metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.134D14, C14.142- (1+4), (C4×Q8)⋊16D7, (Q8×C28)⋊18C2, C4⋊C4.301D14, (C4×Dic14)⋊40C2, C4.50(C4○D28), Dic7.Q8⋊10C2, (C2×Q8).182D14, C28.6Q8⋊27C2, C4.Dic14⋊17C2, C42⋊2D7.2C2, C42⋊D7.5C2, Dic7⋊Q8⋊10C2, C28.121(C4○D4), (C4×C28).179C22, (C2×C28).624C23, (C2×C14).127C24, D14⋊3Q8.10C2, D14⋊2Q8.10C2, D14⋊C4.126C22, Dic7⋊C4.78C22, C4⋊Dic7.370C22, (Q8×C14).227C22, (C2×Dic7).58C23, (C4×Dic7).87C22, (C22×D7).49C23, C22.148(C23×D7), C7⋊2(C22.35C24), C2.24(D4.10D14), C2.15(Q8.10D14), (C2×Dic14).292C22, C4⋊C4⋊D7.1C2, C14.57(C2×C4○D4), C2.66(C2×C4○D28), (C2×C4×D7).77C22, (C7×C4⋊C4).355C22, (C2×C4).290(C22×D7), SmallGroup(448,1036)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 740 in 192 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×13], C22, C22 [×3], C7, C2×C4 [×3], C2×C4 [×4], C2×C4 [×9], Q8 [×4], C23, D7, C14 [×3], C42, C42 [×2], C42 [×3], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×17], C22×C4, C2×Q8, C2×Q8, Dic7 [×7], C28 [×2], C28 [×6], D14 [×3], C2×C14, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8 [×2], C42.C2 [×5], C42⋊2C2 [×4], C4⋊Q8, Dic14 [×2], C4×D7 [×2], C2×Dic7 [×3], C2×Dic7 [×4], C2×C28 [×3], C2×C28 [×4], C7×Q8 [×2], C22×D7, C22.35C24, C4×Dic7, C4×Dic7 [×2], Dic7⋊C4 [×2], Dic7⋊C4 [×10], C4⋊Dic7, C4⋊Dic7 [×4], D14⋊C4 [×2], D14⋊C4 [×4], C4×C28, C4×C28 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×Dic14, C2×C4×D7, Q8×C14, C4×Dic14, C28.6Q8 [×2], C42⋊D7, C42⋊2D7 [×2], Dic7.Q8 [×2], C4.Dic14, D14⋊2Q8, C4⋊C4⋊D7 [×2], Dic7⋊Q8, D14⋊3Q8, Q8×C28, C42.134D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2- (1+4) [×2], C22×D7 [×7], C22.35C24, C4○D28 [×2], C23×D7, C2×C4○D28, Q8.10D14, D4.10D14, C42.134D14
Generators and relations
G = < a,b,c,d | a4=b4=1, c14=a2b2, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=b2c13 >
►On 224 points
Generators in S224
(1 161 85 70)(2 71 86 162)(3 163 87 72)(4 73 88 164)(5 165 89 74)(6 75 90 166)(7 167 91 76)(8 77 92 168)(9 141 93 78)(10 79 94 142)(11 143 95 80)(12 81 96 144)(13 145 97 82)(14 83 98 146)(15 147 99 84)(16 57 100 148)(17 149 101 58)(18 59 102 150)(19 151 103 60)(20 61 104 152)(21 153 105 62)(22 63 106 154)(23 155 107 64)(24 65 108 156)(25 157 109 66)(26 67 110 158)(27 159 111 68)(28 69 112 160)(29 171 206 122)(30 123 207 172)(31 173 208 124)(32 125 209 174)(33 175 210 126)(34 127 211 176)(35 177 212 128)(36 129 213 178)(37 179 214 130)(38 131 215 180)(39 181 216 132)(40 133 217 182)(41 183 218 134)(42 135 219 184)(43 185 220 136)(44 137 221 186)(45 187 222 138)(46 139 223 188)(47 189 224 140)(48 113 197 190)(49 191 198 114)(50 115 199 192)(51 193 200 116)(52 117 201 194)(53 195 202 118)(54 119 203 196)(55 169 204 120)(56 121 205 170)
