metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.171D14, C14.342- (1+4), C4⋊Q8⋊9D7, C4.37(D4×D7), (C4×D7).13D4, C28.69(C2×D4), C4⋊C4.122D14, D14.48(C2×D4), D14⋊Q8⋊47C2, C4.D28⋊26C2, C42⋊D7⋊25C2, (C2×Q8).143D14, Dic7.53(C2×D4), C14.98(C22×D4), Dic7⋊Q8⋊26C2, D14.5D4⋊45C2, C28.23D4⋊25C2, (C2×C14).268C24, (C2×C28).101C23, (C4×C28).209C22, D14⋊C4.49C22, (C2×D28).171C22, (Q8×C14).135C22, C22.289(C23×D7), Dic7⋊C4.165C22, C7⋊5(C23.38C23), (C2×Dic7).140C23, (C4×Dic7).159C22, (C22×D7).230C23, C2.35(Q8.10D14), (C2×Dic14).188C22, (C2×Q8×D7)⋊12C2, C2.71(C2×D4×D7), (C7×C4⋊Q8)⋊10C2, (C2×Q8⋊2D7).7C2, (C2×C4×D7).142C22, (C7×C4⋊C4).211C22, (C2×C4).217(C22×D7), SmallGroup(448,1177)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1292 in 270 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×10], C7, C2×C4, C2×C4 [×6], C2×C4 [×17], D4 [×6], Q8 [×10], C23 [×3], D7 [×4], C14, C14 [×2], C42, C42, C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×6], C22×C4 [×5], C2×D4 [×3], C2×Q8 [×2], C2×Q8 [×7], C4○D4 [×4], Dic7 [×2], Dic7 [×4], C28 [×2], C28 [×6], D14 [×2], D14 [×8], C2×C14, C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8, C4⋊Q8, C22×Q8, C2×C4○D4, Dic14 [×6], C4×D7 [×4], C4×D7 [×8], D28 [×6], C2×Dic7, C2×Dic7 [×4], C2×C28, C2×C28 [×6], C7×Q8 [×4], C22×D7, C22×D7 [×2], C23.38C23, C4×Dic7, Dic7⋊C4 [×6], D14⋊C4 [×10], C4×C28, C7×C4⋊C4 [×4], C2×Dic14, C2×Dic14 [×2], C2×C4×D7, C2×C4×D7 [×4], C2×D28, C2×D28 [×2], Q8×D7 [×4], Q8⋊2D7 [×4], Q8×C14 [×2], C42⋊D7, C4.D28, D14.5D4 [×4], D14⋊Q8 [×4], Dic7⋊Q8, C28.23D4, C7×C4⋊Q8, C2×Q8×D7, C2×Q8⋊2D7, C42.171D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, 2- (1+4) [×2], C22×D7 [×7], C23.38C23, D4×D7 [×2], C23×D7, C2×D4×D7, Q8.10D14 [×2], C42.171D14
Generators and relations
G = < a,b,c,d | a4=b4=1, c14=d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c13 >
►On 224 points
Generators in S224
(1 55 15 41)(2 42 16 56)(3 29 17 43)(4 44 18 30)(5 31 19 45)(6 46 20 32)(7 33 21 47)(8 48 22 34)(9 35 23 49)(10 50 24 36)(11 37 25 51)(12 52 26 38)(13 39 27 53)(14 54 28 40)(57 169 71 183)(58 184 72 170)(59 171 73 185)(60 186 74 172)(61 173 75 187)(62 188 76 174)(63 175 77 189)(64 190 78 176)(65 177 79 191)(66 192 80 178)(67 179 81 193)(68 194 82 180)(69 181 83 195)(70 196 84 182)(85 150 99 164)(86 165 100 151)(87 152 101 166)(88 167 102 153)(89 154 103 168)(90 141 104 155)(91 156 105 142)(92 143 106 157)(93 158 107 144)(94 145 108 159)(95 160 109 146)(96 147 110 161)(97 162 111 148)(98 149 112 163)(113 207 127 221)(114 222 128 208)(115 209 129 223)(116 224 130 210)(117 211 131 197)(118 198 132 212)(119 213 133 199)(120 200 134 214)(121 215 135 201)(122 202 136 216)(123 217 137 203)(124 204 138 218)(125 219 139 205)(126 206 140 220)
