metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.238D14, (C4×D7)⋊8D4, C4.35(D4×D7), C28⋊4(C4○D4), C4⋊1D4⋊10D7, D14.7(C2×D4), C28.66(C2×D4), C4⋊2(D4⋊2D7), C28⋊2D4⋊35C2, C28⋊2Q8⋊34C2, (D4×Dic7)⋊34C2, (D7×C42)⋊13C2, (C2×D4).178D14, Dic7.66(C2×D4), C14.94(C22×D4), C28.17D4⋊27C2, (C4×C28).203C22, (C2×C14).260C24, (C2×C28).508C23, C23.66(C22×D7), (D4×C14).161C22, C4⋊Dic7.248C22, (C22×C14).74C23, C7⋊5(C22.26C24), C22.281(C23×D7), C23.D7.72C22, (C4×Dic7).256C22, (C2×Dic7).135C23, (C22×D7).228C23, (C2×Dic14).185C22, (C22×Dic7).157C22, C2.67(C2×D4×D7), (C7×C4⋊1D4)⋊7C2, C14.96(C2×C4○D4), (C2×D4⋊2D7)⋊21C2, C2.60(C2×D4⋊2D7), (C2×C4×D7).251C22, (C2×C4).597(C22×D7), (C2×C7⋊D4).77C22, SmallGroup(448,1169)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1324 in 310 conjugacy classes, 111 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×6], C4 [×8], C22, C22 [×16], C7, C2×C4, C2×C4 [×2], C2×C4 [×23], D4 [×20], Q8 [×4], C23 [×4], C23, D7 [×2], C14, C14 [×2], C14 [×4], C42, C42 [×3], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×7], C2×D4 [×6], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×8], Dic7 [×2], Dic7 [×6], C28 [×6], D14 [×2], D14 [×2], C2×C14, C2×C14 [×12], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C2×C4○D4 [×2], Dic14 [×4], C4×D7 [×4], C4×D7 [×4], C2×Dic7, C2×Dic7 [×6], C2×Dic7 [×8], C7⋊D4 [×8], C2×C28, C2×C28 [×2], C7×D4 [×12], C22×D7, C22×C14 [×4], C22.26C24, C4×Dic7, C4×Dic7 [×2], C4⋊Dic7 [×4], C23.D7 [×8], C4×C28, C2×Dic14 [×2], C2×C4×D7, C2×C4×D7 [×2], D4⋊2D7 [×8], C22×Dic7 [×4], C2×C7⋊D4 [×4], D4×C14 [×6], C28⋊2Q8, D7×C42, D4×Dic7 [×4], C28.17D4 [×2], C28⋊2D4 [×4], C7×C4⋊1D4, C2×D4⋊2D7 [×2], C42.238D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×4], C24, D14 [×7], C22×D4, C2×C4○D4 [×2], C22×D7 [×7], C22.26C24, D4×D7 [×2], D4⋊2D7 [×4], C23×D7, C2×D4×D7, C2×D4⋊2D7 [×2], C42.238D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b2c-1 >
►On 224 points
Generators in S224
(1 183 221 75)(2 76 222 184)(3 185 223 77)(4 78 224 186)(5 187 211 79)(6 80 212 188)(7 189 213 81)(8 82 214 190)(9 191 215 83)(10 84 216 192)(11 193 217 71)(12 72 218 194)(13 195 219 73)(14 74 220 196)(15 130 97 153)(16 154 98 131)(17 132 85 141)(18 142 86 133)(19 134 87 143)(20 144 88 135)(21 136 89 145)(22 146 90 137)(23 138 91 147)(24 148 92 139)(25 140 93 149)(26 150 94 127)(27 128 95 151)(28 152 96 129)(29 59 210 100)(30 101 197 60)(31 61 198 102)(32 103 199 62)(33 63 200 104)(34 105 201 64)(35 65 202 106)(36 107 203 66)(37 67 204 108)(38 109 205 68)(39 69 206 110)(40 111 207 70)(41 57 208 112)(42 99 209 58)(43 173 116 164)(44 165 117 174)(45 175 118 166)(46 167 119 176)(47 177 120 168)(48 155 121 178)(49 179 122 156)(50 157 123 180)(51 181 124 158)(52 159 125 182)(53 169 126 160)(54 161 113 170)(55 171 114 162)(56 163 115 172)
