metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.138D14, C14.712+ (1+4), C4.4D4⋊7D7, (C2×Q8).82D14, D14⋊D4⋊38C2, (C2×D4).108D14, C42⋊D7⋊35C2, C22⋊C4.72D14, Dic7⋊Q8⋊21C2, Dic7⋊4D4⋊29C2, Dic7⋊D4⋊33C2, C28.23D4⋊19C2, (C2×C14).214C24, (C2×C28).630C23, (C4×C28).239C22, C2.73(D4⋊6D14), C23.36(C22×D7), D14⋊C4.134C22, Dic7.27(C4○D4), Dic7.D4⋊38C2, (D4×C14).208C22, (C2×D28).161C22, (C22×C14).44C23, (Q8×C14).123C22, (C22×D7).94C23, C22.235(C23×D7), C23.D7.51C22, C23.11D14⋊17C2, Dic7⋊C4.141C22, C7⋊4(C22.49C24), (C2×Dic7).251C23, (C4×Dic7).130C22, (C2×Dic14).175C22, (C22×Dic7).139C22, C2.73(D7×C4○D4), (C7×C4.4D4)⋊8C2, C14.185(C2×C4○D4), (C2×C4×D7).214C22, (C2×C4).73(C22×D7), (C2×C7⋊D4).57C22, (C7×C22⋊C4).61C22, SmallGroup(448,1123)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1100 in 236 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×13], C22, C22 [×12], C7, C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×8], Q8 [×2], C23 [×2], C23 [×2], D7 [×2], C14, C14 [×2], C14 [×2], C42, C42 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8, Dic7 [×4], Dic7 [×4], C28 [×5], D14 [×6], C2×C14, C2×C14 [×6], C42⋊C2 [×4], C4×D4 [×2], C4⋊D4 [×4], C4.4D4, C4.4D4 [×3], C4⋊Q8, Dic14, C4×D7 [×4], D28, C2×Dic7 [×6], C2×Dic7 [×4], C7⋊D4 [×6], C2×C28 [×3], C2×C28 [×2], C7×D4, C7×Q8, C22×D7 [×2], C22×C14 [×2], C22.49C24, C4×Dic7 [×2], C4×Dic7 [×2], Dic7⋊C4 [×6], D14⋊C4 [×6], C23.D7 [×2], C4×C28, C7×C22⋊C4 [×4], C2×Dic14, C2×C4×D7 [×2], C2×D28, C22×Dic7 [×2], C2×C7⋊D4 [×4], D4×C14, Q8×C14, C42⋊D7 [×2], C23.11D14 [×2], Dic7⋊4D4 [×2], D14⋊D4 [×2], Dic7.D4 [×2], Dic7⋊D4 [×2], Dic7⋊Q8, C28.23D4, C7×C4.4D4, C42.138D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×4], C24, D14 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D7 [×7], C22.49C24, C23×D7, D4⋊6D14, D7×C4○D4 [×2], C42.138D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=1, d2=b2, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd-1=a2b, dcd-1=c-1 >
►On 224 points
Generators in S224
(1 166 41 217)(2 55 42 91)(3 168 29 219)(4 43 30 93)(5 156 31 221)(6 45 32 95)(7 158 33 223)(8 47 34 97)(9 160 35 211)(10 49 36 85)(11 162 37 213)(12 51 38 87)(13 164 39 215)(14 53 40 89)(15 131 193 198)(16 119 194 103)(17 133 195 200)(18 121 196 105)(19 135 183 202)(20 123 184 107)(21 137 185 204)(22 125 186 109)(23 139 187 206)(24 113 188 111)(25 127 189 208)(26 115 190 99)(27 129 191 210)(28 117 192 101)(44 151 94 176)(46 153 96 178)(48 141 98 180)(50 143 86 182)(52 145 88 170)(54 147 90 172)(56 149 92 174)(57 203 74 136)(58 108 75 124)(59 205 76 138)(60 110 77 126)(61 207 78 140)(62 112 79 114)(63 209 80 128)(64 100 81 116)(65 197 82 130)(66 102 83 118)(67 199 84 132)(68 104 71 120)(69 201 72 134)(70 106 73 122)(142 212 181 161)(144 214 169 163)(146 216 171 165)(148 218 173 167)(150 220 175 155)(152 222 177 157)(154 224 179 159)
