metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.131D14, (C4×Q8)⋊13D7, (C4×D28)⋊40C2, (Q8×C28)⋊15C2, (D7×C42)⋊7C2, C4⋊C4.298D14, D14⋊3Q8⋊46C2, D14.3(C4○D4), C4.48(C4○D28), C28⋊1D4.14C2, C42⋊D7⋊17C2, (C2×Q8).179D14, C4.Dic14⋊47C2, D14.5D4⋊50C2, C28.340(C4○D4), C28.23D4⋊33C2, (C4×C28).176C22, (C2×C28).622C23, (C2×C14).124C24, C4.60(Q8⋊2D7), D14⋊C4.104C22, (C2×D28).217C22, C4⋊Dic7.308C22, (Q8×C14).224C22, C22.145(C23×D7), Dic7⋊C4.156C22, C7⋊5(C23.36C23), (C2×Dic7).217C23, (C4×Dic7).295C22, (C22×D7).181C23, C2.31(D7×C4○D4), C4⋊C4⋊D7⋊51C2, C2.63(C2×C4○D28), C14.146(C2×C4○D4), C2.12(C2×Q8⋊2D7), (C2×C4×D7).296C22, (C7×C4⋊C4).352C22, (C2×C4).170(C22×D7), SmallGroup(448,1033)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1060 in 234 conjugacy classes, 101 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×10], C7, C2×C4 [×3], C2×C4 [×4], C2×C4 [×15], D4 [×6], Q8 [×2], C23 [×3], D7 [×4], C14 [×3], C42, C42 [×2], C42 [×3], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×7], C22×C4 [×5], C2×D4 [×3], C2×Q8, Dic7 [×5], C28 [×4], C28 [×5], D14 [×2], D14 [×8], C2×C14, C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], C4×D7 [×10], D28 [×6], C2×Dic7 [×3], C2×Dic7 [×2], C2×C28 [×3], C2×C28 [×4], C7×Q8 [×2], C22×D7, C22×D7 [×2], C23.36C23, C4×Dic7 [×3], Dic7⋊C4 [×4], C4⋊Dic7, C4⋊Dic7 [×2], D14⋊C4 [×10], C4×C28, C4×C28 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×C4×D7 [×3], C2×C4×D7 [×2], C2×D28, C2×D28 [×2], Q8×C14, D7×C42, C42⋊D7 [×2], C4×D28, C4×D28 [×2], C4.Dic14, D14.5D4 [×2], C28⋊1D4, C4⋊C4⋊D7 [×2], D14⋊3Q8, C28.23D4, Q8×C28, C42.131D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×6], C24, D14 [×7], C2×C4○D4 [×3], C22×D7 [×7], C23.36C23, C4○D28 [×2], Q8⋊2D7 [×2], C23×D7, C2×C4○D28, C2×Q8⋊2D7, D7×C4○D4, C42.131D14
Generators and relations
G = < a,b,c,d | a4=b4=1, c14=d2=a2b2, ab=ba, cac-1=a-1, dad-1=ab2, bc=cb, bd=db, dcd-1=c13 >
►On 224 points
Generators in S224
(1 211 191 61)(2 62 192 212)(3 213 193 63)(4 64 194 214)(5 215 195 65)(6 66 196 216)(7 217 169 67)(8 68 170 218)(9 219 171 69)(10 70 172 220)(11 221 173 71)(12 72 174 222)(13 223 175 73)(14 74 176 224)(15 197 177 75)(16 76 178 198)(17 199 179 77)(18 78 180 200)(19 201 181 79)(20 80 182 202)(21 203 183 81)(22 82 184 204)(23 205 185 83)(24 84 186 206)(25 207 187 57)(26 58 188 208)(27 209 189 59)(28 60 190 210)(29 94 146 140)(30 113 147 95)(31 96 148 114)(32 115 149 97)(33 98 150 116)(34 117 151 99)(35 100 152 118)(36 119 153 101)(37 102 154 120)(38 121 155 103)(39 104 156 122)(40 123 157 105)(41 106 158 124)(42 125 159 107)(43 108 160 126)(44 127 161 109)(45 110 162 128)(46 129 163 111)(47 112 164 130)(48 131 165 85)(49 86 166 132)(50 133 167 87)(51 88 168 134)(52 135 141 89)(53 90 142 136)(54 137 143 91)(55 92 144 138)(56 139 145 93)
