(48 126 222 103)(49 113 223 104)(50 114 224 105)(51 115 211 106)(52 116 212 107)(53 117 213 108)(54 118 214 109)(55 119 215 110)(56 120 216 111)(71 193 87 167)(72 194 88 168)(73 195 89 155)(74 196 90 156)(75 183 91 157)(76 184 92 158)(77 185 93 159)(78 186 94 160)(79 187 95 161)(80 188 96 162)(81 189 97 163)(82 190 98 164)(83 191 85 165)(84 192 86 166)
(1 49 141 113 204 223 30 104)(2 50 142 114 205 224 31 105)(3 51 143 115 206 211 32 106)(4 52 144 116 207 212 33 107)(5 53 145 117 208 213 34 108)(6 54 146 118 209 214 35 109)(7 55 147 119 210 215 36 110)(8 56 148 120 197 216 37 111)(9 43 149 121 198 217 38 112)(10 44 150 122 199 218 39 99)(11 45 151 123 200 219 40 100)(12 46 152 124 201 220 41 101)(13 47 153 125 202 221 42 102)(14 48 154 126 203 222 29 103)(15 187 133 79 70 161 181 95)(16 188 134 80 57 162 182 96)(17 189 135 81 58 163 169 97)(18 190 136 82 59 164 170 98)(19 191 137 83 60 165 171 85)(20 192 138 84 61 166 172 86)(21 193 139 71 62 167 173 87)(22 194 140 72 63 168 174 88)(23 195 127 73 64 155 175 89)(24 196 128 74 65 156 176 90)(25 183 129 75 66 157 177 91)(26 184 130 76 67 158 178 92)(27 185 131 77 68 159 179 93)(28 186 132 78 69 160 180 94)
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 85)(9 86)(10 87)(11 88)(12 89)(13 90)(14 91)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 99)(22 100)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 183)(30 184)(31 185)(32 186)(33 187)(34 188)(35 189)(36 190)(37 191)(38 192)(39 193)(40 194)(41 195)(42 196)(43 172)(44 173)(45 174)(46 175)(47 176)(48 177)(49 178)(50 179)(51 180)(52 181)(53 182)(54 169)(55 170)(56 171)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 113)(68 114)(69 115)(70 116)(71 199)(72 200)(73 201)(74 202)(75 203)(76 204)(77 205)(78 206)(79 207)(80 208)(81 209)(82 210)(83 197)(84 198)(127 220)(128 221)(129 222)(130 223)(131 224)(132 211)(133 212)(134 213)(135 214)(136 215)(137 216)(138 217)(139 218)(140 219)(141 158)(142 159)(143 160)(144 161)(145 162)(146 163)(147 164)(148 165)(149 166)(150 167)(151 168)(152 155)(153 156)(154 157)
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), (1,141,204,30)(2,142,205,31)(3,143,206,32)(4,144,207,33)(5,145,208,34)(6,146,209,35)(7,147,210,36)(8,148,197,37)(9,149,198,38)(10,150,199,39)(11,151,200,40)(12,152,201,41)(13,153,202,42)(14,154,203,29)(15,181,70,133)(16,182,57,134)(17,169,58,135)(18,170,59,136)(19,171,60,137)(20,172,61,138)(21,173,62,139)(22,174,63,140)(23,175,64,127)(24,176,65,128)(25,177,66,129)(26,178,67,130)(27,179,68,131)(28,180,69,132)(43,121,217,112)(44,122,218,99)(45,123,219,100)(46,124,220,101)(47,125,221,102)(48,126,222,103)(49,113,223,104)(50,114,224,105)(51,115,211,106)(52,116,212,107)(53,117,213,108)(54,118,214,109)(55,119,215,110)(56,120,216,111)(71,193,87,167)(72,194,88,168)(73,195,89,155)(74,196,90,156)(75,183,91,157)(76,184,92,158)(77,185,93,159)(78,186,94,160)(79,187,95,161)(80,188,96,162)(81,189,97,163)(82,190,98,164)(83,191,85,165)(84,192,86,166), (1,49,141,113,204,223,30,104)(2,50,142,114,205,224,31,105)(3,51,143,115,206,211,32,106)(4,52,144,116,207,212,33,107)(5,53,145,117,208,213,34,108)(6,54,146,118,209,214,35,109)(7,55,147,119,210,215,36,110)(8,56,148,120,197,216,37,111)(9,43,149,121,198,217,38,112)(10,44,150,122,199,218,39,99)(11,45,151,123,200,219,40,100)(12,46,152,124,201,220,41,101)(13,47,153,125,202,221,42,102)(14,48,154,126,203,222,29,103)(15,187,133,79,70,161,181,95)(16,188,134,80,57,162,182,96)(17,189,135,81,58,163,169,97)(18,190,136,82,59,164,170,98)(19,191,137,83,60,165,171,85)(20,192,138,84,61,166,172,86)(21,193,139,71,62,167,173,87)(22,194,140,72,63,168,174,88)(23,195,127,73,64,155,175,89)(24,196,128,74,65,156,176,90)(25,183,129,75,66,157,177,91)(26,184,130,76,67,158,178,92)(27,185,131