(1 74 172 20)(2 58 173 185)(3 76 174 22)(4 60 175 187)(5 78 176 24)(6 62 177 189)(7 80 178 26)(8 64 179 191)(9 82 180 28)(10 66 181 193)(11 84 182 16)(12 68 169 195)(13 72 170 18)(14 70 171 183)(15 36 83 142)(17 38 71 144)(19 40 73 146)(21 42 75 148)(23 30 77 150)(25 32 79 152)(27 34 81 154)(29 59 149 186)(31 61 151 188)(33 63 153 190)(35 65 141 192)(37 67 143 194)(39 69 145 196)(41 57 147 184)(43 110 155 206)(44 113 156 140)(45 112 157 208)(46 115 158 128)(47 100 159 210)(48 117 160 130)(49 102 161 198)(50 119 162 132)(51 104 163 200)(52 121 164 134)(53 106 165 202)(54 123 166 136)(55 108 167 204)(56 125 168 138)(85 118 212 131)(86 103 213 199)(87 120 214 133)(88 105 215 201)(89 122 216 135)(90 107 217 203)(91 124 218 137)(92 109 219 205)(93 126 220 139)(94 111 221 207)(95 114 222 127)(96 99 223 209)(97 116 224 129)(98 101 211 197)
(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 99 172 209)(2 112 173 208)(3 111 174 207)(4 110 175 206)(5 109 176 205)(6 108 177 204)(7 107 178 203)(8 106 179 202)(9 105 180 201)(10 104 181 200)(11 103 182 199)(12 102 169 198)(13 101 170 197)(14 100 171 210)(15 51 83 163)(16 50 84 162)(17 49 71 161)(18 48 72 160)(19 47 73 159)(20 46 74 158)(21 45 75 157)(22 44 76 156)(23 43 77 155)(24 56 78 168)(25 55 79 167)(26 54 80 166)(27 53 81 165)(28 52 82 164)(29 113 149 140)(30 126 150 139)(31 125 151 138)(32 124 152 137)(33 123 153 136)(34 122 154 135)(35 121 141 134)(36 120 142 133)(37 119 143 132)(38 118 144 131)(39 117 145 130)(40 116 146 129)(41 115 147 128)(42 114 148 127)(57 223 184 96)(58 222 185 95)(59 221 186 94)(60 220 187 93)(61 219 188 92)(62 218 189 91)(63 217 190 90)(64 216 191 89)(65 215 192 88)(66 214 193 87)(67 213 194 86)(68 212 195 85)(69 211 196 98)(70 224 183 97)
G:=sub<Sym(224)| (1,166,41,217)(2,55,42,91)(3,168,29,219)(4,43,30,93)(5,156,31,221)(6,45,32,95)(7,158,33,223)(8,47,34,97)(9,160,35,211)(10,49,36,85)(11,162,37,213)(12,51,38,87)(13,164,39,215)(14,53,40,89)(15,131,193,198)(16,119,194,103)(17,133,195,200)(18,121,196,105)(19,135,183,202)(20,123,184,107)(21,137,185,204)(22,125,186,109)(23,139,187,206)(24,113,188,111)(25,127,189,208)(26,115,190,99)(27,129,191,210)(28,117,192,101)(44,151,94,176)(46,153,96,178)(48,141,98,180)(50,143,86,182)(52,145,88,170)(54,147,90,172)(56,149,92,174)(57,203,74,136)(58,108,75,124)(59,205,76,138)(60,110,77,126)(61,207,78,140)(62,112,79,114)(63,209,80,128)(64,100,81,116)(65,197,82,130)(66,102,83,118)(67,199,84,132)(68,104,71,120)(69,201,72,134)(70,106,73,122)(142,212,181,161)(144,214,169,163)(146,216,171,165)(148,218,173,167)(150,220,175,155)(152,222,177,157)(154,224,179,159), (1,74,172,20)(2,58,173,185)(3,76,174,22)(4,60,175,187)(5,78,176,24)(6,62,177,189)(7,80,178,26)(8,64,179,191)(9,82,180,28)(10,66,181,193)(11,84,182,16)(12,68,169,195)(13,72,170,18)(14,70,171,183)(15,36,83,142)(17,38,71,144)(19,40,73,146)(21,42,75,148)(23,30,77,150)(25,32,79,152)(27,34,81,154)(29,59,149,186)(31,61,151,188