(1 105 16 54)(2 55 17 106)(3 107 18 56)(4 43 19 108)(5 109 20 44)(6 45 21 110)(7 111 22 46)(8 47 23 112)(9 99 24 48)(10 49 25 100)(11 101 26 50)(12 51 27 102)(13 103 28 52)(14 53 15 104)(29 84 179 140)(30 127 180 71)(31 72 181 128)(32 129 182 73)(33 74 169 130)(34 131 170 75)(35 76 171 132)(36 133 172 77)(37 78 173 134)(38 135 174 79)(39 80 175 136)(40 137 176 81)(41 82 177 138)(42 139 178 83)(57 214 120 91)(58 92 121 215)(59 216 122 93)(60 94 123 217)(61 218 124 95)(62 96 125 219)(63 220 126 97)(64 98 113 221)(65 222 114 85)(66 86 115 223)(67 224 116 87)(68 88 117 211)(69 212 118 89)(70 90 119 213)(141 202 184 162)(142 163 185 203)(143 204 186 164)(144 165 187 205)(145 206 188 166)(146 167 189 207)(147 208 190 168)(148 155 191 209)(149 210 192 156)(150 157 193 197)(151 198 194 158)(152 159 195 199)(153 200 196 160)(154 161 183 201)
(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)
(1 220 16 97)(2 96 17 219)(3 218 18 95)(4 94 19 217)(5 216 20 93)(6 92 21 215)(7 214 22 91)(8 90 23 213)(9 212 24 89)(10 88 25 211)(11 224 26 87)(12 86 27 223)(13 222 28 85)(14 98 15 221)(29 165 179 205)(30 204 180 164)(31 163 181 203)(32 202 182 162)(33 161 169 201)(34 200 170 160)(35 159 171 199)(36 198 172 158)(37 157 173 197)(38 210 174 156)(39 155 175 209)(40 208 176 168)(41 167 177 207)(42 206 178 166)(43 60 108 123)(44 122 109 59)(45 58 110 121)(46 120 111 57)(47 70 112 119)(48 118 99 69)(49 68 100 117)(50 116 101 67)(51 66 102 115)(52 114 103 65)(53 64 104 113)(54 126 105 63)(55 62 106 125)(56 124 107 61)(71 186 127 143)(72 142 128 185)(73 184 129 141)(74 154 130 183)(75 196 131 153)(76 152 132 195)(77 194 133 151)(78 150 134 193)(79 192 135 149)(80 148 136 191)(81 190 137 147)(82 146 138 189)(83 188 139 145)(84 144 140 187)
G:=sub<Sym(224)| (1,183,221,75)(2,76,222,184)(3,185,223,77)(4,78,224,186)(5,187,211,79)(6,80,212,188)(7,189,213,81)(8,82,214,190)(9,191,215,83)(10,84,216,192)(11,193,217,71)(12,72,218,194)(13,195,219,73)(14,74,220,196)(15,130,97,153)(16,154,98,131)(17,132,85,141)(18,142,86,133)(19,134,87,143)(20,144,88,135)(21,136,89,145)(22,146,90,137)(23,138,91,147)(24,148,92,139)(25,140,93,149)(26,150,94,127)(27,128,95,151)(28,152,96,129)(29,59,210,100)(30,101,197,60)(31,61,198,102)(32,103,199,62)(33,63,200,104)(34,105,201,64)(35,65,202,106)(36,107,203,66)(37,67,204,108)(38,109,205,68)(39,69,206,110)(40,111,207,70)(41,57,208,112)(42,99,209,58)(43,173,116,164)(44,165,117,174)(45,175,118,166)(46,167,119,176)(47,177,120,168)(48,155,121,178)(49,179,122,156)(50,157,123,180)(51,181,124,158)(52,159,125,182)(53,169,126,160)(54,161,113,170)(55,171,114,162)(56,163,115,172), (1,105,16,54)(2,55,17,106)(3,107,18,56)(4,43,19,108)(5,109,20,44)(6,45,21,110)(7,111