(1 188 132 162)(2 163 133 189)(3 190 134 164)(4 165 135 191)(5 192 136 166)(6 167 137 193)(7 194 138 168)(8 141 139 195)(9 196 140 142)(10 143 113 169)(11 170 114 144)(12 145 115 171)(13 172 116 146)(14 147 117 173)(15 174 118 148)(16 149 119 175)(17 176 120 150)(18 151 121 177)(19 178 122 152)(20 153 123 179)(21 180 124 154)(22 155 125 181)(23 182 126 156)(24 157 127 183)(25 184 128 158)(26 159 129 185)(27 186 130 160)(28 161 131 187)(29 78 214 85)(30 86 215 79)(31 80 216 87)(32 88 217 81)(33 82 218 89)(34 90 219 83)(35 84 220 91)(36 92 221 57)(37 58 222 93)(38 94 223 59)(39 60 224 95)(40 96 197 61)(41 62 198 97)(42 98 199 63)(43 64 200 99)(44 100 201 65)(45 66 202 101)(46 102 203 67)(47 68 204 103)(48 104 205 69)(49 70 206 105)(50 106 207 71)(51 72 208 107)(52 108 209 73)(53 74 210 109)(54 110 211 75)(55 76 212 111)(56 112 213 77)
(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 14 15 28)(2 27 16 13)(3 12 17 26)(4 25 18 11)(5 10 19 24)(6 23 20 9)(7 8 21 22)(29 38 43 52)(30 51 44 37)(31 36 45 50)(32 49 46 35)(33 34 47 48)(39 56 53 42)(40 41 54 55)(57 80 71 66)(58 65 72 79)(59 78 73 64)(60 63 74 77)(61 76 75 62)(67 70 81 84)(68 83 82 69)(85 108 99 94)(86 93 100 107)(87 106 101 92)(88 91 102 105)(89 104 103 90)(95 98 109 112)(96 111 110 97)(113 122 127 136)(114 135 128 121)(115 120 129 134)(116 133 130 119)(117 118 131 132)(123 140 137 126)(124 125 138 139)(141 168 155 154)(142 153 156 167)(143 166 157 152)(144 151 158 165)(145 164 159 150)(146 149 160 163)(147 162 161 148)(169 192 183 178)(170 177 184 191)(171 190 185 176)(172 175 186 189)(173 188 187 174)(179 182 193 196)(180 195 194 181)(197 198 211 212)(199 224 213 210)(200 209 214 223)(201 222 215 208)(202 207 216 221)(203 220 217 206)(204 205 218 219)
G:=sub<Sym(224)| (1,55,15,41)(2,42,16,56)(3,29,17,43)(4,44,18,30)(5,31,19,45)(6,46,20,32)(7,33,21,47)(8,48,22,34)(9,35,23,49)(10,50,24,36)(11,37,25,51)(12,52,26,38)(13,39,27,53)(14,54,28,40)(57,169,71,183)(58,184,72,170)(59,171,73,185)(60,186,74,172)(61,173,75,187)(62,188,76,174)(63,175,77,189)(64,190,78,176)(65,177,79,191)(66,192,80,178)(67,179,81,193)(68,194,82,180)(69,181,83,195)(70,196,84,182)(85,150,99,164)(86,165,100,151)(87,152,101,166)(88,167,102,153)(89,154,103,168)(90,141,104,155)(91,156,105,142)(92,143,106,157)(93,158,107,144)(94,145,108,159)(95,160,109,146)(96,147,110,161)(97,162,111,148)(98,149,112,163)(113,207,127,221)(114,222,128,208)(115,209,129,223)(116,224,130,210)(117,211,131,197)(118,198,132,212)(119,213,133,199)(120,200,134,214)(121,215,135,201)(122,202,136,216)(123,217,137,203)(124,204,138,218)(125,219,139,205)(126,206,140,220), (1,188,132,162)(2,163,133,189)(3,190,134,164)(4,165,135,191)(5,192,136,166)(6,167,137,193)(7,194,138,168)(8,141,139,195)(9,196,140,142)(10,143,113,169)(11,170,114,144)(12,145,115,171)(13,172,116,146)(14,147,117,173)(15,174,118,148)(16,149,119,175)(17,176,120,150)(18,151,121,177)(19,178,122,152)(20,153,123,179)(21,180,124,154)(22,155,125,181)(23,182,126,156)(24,157,127,183)(