)(33,63,153,190)(35,65,141,192)(37,67,143,194)(39,69,145,196)(41,57,147,184)(43,110,155,206)(44,113,156,140)(45,112,157,208)(46,115,158,128)(47,100,159,210)(48,117,160,130)(49,102,161,198)(50,119,162,132)(51,104,163,200)(52,121,164,134)(53,106,165,202)(54,123,166,136)(55,108,167,204)(56,125,168,138)(85,118,212,131)(86,103,213,199)(87,120,214,133)(88,105,215,201)(89,122,216,135)(90,107,217,203)(91,124,218,137)(92,109,219,205)(93,126,220,139)(94,111,221,207)(95,114,222,127)(96,99,223,209)(97,116,224,129)(98,101,211,197), (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), (1,99,172,209)(2,112,173,208)(3,111,174,207)(4,110,175,206)(5,109,176,205)(6,108,177,204)(7,107,178,203)(8,106,179,202)(9,105,180,201)(10,104,181,200)(11,103,182,199)(12,102,169,198)(13,101,170,197)(14,100,171,210)(15,51,83,163)(16,50,84,162)(17,49,71,161)(18,48,72,160)(19,47,73,159)(20,46,74,158)(21,45,75,157)(22,44,76,156)(23,43,77,155)(24,56,78,168)(25,55,79,167)(26,54,80,166)(27,53,81,165)(28,52,82,164)(29,113,149,140)(30,126,150,139)(31,125,151,138)(32,124,152,137)(33,123,153,136)(34,122,154,135)(35,121,141,134)(36,120,142,133)(37,119,143,132)(38,118,144,131)(39,117,145,130)(40,116,146,129)(41,115,147,128)(42,114,148,127)(57,223,184,96)(58,222,185,95)(59,221,186,94)(60,220,187,93)(61,219,188,92)(62,218,189,91)(63,217,190,90)(64,216,191,89)(65,215,192,88)(66,214,193,87)(67,213,194,86)(68,212,195,85)(69,211,196,98)(70,224,183,97)>;
G:=Group( (1,166,41,217)(2,55,42,91)(3,168,29,219)(4,43,30,93)(5,156,31,221)(6,45,32,95)(7,158,33,223)(8,47,34,97)(9,160,35,211)(10,49,36,85)(11,162,37,213)(12,51,38,87)(13,164,39,215)(14,53,40,89)(15,131,193,198)(16,119,194,103)(17,133,195,200)(18,121,196,105)(19,135,183,202)(20,123,184,107)(21,137,185,204)(22,125,186,109)(23,139,187,206)(24,113,188,111)(25,127,189,208)(26,115,190,99)(27,129,191,210)(28,117,192,101)(44,151,94,176)(46,153,96,178)(48,141,98,180)(50,143,86,182)(52,145,88,170)(54,147,90,172)(56,149,92,174)(57,203,74,136)(58,108,75,124)(59,205,76,138)(60,110,77,126)(61,207,78,140)(62,112,79,114)(63,209,80,128)(64,100,81,116)(65,197,82,130)(66,102,83,118)(67,199,84,132)(68,104,71,120)(69,201,72,134)(70,106,73,122)(142,212,181,161)(144,214,169,163)(146,216,171,165)(148,218,173,167)(150,220,175,155)(152,222,177,157)(154,224,179,159), (1,74,172,20)(2,58,173,185)(3,76,174,22)(4,60,175,187)(5,78,176,24)(6,62,177,189)(7,80,178,26)(8,64,179,191)(9,82,180,28)(10,66,181,193)(11,84,182,16)(12,68,169,195)(13,72,170,18)(14,70,171,183)(15,36,83,142)(17,38,71,144)(19,40,73,146)(21,42,75,148)(23,30,77,150)(25,32,79,152)(27,34,81,154)(29,59,149,186)(31,61,151,188)(33,63,153,190)(35,65,141,192)(37,67,143,194)(39,69,145,196)(41,57,147,184)(43,110,155,206)(44,113,156,140)(45,112,157,208)(46,115,158,128)(47,100,159,210)(48,117,160,130)(49,102,161,198)(50,119,162,132)(51,104,163,200)(52,121,164,134)(53,106,165,202)(54,123,166,136)(55