,77,68,159,179,93)(28,186,132,78,69,160,180,94), (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,183)(30,184)(31,185)(32,186)(33,187)(34,188)(35,189)(36,190)(37,191)(38,192)(39,193)(40,194)(41,195)(42,196)(43,172)(44,173)(45,174)(46,175)(47,176)(48,177)(49,178)(50,179)(51,180)(52,181)(53,182)(54,169)(55,170)(56,171)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,113)(68,114)(69,115)(70,116)(71,199)(72,200)(73,201)(74,202)(75,203)(76,204)(77,205)(78,206)(79,207)(80,208)(81,209)(82,210)(83,197)(84,198)(127,220)(128,221)(129,222)(130,223)(131,224)(132,211)(133,212)(134,213)(135,214)(136,215)(137,216)(138,217)(139,218)(140,219)(141,158)(142,159)(143,160)(144,161)(145,162)(146,163)(147,164)(148,165)(149,166)(150,167)(151,168)(152,155)(153,156)(154,157)>;
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), (1,141,204,30)(2,142,205,31)(3,143,206,32)(4,144,207,33)(5,145,208,34)(6,146,209,35)(7,147,210,36)(8,148,197,37)(9,149,198,38)(10,150,199,39)(11,151,200,40)(12,152,201,41)(13,153,202,42)(14,154,203,29)(15,181,70,133)(16,182,57,134)(17,169,58,135)(18,170,59,136)(19,171,60,137)(20,172,61,138)(21,173,62,139)(22,174,63,140)(23,175,64,127)(24,176,65,128)(25,177,66,129)(26,178,67,130)(27,179,68,131)(28,180,69,132)(43,121,217,112)(44,122,218,99)(45,123,219,100)(46,124,220,101)(47,125,221,102)(48,126,222,103)(49,113,223,104)(50,114,224,105)(51,115,211,106)(52,116,212,107)(53,117,213,108)(54,118,214,109)(55,119,215,110)(56,120,216,111)(71,193,87,167)(72,194,88,168)(73,195,89,155)(74,196,90,156)(75,183,91,157)(76,184,92,158)(77,185,93,159)(78,186,94,160)(79,187,95,161)(80,188,96,162)(81,189,97,163)(82,190,98,164)(83,191,85,165)(84,192,86,166), (1,49,141,113,204,223,30,104)(2,50,142,114,205,224,31,105)(3,51,143,115,206,211,32,106)(4,52,144,116,207,212,33,107)(5,53,145,117,208,213,34,108)(6,54,146,118,209,214,35,109)(7,55,147,119,210,215,36,110)(8,56,148,120,197,216,37,111)(9,43,149,121,198,217,38,112)(10,44,150,122,199,218,39,99)(11,45,151,123,200,219,40,100)(12,46,152,124,201,220,41,101)(13,47,153,125,202,221,42,102)(14,48,154,126,203,222,29,103)(15,187,133,79,70,161,181,95)(16,188,134,80,57,162,182,96)(17,189,135,81,58,163,169,97)(18,190,136,82,59,164,170,98)(19,191,137,83,60,165,171,85)(20,192,138,84,61,166,172,86)(21,193,139,71,62,167,173,87)(22,194,140,72,63,168,174,88)(23,195,127,73,64,155,175,89)(24,196,128,74,65,156,176,90)(25,183,129,75,66,157,177,91)(26,184,130,76,67,158,178,92)(27,185,131,77,68,159,179,93)(28,186,132,78,69,160,180,94), (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,183)(30,184)(31,185)(32,186)(33,187)(34,188)(35,189)(36,190)(37,191)(38,192)(39,193)(40,194)(41,195)(42,196)(43,172)(44,173)(45,174)(46,175)(47,176)(48,177)(49,178)(50,179)(51,180)(52,181)(53,182)(54,169)(55,170)(56,171)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,113)(68,114)(69,115)(70,116)(71,199)(72,200)(73,201)(74,202)(75,203)(76,204)(77,205)(78,206)(79,207)(80,208)(81,209)(82,210)(83,197)(84,198)(127,220)(128,221)(129,222)(130,223)(131,224)(132,211)(133,212)(134,213)(135,214)(136,215)(137,216)(138,217)(139,218)(140,219)(141,158)(142,159)(143,160)(144,161)(145,162)(146,163)(147,164)(148,165)(149,166)(150,167)(151,168)(152,155)(153,156)(154,157) );