25,184,128,158)(26,159,129,185)(27,186,130,160)(28,161,131,187)(29,78,214,85)(30,86,215,79)(31,80,216,87)(32,88,217,81)(33,82,218,89)(34,90,219,83)(35,84,220,91)(36,92,221,57)(37,58,222,93)(38,94,223,59)(39,60,224,95)(40,96,197,61)(41,62,198,97)(42,98,199,63)(43,64,200,99)(44,100,201,65)(45,66,202,101)(46,102,203,67)(47,68,204,103)(48,104,205,69)(49,70,206,105)(50,106,207,71)(51,72,208,107)(52,108,209,73)(53,74,210,109)(54,110,211,75)(55,76,212,111)(56,112,213,77), (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), (1,14,15,28)(2,27,16,13)(3,12,17,26)(4,25,18,11)(5,10,19,24)(6,23,20,9)(7,8,21,22)(29,38,43,52)(30,51,44,37)(31,36,45,50)(32,49,46,35)(33,34,47,48)(39,56,53,42)(40,41,54,55)(57,80,71,66)(58,65,72,79)(59,78,73,64)(60,63,74,77)(61,76,75,62)(67,70,81,84)(68,83,82,69)(85,108,99,94)(86,93,100,107)(87,106,101,92)(88,91,102,105)(89,104,103,90)(95,98,109,112)(96,111,110,97)(113,122,127,136)(114,135,128,121)(115,120,129,134)(116,133,130,119)(117,118,131,132)(123,140,137,126)(124,125,138,139)(141,168,155,154)(142,153,156,167)(143,166,157,152)(144,151,158,165)(145,164,159,150)(146,149,160,163)(147,162,161,148)(169,192,183,178)(170,177,184,191)(171,190,185,176)(172,175,186,189)(173,188,187,174)(179,182,193,196)(180,195,194,181)(197,198,211,212)(199,224,213,210)(200,209,214,223)(201,222,215,208)(202,207,216,221)(203,220,217,206)(204,205,218,219)>;
G:=Group( (1,55,15,41)(2,42,16,56)(3,29,17,43)(4,44,18,30)(5,31,19,45)(6,46,20,32)(7,33,21,47)(8,48,22,34)(9,35,23,49)(10,50,24,36)(11,37,25,51)(12,52,26,38)(13,39,27,53)(14,54,28,40)(57,169,71,183)(58,184,72,170)(59,171,73,185)(60,186,74,172)(61,173,75,187)(62,188,76,174)(63,175,77,189)(64,190,78,176)(65,177,79,191)(66,192,80,178)(67,179,81,193)(68,194,82,180)(69,181,83,195)(70,196,84,182)(85,150,99,164)(86,165,100,151)(87,152,101,166)(88,167,102,153)(89,154,103,168)(90,141,104,155)(91,156,105,142)(92,143,106,157)(93,158,107,144)(94,145,108,159)(95,160,109,146)(96,147,110,161)(97,162,111,148)(98,149,112,163)(113,207,127,221)(114,222,128,208)(115,209,129,223)(116,224,130,210)(117,211,131,197)(118,198,132,212)(119,213,133,199)(120,200,134,214)(121,215,135,201)(122,202,136,216)(123,217,137,203)(124,204,138,218)(125,219,139,205)(126,206,140,220), (1,188,132,162)(2,163,133,189)(3,190,134,164)(4,165,135,191)(5,192,136,166)(6,167,137,193)(7,194,138,168)(8,141,139,195)(9,196,140,142)(10,143,113,169)(11,170,114,144)(12,145,115,171)(13,172,116,146)(14,147,117,173)(15,174,118,148)(16,149,119,175)(17,176,120,150)(18,151,121,177)(19,178,122,152)(20,153,123,179)(21,180,124,154)(22,155,125,181)(23,182,126,156)(24,157,127,183)(25,184,128,158)(26,159,129,185)(27,186,130,160)(28,161,131,187)(29,78,214,85)(30,86,215,79)(31,80,216,87)(32,88,217,81)(33,82,218,89)(34,90,219,83)(35,84,220,91)(36,92,221,57)(37,58,222,93)(38,94,223,59)(39,60,224,95)(40,96,197,61)(41,62,198,97)(