,108,167,204)(56,125,168,138)(85,118,212,131)(86,103,213,199)(87,120,214,133)(88,105,215,201)(89,122,216,135)(90,107,217,203)(91,124,218,137)(92,109,219,205)(93,126,220,139)(94,111,221,207)(95,114,222,127)(96,99,223,209)(97,116,224,129)(98,101,211,197), (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), (1,99,172,209)(2,112,173,208)(3,111,174,207)(4,110,175,206)(5,109,176,205)(6,108,177,204)(7,107,178,203)(8,106,179,202)(9,105,180,201)(10,104,181,200)(11,103,182,199)(12,102,169,198)(13,101,170,197)(14,100,171,210)(15,51,83,163)(16,50,84,162)(17,49,71,161)(18,48,72,160)(19,47,73,159)(20,46,74,158)(21,45,75,157)(22,44,76,156)(23,43,77,155)(24,56,78,168)(25,55,79,167)(26,54,80,166)(27,53,81,165)(28,52,82,164)(29,113,149,140)(30,126,150,139)(31,125,151,138)(32,124,152,137)(33,123,153,136)(34,122,154,135)(35,121,141,134)(36,120,142,133)(37,119,143,132)(38,118,144,131)(39,117,145,130)(40,116,146,129)(41,115,147,128)(42,114,148,127)(57,223,184,96)(58,222,185,95)(59,221,186,94)(60,220,187,93)(61,219,188,92)(62,218,189,91)(63,217,190,90)(64,216,191,89)(65,215,192,88)(66,214,193,87)(67,213,194,86)(68,212,195,85)(69,211,196,98)(70,224,183,97) );
G=PermutationGroup([(1,166,41,217),(2,55,42,91),(3,168,29,219),(4,43,30,93),(5,156,31,221),(6,45,32,95),(7,158,33,223),(8,47,34,97),(9,160,35,211),(10,49,36,85),(11,162,37,213),(12,51,38,87),(13,164,39,215),(14,53,40,89),(15,131,193,198),(16,119,194,103),(17,133,195,200),(18,121,196,105),(19,135,183,202),(20,123,184,107),(21,137,185,204),(22,125,186,109),(23,139,187,206),(24,113,188,111),(25,127,189,208),(26,115,190,99),(27,129,191,210),(28,117,192,101),(44,151,94,176),(46,153,96,178),(48,141,98,180),(50,143,86,182),(52,145,88,170),(54,147,90,172),(56,149,92,174),(57,203,74,136),(58,108,75,124),(59,205,76,138),(60,110,77,126),(61,207,78,140),(62,112,79,114),(63,209,80,128),(64,100,81,116),(65,197,82,130),(66,102,83,118),(67,199,84,132),(68,104,71,120),(69,201,72,134),(70,106,73,122),(142,212,181,161),(144,214,169,163),(146,216,171,165),(148,218,173,167),(150,220,175,155),(152,222,177,157),(154,224,179,159)], [(1,74,172,20),(2,58,173,185),(3,76,174,22),(4,60,175,187),(5,78,176,24),(6,62,177,189),(7,80,178,26),(8,64,179,191),(9,82,180,28),(10,66,181,193),(11,84,182,16),(12,68,169,195),(13,72,170,18),(14,70,171,183),(15,36,83,142),(17,38,71,144),(19,40,73,146),(21,42,75,148),(23,30,77,150),(25,32,79,152),(27,34,81,154),(29,59,149,186),(31,61,151,188),(33,63,153,190),(35,65,141,192),(37,67,143,194),(39,69,145,196),(41,57,147,184),(43,110,155,206),(44,113,156,140),(45,112,157,208),(46,115,158,128),(47,100,159,210),(48,117,160,130),(49,102,161,198),(50,119,162,132),(51,104,163,200),(52,121,164,134),(53,106,165,202),(54,123,166,136),(55,108,167,204),(56,125,168,138),(85,118,212,131),(86,103,213,199),(87,120,214,133),(88,105,215,201),(89,122,216,135),(90,107,217,203),(91,124,218,137),(92,109,219