(1 215 99 52)(2 216 100 53)(3 217 101 54)(4 218 102 55)(5 219 103 56)(6 220 104 29)(7 221 105 30)(8 222 106 31)(9 223 107 32)(10 224 108 33)(11 197 109 34)(12 198 110 35)(13 199 111 36)(14 200 112 37)(15 201 85 38)(16 202 86 39)(17 203 87 40)(18 204 88 41)(19 205 89 42)(20 206 90 43)(21 207 91 44)(22 208 92 45)(23 209 93 46)(24 210 94 47)(25 211 95 48)(26 212 96 49)(27 213 97 50)(28 214 98 51)(57 118 162 181)(58 119 163 182)(59 120 164 183)(60 121 165 184)(61 122 166 185)(62 123 167 186)(63 124 168 187)(64 125 141 188)(65 126 142 189)(66 127 143 190)(67 128 144 191)(68 129 145 192)(69 130 146 193)(70 131 147 194)(71 132 148 195)(72 133 149 196)(73 134 150 169)(74 135 151 170)(75 136 152 171)(76 137 153 172)(77 138 154 173)(78 139 155 174)(79 140 156 175)(80 113 157 176)(81 114 158 177)(82 115 159 178)(83 116 160 179)(84 117 161 180)
(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 85 98)(2 97 86 13)(3 12 87 96)(4 95 88 11)(5 10 89 94)(6 93 90 9)(7 8 91 92)(15 28 99 112)(16 111 100 27)(17 26 101 110)(18 109 102 25)(19 24 103 108)(20 107 104 23)(21 22 105 106)(29 32 206 209)(30 208 207 31)(33 56 210 205)(34 204 211 55)(35 54 212 203)(36 202 213 53)(37 52 214 201)(38 200 215 51)(39 50 216 199)(40 198 217 49)(41 48 218 197)(42 224 219 47)(43 46 220 223)(44 222 221 45)(57 159 148 68)(58 67 149 158)(59 157 150 66)(60 65 151 156)(61 155 152 64)(62 63 153 154)(69 147 160 84)(70 83 161 146)(71 145 162 82)(72 81 163 144)(73 143 164 80)(74 79 165 142)(75 141 166 78)(76 77 167 168)(113 183 190 134)(114 133 191 182)(115 181 192 132)(116 131 193 180)(117 179 194 130)(118 129 195 178)(119 177 196 128)(120 127 169 176)(121 175 170 126)(122 125 171 174)(123 173 172 124)(135 189 184 140)(136 139 185 188)(137 187 186 138)
G:=sub<Sym(224)| (1,161,85,70)(2,71,86,162)(3,163,87,72)(4,73,88,164)(5,165,89,74)(6,75,90,166)(7,167,91,76)(8,77,92,168)(9,141,93,78)(10,79,94,142)(11,143,95,80)(12,81,96,144)(13,145,97,82)(14,83,98,146)(15,147,99,84)(16,57,100,148)(17,149,101,58)(18,59,102,150)(19,151,103,60)(20,61,104,152)(21,153,105,62)(22,63,106,154)(23,155,107,64)(24,65,108,156)(25,157,109,66)(26,67,110,158)(27,159,111,68)(28,69,112,160)(29,171,206,122)(30,123,207,172)(31,173,208,124)(32,125,209,174)(33,175,210,126)(34,127,211,176)(35,177,212,128)(36,129,213,178)(37,179,214,130)(38,131,215,180)(39,181,216,132)(40,133,217,182)(41,183,218,134)(42,135,219,184)(43,185,220,136)(44,137,221,186)(45,187,222,138)(46,139,223,188)(47,189,224,140)(48,113,197,190)(49,191,198,114)(50,115,199,192)(51,193,200,116)(52,117,201,194)(53,195,202,118)(54,119,203,196)(55,169,204,120)(56,121,205,170), (1,215,99,52)(2,216,100,53)(3,217,101,54)(4,218,102,55)(5,219,103,56)(6,220,104,29)(7,221,105,30)(8,222,106,31)(9,223,107,32)(10,224,108,33)(11,197,109,34)(12,198,110,35)(13,199,111,36)(14,200,112,37)(15,201,85,38)(16,202,86,39)(17,203,87,40)(18,204,88,41)(19,205,89,42)(20,206,90,43)(21,207,91,44)(22,208,92,45)(23,209,93,46)(24,210,94,47)(25,211,95,48)(26,212,96,49)(27,213,97,50)(28,214,98,51)(57,118,162,181)(58,119,163,182)(59,120,164,183)(60,121,165,184)(61,122