metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.174D14, C14.372- (1+4), C14.822+ (1+4), C4⋊Q8⋊12D7, (C4×D7)⋊2Q8, C28⋊Q8⋊45C2, C4.41(Q8×D7), D14.6(C2×Q8), C28.55(C2×Q8), C4⋊C4.124D14, (C2×Q8).86D14, Dic7.7(C2×Q8), Dic7.Q8⋊41C2, D14⋊Q8.4C2, C28.6Q8⋊24C2, C4.Dic14⋊43C2, C42⋊D7.8C2, Dic7⋊Q8⋊27C2, C14.49(C22×Q8), (C2×C28).106C23, (C2×C14).273C24, (C4×C28).214C22, D14⋊C4.52C22, D14⋊3Q8.12C2, C2.86(D4⋊6D14), C4⋊Dic7.252C22, (Q8×C14).140C22, C22.294(C23×D7), Dic7⋊C4.167C22, C7⋊5(C23.41C23), (C2×Dic7).144C23, (C4×Dic7).162C22, (C22×D7).234C23, C2.38(Q8.10D14), (C2×Dic14).191C22, C2.32(C2×Q8×D7), (C7×C4⋊Q8)⋊15C2, (D7×C4⋊C4).13C2, C4⋊C4⋊7D7.15C2, (C2×C4×D7).146C22, (C7×C4⋊C4).216C22, (C2×C4).219(C22×D7), SmallGroup(448,1182)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 844 in 206 conjugacy classes, 103 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×14], C22, C22 [×4], C7, C2×C4 [×3], C2×C4 [×4], C2×C4 [×13], Q8 [×4], C23, D7 [×2], C14 [×3], C42, C42 [×3], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×16], C22×C4 [×3], C2×Q8 [×2], C2×Q8 [×2], Dic7 [×2], Dic7 [×6], C28 [×2], C28 [×6], D14 [×2], D14 [×2], C2×C14, C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8, C4⋊Q8 [×3], Dic14 [×2], C4×D7 [×4], C4×D7 [×2], C2×Dic7 [×3], C2×Dic7 [×4], C2×C28 [×3], C2×C28 [×4], C7×Q8 [×2], C22×D7, C23.41C23, C4×Dic7, C4×Dic7 [×2], Dic7⋊C4 [×2], Dic7⋊C4 [×10], C4⋊Dic7 [×2], C4⋊Dic7 [×2], D14⋊C4 [×2], D14⋊C4 [×2], C4×C28, C7×C4⋊C4 [×2], C7×C4⋊C4 [×2], C2×Dic14 [×2], C2×C4×D7, C2×C4×D7 [×2], Q8×C14 [×2], C28.6Q8, C42⋊D7, C28⋊Q8, Dic7.Q8 [×2], C4.Dic14, D7×C4⋊C4, C4⋊C4⋊7D7, D14⋊Q8 [×2], Dic7⋊Q8 [×2], D14⋊3Q8 [×2], C7×C4⋊Q8, C42.174D14
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D7, C2×Q8 [×6], C24, D14 [×7], C22×Q8, 2+ (1+4), 2- (1+4), C22×D7 [×7], C23.41C23, Q8×D7 [×2], C23×D7, D4⋊6D14, C2×Q8×D7, Q8.10D14, C42.174D14
Generators and relations
G = < a,b,c,d | a4=b4=1, c14=d2=a2b2, 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
(1 159 89 78)(2 79 90 160)(3 161 91 80)(4 81 92 162)(5 163 93 82)(6 83 94 164)(7 165 95 84)(8 57 96 166)(9 167 97 58)(10 59 98 168)(11 141 99 60)(12 61 100 142)(13 143 101 62)(14 63 102 144)(15 145 103 64)(16 65 104 146)(17 147 105 66)(18 67 106 148)(19 149 107 68)(20 69 108 150)(21 151 109 70)(22 71 110 152)(23 153 111 72)(24 73 112 154)(25 155 85 74)(26 75 86 156)(27 157 87 76)(28 77 88 158)(29 177 123 198)(30 199 124 178)(31 179 125 200)(32 201 126 180)(33 181 127 202)(34 203 128 182)(35 183 129 204)(36 205 130 184)(37 185 131 206)(38 207 132 186)(39 187 133 208)(40 209 134 188)(41 189 135 210)(42 211 136 190)(43 191 137 212)(44 213 138 192)(45 193 139 214)(46 215 140 194)(47 195 113 216)(48 217 114 196)(49 169 115 218)(50 219 116 170)(51 171 117 220)(52 221 118 172)(53 173 119 222)(54 223 120 174)(55 175 121 224)(56 197 122 176)