,22,46)(8,47,23,112)(9,99,24,48)(10,49,25,100)(11,101,26,50)(12,51,27,102)(13,103,28,52)(14,53,15,104)(29,84,179,140)(30,127,180,71)(31,72,181,128)(32,129,182,73)(33,74,169,130)(34,131,170,75)(35,76,171,132)(36,133,172,77)(37,78,173,134)(38,135,174,79)(39,80,175,136)(40,137,176,81)(41,82,177,138)(42,139,178,83)(57,214,120,91)(58,92,121,215)(59,216,122,93)(60,94,123,217)(61,218,124,95)(62,96,125,219)(63,220,126,97)(64,98,113,221)(65,222,114,85)(66,86,115,223)(67,224,116,87)(68,88,117,211)(69,212,118,89)(70,90,119,213)(141,202,184,162)(142,163,185,203)(143,204,186,164)(144,165,187,205)(145,206,188,166)(146,167,189,207)(147,208,190,168)(148,155,191,209)(149,210,192,156)(150,157,193,197)(151,198,194,158)(152,159,195,199)(153,200,196,160)(154,161,183,201), (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), (1,220,16,97)(2,96,17,219)(3,218,18,95)(4,94,19,217)(5,216,20,93)(6,92,21,215)(7,214,22,91)(8,90,23,213)(9,212,24,89)(10,88,25,211)(11,224,26,87)(12,86,27,223)(13,222,28,85)(14,98,15,221)(29,165,179,205)(30,204,180,164)(31,163,181,203)(32,202,182,162)(33,161,169,201)(34,200,170,160)(35,159,171,199)(36,198,172,158)(37,157,173,197)(38,210,174,156)(39,155,175,209)(40,208,176,168)(41,167,177,207)(42,206,178,166)(43,60,108,123)(44,122,109,59)(45,58,110,121)(46,120,111,57)(47,70,112,119)(48,118,99,69)(49,68,100,117)(50,116,101,67)(51,66,102,115)(52,114,103,65)(53,64,104,113)(54,126,105,63)(55,62,106,125)(56,124,107,61)(71,186,127,143)(72,142,128,185)(73,184,129,141)(74,154,130,183)(75,196,131,153)(76,152,132,195)(77,194,133,151)(78,150,134,193)(79,192,135,149)(80,148,136,191)(81,190,137,147)(82,146,138,189)(83,188,139,145)(84,144,140,187)>;
G:=Group( (1,183,221,75)(2,76,222,184)(3,185,223,77)(4,78,224,186)(5,187,211,79)(6,80,212,188)(7,189,213,81)(8,82,214,190)(9,191,215,83)(10,84,216,192)(11,193,217,71)(12,72,218,194)(13,195,219,73)(14,74,220,196)(15,130,97,153)(16,154,98,131)(17,132,85,141)(18,142,86,133)(19,134,87,143)(20,144,88,135)(21,136,89,145)(22,146,90,137)(23,138,91,147)(24,148,92,139)(25,140,93,149)(26,150,94,127)(27,128,95,151)(28,152,96,129)(29,59,210,100)(30,101,197,60)(31,61,198,102)(32,103,199,62)(33,63,200,104)(34,105,201,64)(35,65,202,106)(36,107,203,66)(37,67,204,108)(38,109,205,68)(39,69,206,110)(40,111,207,70)(41,57,208,112)(42,99,209,58)(43,173,116,164)(44,165,117,174)(45,175,118,166)(46,167,119,176)(47,177,120,168)(48,155,121,178)(49,179,122,156)(50,157,123,180)(51,181,124,158)(52,159,125,182)(53,169,126,160)(54,161,113,170)(55,171,114,162)(56,163,115,172), (1,105,16,54)(2,55,17,106)(3,107,18,56)(4,43,19,108)(5,109,20,44)(6,45,21,110)(7,111,22,46)(8,47,23,112)(9,99,24,48)(10,49,25,100)(11,101,26,50)(12,51,27,102)(13,103,28,52)(14,53,15,104)(29,84,179,140)(30,127,180,71)(31