42,98,199,63)(43,64,200,99)(44,100,201,65)(45,66,202,101)(46,102,203,67)(47,68,204,103)(48,104,205,69)(49,70,206,105)(50,106,207,71)(51,72,208,107)(52,108,209,73)(53,74,210,109)(54,110,211,75)(55,76,212,111)(56,112,213,77), (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), (1,14,15,28)(2,27,16,13)(3,12,17,26)(4,25,18,11)(5,10,19,24)(6,23,20,9)(7,8,21,22)(29,38,43,52)(30,51,44,37)(31,36,45,50)(32,49,46,35)(33,34,47,48)(39,56,53,42)(40,41,54,55)(57,80,71,66)(58,65,72,79)(59,78,73,64)(60,63,74,77)(61,76,75,62)(67,70,81,84)(68,83,82,69)(85,108,99,94)(86,93,100,107)(87,106,101,92)(88,91,102,105)(89,104,103,90)(95,98,109,112)(96,111,110,97)(113,122,127,136)(114,135,128,121)(115,120,129,134)(116,133,130,119)(117,118,131,132)(123,140,137,126)(124,125,138,139)(141,168,155,154)(142,153,156,167)(143,166,157,152)(144,151,158,165)(145,164,159,150)(146,149,160,163)(147,162,161,148)(169,192,183,178)(170,177,184,191)(171,190,185,176)(172,175,186,189)(173,188,187,174)(179,182,193,196)(180,195,194,181)(197,198,211,212)(199,224,213,210)(200,209,214,223)(201,222,215,208)(202,207,216,221)(203,220,217,206)(204,205,218,219) );
G=PermutationGroup([(1,55,15,41),(2,42,16,56),(3,29,17,43),(4,44,18,30),(5,31,19,45),(6,46,20,32),(7,33,21,47),(8,48,22,34),(9,35,23,49),(10,50,24,36),(11,37,25,51),(12,52,26,38),(13,39,27,53),(14,54,28,40),(57,169,71,183),(58,184,72,170),(59,171,73,185),(60,186,74,172),(61,173,75,187),(62,188,76,174),(63,175,77,189),(64,190,78,176),(65,177,79,191),(66,192,80,178),(67,179,81,193),(68,194,82,180),(69,181,83,195),(70,196,84,182),(85,150,99,164),(86,165,100,151),(87,152,101,166),(88,167,102,153),(89,154,103,168),(90,141,104,155),(91,156,105,142),(92,143,106,157),(93,158,107,144),(94,145,108,159),(95,160,109,146),(96,147,110,161),(97,162,111,148),(98,149,112,163),(113,207,127,221),(114,222,128,208),(115,209,129,223),(116,224,130,210),(117,211,131,197),(118,198,132,212),(119,213,133,199),(120,200,134,214),(121,215,135,201),(122,202,136,216),(123,217,137,203),(124,204,138,218),(125,219,139,205),(126,206,140,220)], [(1,188,132,162),(2,163,133,189),(3,190,134,164),(4,165,135,191),(5,192,136,166),(6,167,137,193),(7,194,138,168),(8,141,139,195),(9,196,140,142),(10,143,113,169),(11,170,114,144),(12,145,115,171),(13,172,116,146),(14,147,117,173),(15,174,118,148),(16,149,119,175),(17,176,120,150),(18,151,121,177),(19,178,122,152),(20,153,123,179),(21,180,124,154),(22,155,125,181),(23,182,126,156),(24,157,127,183),(25,184,128,158),(26,159,129,185),(27,186,130,160),(28,161,131,187),(29,78,214,85),(30,86,215,79),(31,80,216,87),(32,88,217,81),(33,82,218,89),(34,90,219,83),(35,84,220,91),(36,92,221,57),(37,58,222,93),(38,94,223,59),(39,60,224,95),(40,96,197,61),(41,62,198,97),(42,98,199,63),(43,64,200,99),(44,100,201,65),(45,66,202,101),(46,102,203,67),(47,68,204,103),(48,104,205,69),(49,70,206,105),(50,106,207