,166,185)(62,123,167,186)(63,124,168,187)(64,125,141,188)(65,126,142,189)(66,127,143,190)(67,128,144,191)(68,129,145,192)(69,130,146,193)(70,131,147,194)(71,132,148,195)(72,133,149,196)(73,134,150,169)(74,135,151,170)(75,136,152,171)(76,137,153,172)(77,138,154,173)(78,139,155,174)(79,140,156,175)(80,113,157,176)(81,114,158,177)(82,115,159,178)(83,116,160,179)(84,117,161,180), (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), (1,14,85,98)(2,97,86,13)(3,12,87,96)(4,95,88,11)(5,10,89,94)(6,93,90,9)(7,8,91,92)(15,28,99,112)(16,111,100,27)(17,26,101,110)(18,109,102,25)(19,24,103,108)(20,107,104,23)(21,22,105,106)(29,32,206,209)(30,208,207,31)(33,56,210,205)(34,204,211,55)(35,54,212,203)(36,202,213,53)(37,52,214,201)(38,200,215,51)(39,50,216,199)(40,198,217,49)(41,48,218,197)(42,224,219,47)(43,46,220,223)(44,222,221,45)(57,159,148,68)(58,67,149,158)(59,157,150,66)(60,65,151,156)(61,155,152,64)(62,63,153,154)(69,147,160,84)(70,83,161,146)(71,145,162,82)(72,81,163,144)(73,143,164,80)(74,79,165,142)(75,141,166,78)(76,77,167,168)(113,183,190,134)(114,133,191,182)(115,181,192,132)(116,131,193,180)(117,179,194,130)(118,129,195,178)(119,177,196,128)(120,127,169,176)(121,175,170,126)(122,125,171,174)(123,173,172,124)(135,189,184,140)(136,139,185,188)(137,187,186,138)>;
G:=Group( (1,161,85,70)(2,71,86,162)(3,163,87,72)(4,73,88,164)(5,165,89,74)(6,75,90,166)(7,167,91,76)(8,77,92,168)(9,141,93,78)(10,79,94,142)(11,143,95,80)(12,81,96,144)(13,145,97,82)(14,83,98,146)(15,147,99,84)(16,57,100,148)(17,149,101,58)(18,59,102,150)(19,151,103,60)(20,61,104,152)(21,153,105,62)(22,63,106,154)(23,155,107,64)(24,65,108,156)(25,157,109,66)(26,67,110,158)(27,159,111,68)(28,69,112,160)(29,171,206,122)(30,123,207,172)(31,173,208,124)(32,125,209,174)(33,175,210,126)(34,127,211,176)(35,177,212,128)(36,129,213,178)(37,179,214,130)(38,131,215,180)(39,181,216,132)(40,133,217,182)(41,183,218,134)(42,135,219,184)(43,185,220,136)(44,137,221,186)(45,187,222,138)(46,139,223,188)(47,189,224,140)(48,113,197,190)(49,191,198,114)(50,115,199,192)(51,193,200,116)(52,117,201,194)(53,195,202,118)(54,119,203,196)(55,169,204,120)(56,121,205,170), (1,215,99,52)(2,216,100,53)(3,217,101,54)(4,218,102,55)(5,219,103,56)(6,220,104,29)(7,221,105,30)(8,222,106,31)(9,223,107,32)(10,224,108,33)(11,197,109,34)(12,198,110,35)(13,199,111,36)(14,200,112,37)(15,201,85,38)(16,202,86,39)(17,203,87,40)(18,204,88,41)(19,205,89,42)(20,206,90,43)(21,207,91,44)(22,208,92,45)(23,209,93,46)(24,210,94,47)(25,211,95,48)(26,212,96,49)(27,213,97,50)(28,214,98,51)(57,118,162,181)(58,119,163,182)(59,120,164,183)(60,121,165,184)(61,122,166,185)(62,123,167,186)(63,124,168,187)(64,125,141,188)(65,126,142,189)(66,127,143,190)(67,128,144,191)(68,129,145,192)(69,130,146,193)(70,131,147,194)(71,132,148,195)(72,133,149,196)(73,134,150,169)(74,135,151,170)(75,136,152,171)(76,137,153,172)(77,138,154,173)(78,139,155,174)(79,140