,72,181,128)(32,129,182,73)(33,74,169,130)(34,131,170,75)(35,76,171,132)(36,133,172,77)(37,78,173,134)(38,135,174,79)(39,80,175,136)(40,137,176,81)(41,82,177,138)(42,139,178,83)(57,214,120,91)(58,92,121,215)(59,216,122,93)(60,94,123,217)(61,218,124,95)(62,96,125,219)(63,220,126,97)(64,98,113,221)(65,222,114,85)(66,86,115,223)(67,224,116,87)(68,88,117,211)(69,212,118,89)(70,90,119,213)(141,202,184,162)(142,163,185,203)(143,204,186,164)(144,165,187,205)(145,206,188,166)(146,167,189,207)(147,208,190,168)(148,155,191,209)(149,210,192,156)(150,157,193,197)(151,198,194,158)(152,159,195,199)(153,200,196,160)(154,161,183,201), (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), (1,220,16,97)(2,96,17,219)(3,218,18,95)(4,94,19,217)(5,216,20,93)(6,92,21,215)(7,214,22,91)(8,90,23,213)(9,212,24,89)(10,88,25,211)(11,224,26,87)(12,86,27,223)(13,222,28,85)(14,98,15,221)(29,165,179,205)(30,204,180,164)(31,163,181,203)(32,202,182,162)(33,161,169,201)(34,200,170,160)(35,159,171,199)(36,198,172,158)(37,157,173,197)(38,210,174,156)(39,155,175,209)(40,208,176,168)(41,167,177,207)(42,206,178,166)(43,60,108,123)(44,122,109,59)(45,58,110,121)(46,120,111,57)(47,70,112,119)(48,118,99,69)(49,68,100,117)(50,116,101,67)(51,66,102,115)(52,114,103,65)(53,64,104,113)(54,126,105,63)(55,62,106,125)(56,124,107,61)(71,186,127,143)(72,142,128,185)(73,184,129,141)(74,154,130,183)(75,196,131,153)(76,152,132,195)(77,194,133,151)(78,150,134,193)(79,192,135,149)(80,148,136,191)(81,190,137,147)(82,146,138,189)(83,188,139,145)(84,144,140,187) );
G=PermutationGroup([(1,183,221,75),(2,76,222,184),(3,185,223,77),(4,78,224,186),(5,187,211,79),(6,80,212,188),(7,189,213,81),(8,82,214,190),(9,191,215,83),(10,84,216,192),(11,193,217,71),(12,72,218,194),(13,195,219,73),(14,74,220,196),(15,130,97,153),(16,154,98,131),(17,132,85,141),(18,142,86,133),(19,134,87,143),(20,144,88,135),(21,136,89,145),(22,146,90,137),(23,138,91,147),(24,148,92,139),(25,140,93,149),(26,150,94,127),(27,128,95,151),(28,152,96,129),(29,59,210,100),(30,101,197,60),(31,61,198,102),(32,103,199,62),(33,63,200,104),(34,105,201,64),(35,65,202,106),(36,107,203,66),(37,67,204,108),(38,109,205,68),(39,69,206,110),(40,111,207,70),(41,57,208,112),(42,99,209,58),(43,173,116,164),(44,165,117,174),(45,175,118,166),(46,167,119,176),(47,177,120,168),(48,155,121,178),(49,179,122,156),(50,157,123,180),(51,181,124,158),(52,159,125,182),(53,169,126,160),(54,161,113,170),(55,171,114,162),(56,163,115,172)], [(1,105,16,54),(2,55,17,106),(3,107,18,56),(4,43,19,108),(5,109,20,44),(6,45,21,110),(7,111,22,46),(8,47,23,112),(9,99,24,48),(10,49,25,100),(11,101,26,50),(12,51,27,102),(13,103,28,52),(14,53,15,104),(29,84,179,140),(30,127,180,71),(31,72,181,128),(32,129,182,73),(33,74,169,130),(34,131