(1 30 177 161)(2 31 178 162)(3 32 179 163)(4 33 180 164)(5 34 181 165)(6 35 182 166)(7 36 183 167)(8 37 184 168)(9 38 185 141)(10 39 186 142)(11 40 187 143)(12 41 188 144)(13 42 189 145)(14 43 190 146)(15 44 191 147)(16 45 192 148)(17 46 193 149)(18 47 194 150)(19 48 195 151)(20 49 196 152)(21 50 169 153)(22 51 170 154)(23 52 171 155)(24 53 172 156)(25 54 173 157)(26 55 174 158)(27 56 175 159)(28 29 176 160)(57 91 221 123)(58 92 222 124)(59 93 223 125)(60 94 224 126)(61 95 197 127)(62 96 198 128)(63 97 199 129)(64 98 200 130)(65 99 201 131)(66 100 202 132)(67 101 203 133)(68 102 204 134)(69 103 205 135)(70 104 206 136)(71 105 207 137)(72 106 208 138)(73 107 209 139)(74 108 210 140)(75 109 211 113)(76 110 212 114)(77 111 213 115)(78 112 214 116)(79 85 215 117)(80 86 216 118)(81 87 217 119)(82 88 218 120)(83 89 219 121)(84 90 220 122)
(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)
(1 190 15 176)(2 175 16 189)(3 188 17 174)(4 173 18 187)(5 186 19 172)(6 171 20 185)(7 184 21 170)(8 169 22 183)(9 182 23 196)(10 195 24 181)(11 180 25 194)(12 193 26 179)(13 178 27 192)(14 191 28 177)(29 161 43 147)(30 146 44 160)(31 159 45 145)(32 144 46 158)(33 157 47 143)(34 142 48 156)(35 155 49 141)(36 168 50 154)(37 153 51 167)(38 166 52 152)(39 151 53 165)(40 164 54 150)(41 149 55 163)(42 162 56 148)(57 200 71 214)(58 213 72 199)(59 198 73 212)(60 211 74 197)(61 224 75 210)(62 209 76 223)(63 222 77 208)(64 207 78 221)(65 220 79 206)(66 205 80 219)(67 218 81 204)(68 203 82 217)(69 216 83 202)(70 201 84 215)(85 136 99 122)(86 121 100 135)(87 134 101 120)(88 119 102 133)(89 132 103 118)(90 117 104 131)(91 130 105 116)(92 115 106 129)(93 128 107 114)(94 113 108 127)(95 126 109 140)(96 139 110 125)(97 124 111 138)(98 137 112 123)
G:=sub<Sym(224)| (1,211,191,61)(2,62,192,212)(3,213,193,63)(4,64,194,214)(5,215,195,65)(6,66,196,216)(7,217,169,67)(8,68,170,218)(9,219,171,69)(10,70,172,220)(11,221,173,71)(12,72,174,222)(13,223,175,73)(14,74,176,224)(15,197,177,75)(16,76,178,198)(17,199,179,77)(18,78,180,200)(19,201,181,79)(20,80,182,202)(21,203,183,81)(22,82,184,204)(23,205,185,83)(24,84,186,206)(25,207,187,57)(26,58,188,208)(27,209,189,59)(28,60,190,210)(29,94,146,140)(30,113,147,95)(31,96,148,114)(32,115,149,97)(33,98,150,116)(34,117,151,99)(35,100,152,118)(36,119,153,101)(37,102,154,120)(38,121,155,103)(39,104,156,122)(40,123,157,105)(41,106,158,124)(42,125,159,107)(43,108,160,126)(44,127,161,109)(45,110,162,128)(46,129,163,111)(47,112,164,130)(48,131,165,85)(49,86,166,132)(50,133,167,87)(51,88,168,134)(52,135,141,89)(53,90,142,136)(54,137,143,91)(55,92,144,138)(56,139,145,93), (1,30,177,161)(2,31,178,162)(3,32,179,163)(4,33,180,164)(5,34,181,165)(6,35,182,166)(7,36,183,167)(8,37,184,168)(9,38,185,141)(10,39,186,142)(11,40,187,143)(12,41,188,144)(13,42,189,145)(14,43,190,146)(15,44,191,147)(16,45,192,148)(17,46,193,149)(18,47,194,150)(19,48,195,151)(20,49,196,152)(21,50,169,153)(22,51,170,154)(23,52,171,