(1 31 103 139)(2 140 104 32)(3 33 105 113)(4 114 106 34)(5 35 107 115)(6 116 108 36)(7 37 109 117)(8 118 110 38)(9 39 111 119)(10 120 112 40)(11 41 85 121)(12 122 86 42)(13 43 87 123)(14 124 88 44)(15 45 89 125)(16 126 90 46)(17 47 91 127)(18 128 92 48)(19 49 93 129)(20 130 94 50)(21 51 95 131)(22 132 96 52)(23 53 97 133)(24 134 98 54)(25 55 99 135)(26 136 100 56)(27 29 101 137)(28 138 102 30)(57 172 152 207)(58 208 153 173)(59 174 154 209)(60 210 155 175)(61 176 156 211)(62 212 157 177)(63 178 158 213)(64 214 159 179)(65 180 160 215)(66 216 161 181)(67 182 162 217)(68 218 163 183)(69 184 164 219)(70 220 165 185)(71 186 166 221)(72 222 167 187)(73 188 168 223)(74 224 141 189)(75 190 142 197)(76 198 143 191)(77 192 144 199)(78 200 145 193)(79 194 146 201)(80 202 147 195)(81 196 148 203)(82 204 149 169)(83 170 150 205)(84 206 151 171)
(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)(29 140 43 126)(30 125 44 139)(31 138 45 124)(32 123 46 137)(33 136 47 122)(34 121 48 135)(35 134 49 120)(36 119 50 133)(37 132 51 118)(38 117 52 131)(39 130 53 116)(40 115 54 129)(41 128 55 114)(42 113 56 127)(57 70 71 84)(58 83 72 69)(59 68 73 82)(60 81 74 67)(61 66 75 80)(62 79 76 65)(63 64 77 78)(85 106 99 92)(86 91 100 105)(87 104 101 90)(88 89 102 103)(93 98 107 112)(94 111 108 97)(95 96 109 110)(141 162 155 148)(142 147 156 161)(143 160 157 146)(144 145 158 159)(149 154 163 168)(150 167 164 153)(151 152 165 166)(169 223 183 209)(170 208 184 222)(171 221 185 207)(172 206 186 220)(173 219 187 205)(174 204 188 218)(175 217 189 203)(176 202 190 216)(177 215 191 201)(178 200 192 214)(179 213 193 199)(180 198 194 212)(181 211 195 197)(182 224 196 210)
G:=sub<Sym(224)| (1,159,89,78)(2,79,90,160)(3,161,91,80)(4,81,92,162)(5,163,93,82)(6,83,94,164)(7,165,95,84)(8,57,96,166)(9,167,97,58)(10,59,98,168)(11,141,99,60)(12,61,100,142)(13,143,101,62)(14,63,102,144)(15,145,103,64)(16,65,104,146)(17,147,105,66)(18,67,106,148)(19,149,107,68)(20,69,108,150)(21,151,109,70)(22,71,110,152)(23,153,111,72)(24,73,112,154)(25,155,85,74)(26,75,86,156)(27,157,87,76)(28,77,88,158)(29,177,123,198)(30,199,124,178)(31,179,125,200)(32,201,126,180)(33,181,127,202)(34,203,128,182)(35,183,129,204)(36,205,130,184)(37,185,131,206)(38,207,132,186)(39,187,133,208)(40,209,134,188)(41,189,135,210)(42,211,136,190)(43,191,137,212)(44,213,138,192)(45,193,139,214)(46,215,140,194)(47,195,113,216)(48,217,114,196)(49,169,115,218)(50,219,116,170)(51,171,117,220)(52,221,118,172)(53,173,119,222)(54,223,120,174)(55,175,121,224)(56,197,122,176), (1,31,103,139)(2,140,104,32)(3,33,105,113)(4,114,106,34)(5,35,107,115)(6,116,108,36)(7,37,109,117)(8,118,110,38)(9,39,111,119)(10,120,112,40)(11,41,85,121)(12,122,86,42)(13,43,87,123)(14,124,88,44)(15,45,89,125)(16,126,90,46)(17,47,91,127)(18,128,92,48)(19,49,93,129)(20,130,94,50)(21,51,95,131)(22,132,96,52)(23,53,97,133)(24,134,98,54)(25,55,99,135)(26,136,100,56)(27,29,101,137)(28,138,102,30)(57,172,152,207)(58,208,153,173)(59,174,154,209)(60,210,155,175)(61,176,156,211)(62,212,157,177)(63,178,158,213)(64,214,159,179)(65,180,160,215)(66,216,161,181)(67,182,162,217)(68,218,163,183)(69,184,164,219)(70,220,165,185)(71,186,166,221)(72,222,167,187)(73,188,168,223)(74,224,141,189)(75,190,142,197)(76,198,143,191)(77,192,144,199)(78,200,145,193)(79,194,146,201)(80,202,147,195)(81,196,148,203)(82,204,149,169)(83,170,150,205)(84,206,151,171), (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)(29,140,43,126)(30,125,44,139)(31,138,45,124)(32,123,46,137)(33,136,47,122)(34,121,48,135)(35,134,49,120)(36,119,50,133)(37,132,51,118)(38,117,52,131)(39,130,53,116)(40,115,54,129)(41,128