,71),(51,72,208,107),(52,108,209,73),(53,74,210,109),(54,110,211,75),(55,76,212,111),(56,112,213,77)], [(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)], [(1,14,15,28),(2,27,16,13),(3,12,17,26),(4,25,18,11),(5,10,19,24),(6,23,20,9),(7,8,21,22),(29,38,43,52),(30,51,44,37),(31,36,45,50),(32,49,46,35),(33,34,47,48),(39,56,53,42),(40,41,54,55),(57,80,71,66),(58,65,72,79),(59,78,73,64),(60,63,74,77),(61,76,75,62),(67,70,81,84),(68,83,82,69),(85,108,99,94),(86,93,100,107),(87,106,101,92),(88,91,102,105),(89,104,103,90),(95,98,109,112),(96,111,110,97),(113,122,127,136),(114,135,128,121),(115,120,129,134),(116,133,130,119),(117,118,131,132),(123,140,137,126),(124,125,138,139),(141,168,155,154),(142,153,156,167),(143,166,157,152),(144,151,158,165),(145,164,159,150),(146,149,160,163),(147,162,161,148),(169,192,183,178),(170,177,184,191),(171,190,185,176),(172,175,186,189),(173,188,187,174),(179,182,193,196),(180,195,194,181),(197,198,211,212),(199,224,213,210),(200,209,214,223),(201,222,215,208),(202,207,216,221),(203,220,217,206),(204,205,218,219)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 13 | 2 | 22 |
| 0 | 0 | 18 | 9 | 13 | 19 |
| 0 | 0 | 20 | 2 | 5 | 16 |
| 0 | 0 | 8 | 11 | 16 | 5 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 16 | 0 | 4 |
| 0 | 0 | 24 | 20 | 25 | 11 |
| 0 | 0 | 24 | 15 | 24 | 13 |
| 0 | 0 | 14 | 0 | 16 | 5 |
| 5 | 18 | 0 | 0 | 0 | 0 |
| 18 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 5 | 13 | 9 |
| 0 | 0 | 24 | 2 | 1 | 16 |
| 0 | 0 | 21 | 25 | 19 | 24 |
| 0 | 0 | 0 | 10 | 10 | 12 |
| 24 | 11 | 0 | 0 | 0 | 0 |
| 11 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 12 | 20 | 16 |
| 0 | 0 | 3 | 14 | 13 | 28 |
| 0 | 0 | 17 | 19 | 5 | 10 |
| 0 | 0 | 4 | 16 | 17 | 19 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,18,20,8,0,0,13,9,2,11,0,0,2,13,5,16,0,0,22,19,16,5],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,9,24,24,14,0,0,16,20,15,0,0,0,0,25,24,16,0,0,4,11,13,5],[5,18,0,0,0,0,18,24,0,0,0,0,0,0,25,24,21,0,0,0,5,2,25,10,0,0,13,1,19,10,0,0,9,16,24,12],[24,11,0,0,0,0,11,5,0,0,0,0,0,0,20,3,17,4,0,0,12,14,19,16,0,0,20,13,5,17,0,0,16,28,10,19] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | 2- (1+4) | D4×D7 | Q8.10D14 |
| kernel | C42.171D14 | C42⋊D7 | C4.D28 | D14.5D4 | D14⋊Q8 | Dic7⋊Q8 | C28.23D4 | C7×C4⋊Q8 | C2×Q8×D7 | C2×Q8⋊2D7 | C4×D7 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C14 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 3 | 12 | 6 | 2 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_4^2._{171}D_{14} % in TeX
G:=Group("C4^2.171D14"); // GroupNames label
G:=SmallGroup(448,1177);
// by ID
G=gap.SmallGroup(448,1177);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,100,675,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^13>;
// generators/relations
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