155)(24,53,172,156)(25,54,173,157)(26,55,174,158)(27,56,175,159)(28,29,176,160)(57,91,221,123)(58,92,222,124)(59,93,223,125)(60,94,224,126)(61,95,197,127)(62,96,198,128)(63,97,199,129)(64,98,200,130)(65,99,201,131)(66,100,202,132)(67,101,203,133)(68,102,204,134)(69,103,205,135)(70,104,206,136)(71,105,207,137)(72,106,208,138)(73,107,209,139)(74,108,210,140)(75,109,211,113)(76,110,212,114)(77,111,213,115)(78,112,214,116)(79,85,215,117)(80,86,216,118)(81,87,217,119)(82,88,218,120)(83,89,219,121)(84,90,220,122), (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), (1,190,15,176)(2,175,16,189)(3,188,17,174)(4,173,18,187)(5,186,19,172)(6,171,20,185)(7,184,21,170)(8,169,22,183)(9,182,23,196)(10,195,24,181)(11,180,25,194)(12,193,26,179)(13,178,27,192)(14,191,28,177)(29,161,43,147)(30,146,44,160)(31,159,45,145)(32,144,46,158)(33,157,47,143)(34,142,48,156)(35,155,49,141)(36,168,50,154)(37,153,51,167)(38,166,52,152)(39,151,53,165)(40,164,54,150)(41,149,55,163)(42,162,56,148)(57,200,71,214)(58,213,72,199)(59,198,73,212)(60,211,74,197)(61,224,75,210)(62,209,76,223)(63,222,77,208)(64,207,78,221)(65,220,79,206)(66,205,80,219)(67,218,81,204)(68,203,82,217)(69,216,83,202)(70,201,84,215)(85,136,99,122)(86,121,100,135)(87,134,101,120)(88,119,102,133)(89,132,103,118)(90,117,104,131)(91,130,105,116)(92,115,106,129)(93,128,107,114)(94,113,108,127)(95,126,109,140)(96,139,110,125)(97,124,111,138)(98,137,112,123)>;
G:=Group( (1,211,191,61)(2,62,192,212)(3,213,193,63)(4,64,194,214)(5,215,195,65)(6,66,196,216)(7,217,169,67)(8,68,170,218)(9,219,171,69)(10,70,172,220)(11,221,173,71)(12,72,174,222)(13,223,175,73)(14,74,176,224)(15,197,177,75)(16,76,178,198)(17,199,179,77)(18,78,180,200)(19,201,181,79)(20,80,182,202)(21,203,183,81)(22,82,184,204)(23,205,185,83)(24,84,186,206)(25,207,187,57)(26,58,188,208)(27,209,189,59)(28,60,190,210)(29,94,146,140)(30,113,147,95)(31,96,148,114)(32,115,149,97)(33,98,150,116)(34,117,151,99)(35,100,152,118)(36,119,153,101)(37,102,154,120)(38,121,155,103)(39,104,156,122)(40,123,157,105)(41,106,158,124)(42,125,159,107)(43,108,160,126)(44,127,161,109)(45,110,162,128)(46,129,163,111)(47,112,164,130)(48,131,165,85)(49,86,166,132)(50,133,167,87)(51,88,168,134)(52,135,141,89)(53,90,142,136)(54,137,143,91)(55,92,144,138)(56,139,145,93), (1,30,177,161)(2,31,178,162)(3,32,179,163)(4,33,180,164)(5,34,181,165)(6,35,182,166)(7,36,183,167)(8,37,184,168)(9,38,185,141)(10,39,186,142)(11,40,187,143)(12,41,188,144)(13,42,189,145)(14,43,190,146)(15,44,191,147)(16,45,192,148)(17,46,193,149)(18,47,194,150)(19,48,195,151)(20,49,196,152)(21,50,169,153)(22,51,170,154)(23,52,171,155)(24,53,172,156)(25,54,173,157)(26,55,174,158)(27,56,175,159)(28,29,176,160)(57,91,221,123)(58,92,222,124)(59,93,223,125)(60,94,224,126)(61,95,197,127)(62,96,198,128)(63,97,199,129)(64,98,200,130)(65,99,201,131)(66,100,202,132)(67,101,203,133)(68