,156,175)(80,113,157,176)(81,114,158,177)(82,115,159,178)(83,116,160,179)(84,117,161,180), (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), (1,14,85,98)(2,97,86,13)(3,12,87,96)(4,95,88,11)(5,10,89,94)(6,93,90,9)(7,8,91,92)(15,28,99,112)(16,111,100,27)(17,26,101,110)(18,109,102,25)(19,24,103,108)(20,107,104,23)(21,22,105,106)(29,32,206,209)(30,208,207,31)(33,56,210,205)(34,204,211,55)(35,54,212,203)(36,202,213,53)(37,52,214,201)(38,200,215,51)(39,50,216,199)(40,198,217,49)(41,48,218,197)(42,224,219,47)(43,46,220,223)(44,222,221,45)(57,159,148,68)(58,67,149,158)(59,157,150,66)(60,65,151,156)(61,155,152,64)(62,63,153,154)(69,147,160,84)(70,83,161,146)(71,145,162,82)(72,81,163,144)(73,143,164,80)(74,79,165,142)(75,141,166,78)(76,77,167,168)(113,183,190,134)(114,133,191,182)(115,181,192,132)(116,131,193,180)(117,179,194,130)(118,129,195,178)(119,177,196,128)(120,127,169,176)(121,175,170,126)(122,125,171,174)(123,173,172,124)(135,189,184,140)(136,139,185,188)(137,187,186,138) );
G=PermutationGroup([(1,161,85,70),(2,71,86,162),(3,163,87,72),(4,73,88,164),(5,165,89,74),(6,75,90,166),(7,167,91,76),(8,77,92,168),(9,141,93,78),(10,79,94,142),(11,143,95,80),(12,81,96,144),(13,145,97,82),(14,83,98,146),(15,147,99,84),(16,57,100,148),(17,149,101,58),(18,59,102,150),(19,151,103,60),(20,61,104,152),(21,153,105,62),(22,63,106,154),(23,155,107,64),(24,65,108,156),(25,157,109,66),(26,67,110,158),(27,159,111,68),(28,69,112,160),(29,171,206,122),(30,123,207,172),(31,173,208,124),(32,125,209,174),(33,175,210,126),(34,127,211,176),(35,177,212,128),(36,129,213,178),(37,179,214,130),(38,131,215,180),(39,181,216,132),(40,133,217,182),(41,183,218,134),(42,135,219,184),(43,185,220,136),(44,137,221,186),(45,187,222,138),(46,139,223,188),(47,189,224,140),(48,113,197,190),(49,191,198,114),(50,115,199,192),(51,193,200,116),(52,117,201,194),(53,195,202,118),(54,119,203,196),(55,169,204,120),(56,121,205,170)], [(1,215,99,52),(2,216,100,53),(3,217,101,54),(4,218,102,55),(5,219,103,56),(6,220,104,29),(7,221,105,30),(8,222,106,31),(9,223,107,32),(10,224,108,33),(11,197,109,34),(12,198,110,35),(13,199,111,36),(14,200,112,37),(15,201,85,38),(16,202,86,39),(17,203,87,40),(18,204,88,41),(19,205,89,42),(20,206,90,43),(21,207,91,44),(22,208,92,45),(23,209,93,46),(24,210,94,47),(25,211,95,48),(26,212,96,49),(27,213,97,50),(28,214,98,51),(57,118,162,181),(58,119,163,182),(59,120,164,183),(60,121,165,184),(61,122,166,185),(62,123,167,186),(63,124,168,187),(64,125,141,188),(65,126,142,189),(66,127,143,190),(67,128,144,191),(68,129,145,192),(69,130,146,193),(70,131,147,194),(71,132,148,195),(72,133,149,196),(73,134,150,169),(74,135,151,170),(75,136,152,171),(76,137,153,172),(77,138,154,173),(78,139,155,174),(79,140,156,175),(80,113,157,176),(81,114,158,177),(82,115,159,178),(83,116,160,179),(84,117,161,180)], [(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)], [(1,14,85,98),