,170,75),(35,76,171,132),(36,133,172,77),(37,78,173,134),(38,135,174,79),(39,80,175,136),(40,137,176,81),(41,82,177,138),(42,139,178,83),(57,214,120,91),(58,92,121,215),(59,216,122,93),(60,94,123,217),(61,218,124,95),(62,96,125,219),(63,220,126,97),(64,98,113,221),(65,222,114,85),(66,86,115,223),(67,224,116,87),(68,88,117,211),(69,212,118,89),(70,90,119,213),(141,202,184,162),(142,163,185,203),(143,204,186,164),(144,165,187,205),(145,206,188,166),(146,167,189,207),(147,208,190,168),(148,155,191,209),(149,210,192,156),(150,157,193,197),(151,198,194,158),(152,159,195,199),(153,200,196,160),(154,161,183,201)], [(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)], [(1,220,16,97),(2,96,17,219),(3,218,18,95),(4,94,19,217),(5,216,20,93),(6,92,21,215),(7,214,22,91),(8,90,23,213),(9,212,24,89),(10,88,25,211),(11,224,26,87),(12,86,27,223),(13,222,28,85),(14,98,15,221),(29,165,179,205),(30,204,180,164),(31,163,181,203),(32,202,182,162),(33,161,169,201),(34,200,170,160),(35,159,171,199),(36,198,172,158),(37,157,173,197),(38,210,174,156),(39,155,175,209),(40,208,176,168),(41,167,177,207),(42,206,178,166),(43,60,108,123),(44,122,109,59),(45,58,110,121),(46,120,111,57),(47,70,112,119),(48,118,99,69),(49,68,100,117),(50,116,101,67),(51,66,102,115),(52,114,103,65),(53,64,104,113),(54,126,105,63),(55,62,106,125),(56,124,107,61),(71,186,127,143),(72,142,128,185),(73,184,129,141),(74,154,130,183),(75,196,131,153),(76,152,132,195),(77,194,133,151),(78,150,134,193),(79,192,135,149),(80,148,136,191),(81,190,137,147),(82,146,138,189),(83,188,139,145),(84,144,140,187)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 0 | 21 | 0 | 0 | 0 | 0 |
| 11 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 11 | 21 | 0 | 0 | 0 | 0 |
| 15 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,17,0,0,0,0,0,0,12,0,0,0,0,0,0,17],[0,11,0,0,0,0,21,18,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[11,15,0,0,0,0,21,18,0,0,0,0,0,0,0,1,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,28,0] >;
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 14 | 14 | 2 | ··· | 2 | 7 | 7 | 7 | 7 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D4×D7 | D4⋊2D7 |
| kernel | C42.238D14 | C28⋊2Q8 | D7×C42 | D4×Dic7 | C28.17D4 | C28⋊2D4 | C7×C4⋊1D4 | C2×D4⋊2D7 | C4×D7 | C4⋊1D4 | C28 | C42 | C2×D4 | C4 | C4 |
| # reps | 1 | 1 | 1 | 4 | 2 | 4 | 1 | 2 | 4 | 3 | 8 | 3 | 18 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_4^2._{238}D_{14} % in TeX
G:=Group("C4^2.238D14"); // GroupNames label
G:=SmallGroup(448,1169);
// by ID
G=gap.SmallGroup(448,1169);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,100,675,570,185,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations
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