,102,204,134)(69,103,205,135)(70,104,206,136)(71,105,207,137)(72,106,208,138)(73,107,209,139)(74,108,210,140)(75,109,211,113)(76,110,212,114)(77,111,213,115)(78,112,214,116)(79,85,215,117)(80,86,216,118)(81,87,217,119)(82,88,218,120)(83,89,219,121)(84,90,220,122), (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), (1,190,15,176)(2,175,16,189)(3,188,17,174)(4,173,18,187)(5,186,19,172)(6,171,20,185)(7,184,21,170)(8,169,22,183)(9,182,23,196)(10,195,24,181)(11,180,25,194)(12,193,26,179)(13,178,27,192)(14,191,28,177)(29,161,43,147)(30,146,44,160)(31,159,45,145)(32,144,46,158)(33,157,47,143)(34,142,48,156)(35,155,49,141)(36,168,50,154)(37,153,51,167)(38,166,52,152)(39,151,53,165)(40,164,54,150)(41,149,55,163)(42,162,56,148)(57,200,71,214)(58,213,72,199)(59,198,73,212)(60,211,74,197)(61,224,75,210)(62,209,76,223)(63,222,77,208)(64,207,78,221)(65,220,79,206)(66,205,80,219)(67,218,81,204)(68,203,82,217)(69,216,83,202)(70,201,84,215)(85,136,99,122)(86,121,100,135)(87,134,101,120)(88,119,102,133)(89,132,103,118)(90,117,104,131)(91,130,105,116)(92,115,106,129)(93,128,107,114)(94,113,108,127)(95,126,109,140)(96,139,110,125)(97,124,111,138)(98,137,112,123) );
G=PermutationGroup([(1,211,191,61),(2,62,192,212),(3,213,193,63),(4,64,194,214),(5,215,195,65),(6,66,196,216),(7,217,169,67),(8,68,170,218),(9,219,171,69),(10,70,172,220),(11,221,173,71),(12,72,174,222),(13,223,175,73),(14,74,176,224),(15,197,177,75),(16,76,178,198),(17,199,179,77),(18,78,180,200),(19,201,181,79),(20,80,182,202),(21,203,183,81),(22,82,184,204),(23,205,185,83),(24,84,186,206),(25,207,187,57),(26,58,188,208),(27,209,189,59),(28,60,190,210),(29,94,146,140),(30,113,147,95),(31,96,148,114),(32,115,149,97),(33,98,150,116),(34,117,151,99),(35,100,152,118),(36,119,153,101),(37,102,154,120),(38,121,155,103),(39,104,156,122),(40,123,157,105),(41,106,158,124),(42,125,159,107),(43,108,160,126),(44,127,161,109),(45,110,162,128),(46,129,163,111),(47,112,164,130),(48,131,165,85),(49,86,166,132),(50,133,167,87),(51,88,168,134),(52,135,141,89),(53,90,142,136),(54,137,143,91),(55,92,144,138),(56,139,145,93)], [(1,30,177,161),(2,31,178,162),(3,32,179,163),(4,33,180,164),(5,34,181,165),(6,35,182,166),(7,36,183,167),(8,37,184,168),(9,38,185,141),(10,39,186,142),(11,40,187,143),(12,41,188,144),(13,42,189,145),(14,43,190,146),(15,44,191,147),(16,45,192,148),(17,46,193,149),(18,47,194,150),(19,48,195,151),(20,49,196,152),(21,50,169,153),(22,51,170,154),(23,52,171,155),(24,53,172,156),(25,54,173,157),(26,55,174,158),(27,56,175,159),(28,29,176,160),(57,91,221,123),(58,92,222,124),(59,93,223,125),(60,94,224,126),(61,95,197,127),(62,96,198,128),(63,97,199,129),(64,98,200,130),(65,99,201,131),(66,100,202,132),(67,101,203,133),(68,102,204,134),(69,103,205,135),(70,104,206,136),(71,105,207,137),(72,106,208,138),(73,107,209,139),(74,108,210,140),(75,109,211,113),(76,110,212