(2,97,86,13),(3,12,87,96),(4,95,88,11),(5,10,89,94),(6,93,90,9),(7,8,91,92),(15,28,99,112),(16,111,100,27),(17,26,101,110),(18,109,102,25),(19,24,103,108),(20,107,104,23),(21,22,105,106),(29,32,206,209),(30,208,207,31),(33,56,210,205),(34,204,211,55),(35,54,212,203),(36,202,213,53),(37,52,214,201),(38,200,215,51),(39,50,216,199),(40,198,217,49),(41,48,218,197),(42,224,219,47),(43,46,220,223),(44,222,221,45),(57,159,148,68),(58,67,149,158),(59,157,150,66),(60,65,151,156),(61,155,152,64),(62,63,153,154),(69,147,160,84),(70,83,161,146),(71,145,162,82),(72,81,163,144),(73,143,164,80),(74,79,165,142),(75,141,166,78),(76,77,167,168),(113,183,190,134),(114,133,191,182),(115,181,192,132),(116,131,193,180),(117,179,194,130),(118,129,195,178),(119,177,196,128),(120,127,169,176),(121,175,170,126),(122,125,171,174),(123,173,172,124),(135,189,184,140),(136,139,185,188),(137,187,186,138)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 17 | 26 | 5 |
| 0 | 0 | 12 | 0 | 24 | 3 |
| 0 | 0 | 3 | 12 | 3 | 12 |
| 0 | 0 | 17 | 0 | 17 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 6 | 0 | 0 |
| 0 | 0 | 23 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 6 |
| 0 | 0 | 0 | 0 | 23 | 21 |
| 16 | 20 | 0 | 0 | 0 | 0 |
| 28 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 19 | 9 | 9 |
| 0 | 0 | 10 | 7 | 20 | 14 |
| 0 | 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 0 | 19 | 22 |
| 3 | 8 | 0 | 0 | 0 | 0 |
| 28 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 20 | 20 |
| 0 | 0 | 22 | 19 | 15 | 9 |
| 0 | 0 | 19 | 19 | 19 | 19 |
| 0 | 0 | 7 | 10 | 7 | 10 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,26,12,3,17,0,0,17,0,12,0,0,0,26,24,3,17,0,0,5,3,12,0],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,8,23,0,0,0,0,6,21,0,0,0,0,0,0,8,23,0,0,0,0,6,21],[16,28,0,0,0,0,20,26,0,0,0,0,0,0,19,10,0,0,0,0,19,7,0,0,0,0,9,20,10,19,0,0,9,14,10,22],[3,28,0,0,0,0,8,26,0,0,0,0,0,0,10,22,19,7,0,0,10,19,19,10,0,0,20,15,19,7,0,0,20,9,19,10] >;
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4Q | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 28 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | C4○D28 | 2- (1+4) | Q8.10D14 | D4.10D14 |
| kernel | C42.134D14 | C4×Dic14 | C28.6Q8 | C42⋊D7 | C42⋊2D7 | Dic7.Q8 | C4.Dic14 | D14⋊2Q8 | C4⋊C4⋊D7 | Dic7⋊Q8 | D14⋊3Q8 | Q8×C28 | C4×Q8 | C28 | C42 | C4⋊C4 | C2×Q8 | C4 | C14 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 3 | 4 | 9 | 9 | 3 | 24 | 2 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_4^2._{134}D_{14} % in TeX
G:=Group("C4^2.134D14"); // GroupNames label
G:=SmallGroup(448,1036);
// by ID
G=gap.SmallGroup(448,1036);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,758,219,268,675,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=a^2*b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^13>;
// generators/relations
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