,205),(93,126,220,139),(94,111,221,207),(95,114,222,127),(96,99,223,209),(97,116,224,129),(98,101,211,197)], [(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)], [(1,99,172,209),(2,112,173,208),(3,111,174,207),(4,110,175,206),(5,109,176,205),(6,108,177,204),(7,107,178,203),(8,106,179,202),(9,105,180,201),(10,104,181,200),(11,103,182,199),(12,102,169,198),(13,101,170,197),(14,100,171,210),(15,51,83,163),(16,50,84,162),(17,49,71,161),(18,48,72,160),(19,47,73,159),(20,46,74,158),(21,45,75,157),(22,44,76,156),(23,43,77,155),(24,56,78,168),(25,55,79,167),(26,54,80,166),(27,53,81,165),(28,52,82,164),(29,113,149,140),(30,126,150,139),(31,125,151,138),(32,124,152,137),(33,123,153,136),(34,122,154,135),(35,121,141,134),(36,120,142,133),(37,119,143,132),(38,118,144,131),(39,117,145,130),(40,116,146,129),(41,115,147,128),(42,114,148,127),(57,223,184,96),(58,222,185,95),(59,221,186,94),(60,220,187,93),(61,219,188,92),(62,218,189,91),(63,217,190,90),(64,216,191,89),(65,215,192,88),(66,214,193,87),(67,213,194,86),(68,212,195,85),(69,211,196,98),(70,224,183,97)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 19 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 24 |
| 0 | 0 | 0 | 0 | 7 | 6 |
| 26 | 18 | 0 | 0 | 0 | 0 |
| 6 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 | 0 | 0 |
| 0 | 0 | 21 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 27 |
| 0 | 0 | 0 | 0 | 25 | 14 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 | 0 | 0 |
| 0 | 0 | 3 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 27 |
| 0 | 0 | 0 | 0 | 25 | 14 |
G:=sub<GL(6,GF(29))| [28,19,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,23,7,0,0,0,0,24,6],[26,6,0,0,0,0,18,3,0,0,0,0,0,0,8,21,0,0,0,0,8,3,0,0,0,0,0,0,15,25,0,0,0,0,27,14],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,3,0,0,0,0,8,21,0,0,0,0,0,0,15,25,0,0,0,0,27,14] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4O | 4P | 4Q | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28R | 28S | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 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 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4⋊6D14 | D7×C4○D4 |
| kernel | C42.138D14 | C42⋊D7 | C23.11D14 | Dic7⋊4D4 | D14⋊D4 | Dic7.D4 | Dic7⋊D4 | Dic7⋊Q8 | C28.23D4 | C7×C4.4D4 | C4.4D4 | Dic7 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C14 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 3 | 8 | 3 | 12 | 3 | 3 | 1 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_4^2._{138}D_{14} % in TeX
G:=Group("C4^2.138D14"); // GroupNames label
G:=SmallGroup(448,1123);
// by ID
G=gap.SmallGroup(448,1123);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,758,387,100,346,136,18822]);
// 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=a*b^2,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations
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