,114),(77,111,213,115),(78,112,214,116),(79,85,215,117),(80,86,216,118),(81,87,217,119),(82,88,218,120),(83,89,219,121),(84,90,220,122)], [(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)], [(1,190,15,176),(2,175,16,189),(3,188,17,174),(4,173,18,187),(5,186,19,172),(6,171,20,185),(7,184,21,170),(8,169,22,183),(9,182,23,196),(10,195,24,181),(11,180,25,194),(12,193,26,179),(13,178,27,192),(14,191,28,177),(29,161,43,147),(30,146,44,160),(31,159,45,145),(32,144,46,158),(33,157,47,143),(34,142,48,156),(35,155,49,141),(36,168,50,154),(37,153,51,167),(38,166,52,152),(39,151,53,165),(40,164,54,150),(41,149,55,163),(42,162,56,148),(57,200,71,214),(58,213,72,199),(59,198,73,212),(60,211,74,197),(61,224,75,210),(62,209,76,223),(63,222,77,208),(64,207,78,221),(65,220,79,206),(66,205,80,219),(67,218,81,204),(68,203,82,217),(69,216,83,202),(70,201,84,215),(85,136,99,122),(86,121,100,135),(87,134,101,120),(88,119,102,133),(89,132,103,118),(90,117,104,131),(91,130,105,116),(92,115,106,129),(93,128,107,114),(94,113,108,127),(95,126,109,140),(96,139,110,125),(97,124,111,138),(98,137,112,123)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 8 | 7 | 0 | 0 | 0 | 0 |
| 20 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 | 17 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 19 | 0 | 0 |
| 0 | 0 | 10 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 24 |
| 0 | 0 | 0 | 0 | 17 | 12 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 15 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 12 | 17 |
G:=sub<GL(6,GF(29))| [8,20,0,0,0,0,7,21,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,12,0,0,0,0,0,17],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,7,10,0,0,0,0,19,19,0,0,0,0,0,0,17,17,0,0,0,0,24,12],[17,15,0,0,0,0,0,12,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,12,12,0,0,0,0,5,17] >;
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4R | 4S | 4T | 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 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 28 | 28 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | C2 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | C4○D28 | Q8⋊2D7 | D7×C4○D4 |
| kernel | C42.131D14 | D7×C42 | C42⋊D7 | C4×D28 | C4.Dic14 | D14.5D4 | C28⋊1D4 | C4⋊C4⋊D7 | D14⋊3Q8 | C28.23D4 | Q8×C28 | C4×Q8 | C28 | D14 | C42 | C4⋊C4 | C2×Q8 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 3 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 3 | 8 | 4 | 9 | 9 | 3 | 24 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_4^2._{131}D_{14} % in TeX
G:=Group("C4^2.131D14"); // GroupNames label
G:=SmallGroup(448,1033);
// by ID
G=gap.SmallGroup(448,1033);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,100,1123,794,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^13>;
// generators/relations
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