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)], [(1,141,204,30),(2,142,205,31),(3,143,206,32),(4,144,207,33),(5,145,208,34),(6,146,209,35),(7,147,210,36),(8,148,197,37),(9,149,198,38),(10,150,199,39),(11,151,200,40),(12,152,201,41),(13,153,202,42),(14,154,203,29),(15,181,70,133),(16,182,57,134),(17,169,58,135),(18,170,59,136),(19,171,60,137),(20,172,61,138),(21,173,62,139),(22,174,63,140),(23,175,64,127),(24,176,65,128),(25,177,66,129),(26,178,67,130),(27,179,68,131),(28,180,69,132),(43,121,217,112),(44,122,218,99),(45,123,219,100),(46,124,220,101),(47,125,221,102),(48,126,222,103),(49,113,223,104),(50,114,224,105),(51,115,211,106),(52,116,212,107),(53,117,213,108),(54,118,214,109),(55,119,215,110),(56,120,216,111),(71,193,87,167),(72,194,88,168),(73,195,89,155),(74,196,90,156),(75,183,91,157),(76,184,92,158),(77,185,93,159),(78,186,94,160),(79,187,95,161),(80,188,96,162),(81,189,97,163),(82,190,98,164),(83,191,85,165),(84,192,86,166)], [(1,49,141,113,204,223,30,104),(2,50,142,114,205,224,31,105),(3,51,143,115,206,211,32,106),(4,52,144,116,207,212,33,107),(5,53,145,117,208,213,34,108),(6,54,146,118,209,214,35,109),(7,55,147,119,210,215,36,110),(8,56,148,120,197,216,37,111),(9,43,149,121,198,217,38,112),(10,44,150,122,199,218,39,99),(11,45,151,123,200,219,40,100),(12,46,152,124,201,220,41,101),(13,47,153,125,202,221,42,102),(14,48,154,126,203,222,29,103),(15,187,133,79,70,161,181,95),(16,188,134,80,57,162,182,96),(17,189,135,81,58,163,169,97),(18,190,136,82,59,164,170,98),(19,191,137,83,60,165,171,85),(20,192,138,84,61,166,172,86),(21,193,139,71,62,167,173,87),(22,194,140,72,63,168,174,88),(23,195,127,73,64,155,175,89),(24,196,128,74,65,156,176,90),(25,183,129,75,66,157,177,91),(26,184,130,76,67,158,178,92),(27,185,131,77,68,159,179,93),(28,186,132,78,69,160,180,94)], [(1,92),(2,93),(3,94),(4,95),(5,96),(6,97),(7,98),(8,85),(9,86),(10,87),(11,88),(12,89),(13,90),(14,91),(15,107),(16,108),(17,109),(18,110),(19,111),(20,112),(21,99),(22,100),(23,101),(24,102),(25,103),(26,104),(27,105),(28,106),(29,183),(30,184),(31,185),(32,186),(33,187),(34,188),(35,189),(36,190),(37,191),(38,192),(39,193),(40,194),(41,195),(42,196),(43,172),(44,173),(45,174),(46,175),(47,176),(48,177),(49,178),(50,179),(51,180),(52,181),(53,182),(54,169),(55,170),(56,171),(57,117),(58,118),(59,119),(60,120),(61,121),(62,122),(63,123),(64,124),(65,125),(66,126),(67,113),(68,114),(69,115),(70,116),(71,199),(72,200),(73,201),(74,202),(75,203),(76,204),(77,205),(78,206),(79,207),(80,208),(81,209),(82,210),(83,197),(84,198),(127,220),(128,221),(129,222),(130,223),(131,224),(132,211),(133,212),(134,213),(135,214),(136,215),(137,216),(138,217),(139,218),(140,219),(141,158),(142,159),(143,160),(144,161),(145,162),(146,163),(147,164),(148,165),(149,166),(150,167),(151,168),(152,155),(153,156),(154,157)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 5 | 27 | 0 | 0 |
| 66 | 108 | 0 | 0 |
| 0 | 0 | 44 | 0 |
| 0 | 0 | 13 | 18 |
| 43 | 29 | 0 | 0 |
| 22 | 70 | 0 | 0 |
| 0 | 0 | 95 | 36 |
| 0 | 0 | 13 | 18 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,16,0,0,0,0,16],[112,0,0,0,0,112,0,0,0,0,15,0,0,0,0,15],[5,66,0,0,27,108,0,0,0,0,44,13,0,0,0,18],[43,22,0,0,29,70,0,0,0,0,95,13,0,0,36,18] >;
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14BB | 28A | ··· | 28X | 28Y | ··· | 28AJ | 28AK | ··· | 28BH | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
196 irreducible representations
| dim | 1 | 1 | 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 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | C4○D8 | C7×D4 | C7×D4 | C7×C4○D8 |
| kernel | C14×C4○D8 | C22×C56 | C14×D8 | C14×SD16 | C14×Q16 | C7×C4○D8 | C14×C4○D4 | C2×C4○D8 | C22×C8 | C2×D8 | C2×SD16 | C2×Q16 | C4○D8 | C2×C4○D4 | C2×C28 | C22×C14 | C14 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 8 | 2 | 6 | 6 | 6 | 12 | 6 | 48 | 12 | 3 | 1 | 8 | 18 | 6 | 48 |
In GAP, Magma, Sage, TeX
C_{14}\times C_4\circ D_8 % in TeX
G:=Group("C14xC4oD8"); // GroupNames label
G:=SmallGroup(448,1355);
// by ID
G=gap.SmallGroup(448,1355);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,1192,14117,7068,124]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations
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