,55,114)(42,113,56,127)(57,70,71,84)(58,83,72,69)(59,68,73,82)(60,81,74,67)(61,66,75,80)(62,79,76,65)(63,64,77,78)(85,106,99,92)(86,91,100,105)(87,104,101,90)(88,89,102,103)(93,98,107,112)(94,111,108,97)(95,96,109,110)(141,162,155,148)(142,147,156,161)(143,160,157,146)(144,145,158,159)(149,154,163,168)(150,167,164,153)(151,152,165,166)(169,223,183,209)(170,208,184,222)(171,221,185,207)(172,206,186,220)(173,219,187,205)(174,204,188,218)(175,217,189,203)(176,202,190,216)(177,215,191,201)(178,200,192,214)(179,213,193,199)(180,198,194,212)(181,211,195,197)(182,224,196,210)>;
G:=Group( (1,159,89,78)(2,79,90,160)(3,161,91,80)(4,81,92,162)(5,163,93,82)(6,83,94,164)(7,165,95,84)(8,57,96,166)(9,167,97,58)(10,59,98,168)(11,141,99,60)(12,61,100,142)(13,143,101,62)(14,63,102,144)(15,145,103,64)(16,65,104,146)(17,147,105,66)(18,67,106,148)(19,149,107,68)(20,69,108,150)(21,151,109,70)(22,71,110,152)(23,153,111,72)(24,73,112,154)(25,155,85,74)(26,75,86,156)(27,157,87,76)(28,77,88,158)(29,177,123,198)(30,199,124,178)(31,179,125,200)(32,201,126,180)(33,181,127,202)(34,203,128,182)(35,183,129,204)(36,205,130,184)(37,185,131,206)(38,207,132,186)(39,187,133,208)(40,209,134,188)(41,189,135,210)(42,211,136,190)(43,191,137,212)(44,213,138,192)(45,193,139,214)(46,215,140,194)(47,195,113,216)(48,217,114,196)(49,169,115,218)(50,219,116,170)(51,171,117,220)(52,221,118,172)(53,173,119,222)(54,223,120,174)(55,175,121,224)(56,197,122,176), (1,31,103,139)(2,140,104,32)(3,33,105,113)(4,114,106,34)(5,35,107,115)(6,116,108,36)(7,37,109,117)(8,118,110,38)(9,39,111,119)(10,120,112,40)(11,41,85,121)(12,122,86,42)(13,43,87,123)(14,124,88,44)(15,45,89,125)(16,126,90,46)(17,47,91,127)(18,128,92,48)(19,49,93,129)(20,130,94,50)(21,51,95,131)(22,132,96,52)(23,53,97,133)(24,134,98,54)(25,55,99,135)(26,136,100,56)(27,29,101,137)(28,138,102,30)(57,172,152,207)(58,208,153,173)(59,174,154,209)(60,210,155,175)(61,176,156,211)(62,212,157,177)(63,178,158,213)(64,214,159,179)(65,180,160,215)(66,216,161,181)(67,182,162,217)(68,218,163,183)(69,184,164,219)(70,220,165,185)(71,186,166,221)(72,222,167,187)(73,188,168,223)(74,224,141,189)(75,190,142,197)(76,198,143,191)(77,192,144,199)(78,200,145,193)(79,194,146,201)(80,202,147,195)(81,196,148,203)(82,204,149,169)(83,170,150,205)(84,206,151,171), (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)(29,140,43,126)(30,125,44,139)(31,138,45,124)(32,123,46,137)(33,136,47,122)(34,121,48,135)(35,134,49,120)(36,119,50,133)(37,132,51,118)(38,117,52,131)(39,130,53,116)(40,115,54,129)(41,128,55,114)(42,113,56,127)(57,70,71,84)(58,83,72,69)(59,68,73,82)(60,81,74,67)(61,66,75,80)(62,79,76,65)(63,64,77,78)(85,106,99,92)(86,91,100,105)(87,104,101,90)(88,89,102,103)(93,98,107,112)(94,111,108,97)(95,96,109,110)(141,162,155,148)(142,147,156,161)(143,160,157,146)(144,145,158,159)(149,154,163,168)(150,167,164,153)(151,152,165,166)(169,223,183,209)(170,208,184,222)(171,221,185,207)(172,206,186,220)(173,219,187,205)(174,204,188,218)(175,217,189,203)(176,202,190,216)(177,215,191,201)(178,200,192,214)(179,213,193,199)(180,198,194,212)(181,211,195,197)(182,224,196,210) );
G=PermutationGroup([(1,159,89,78),(2,79,90,160),(3,161,91,80),(4,81,92,162),(5,163,93,82),(6,83,94,164),(7,165,95,84),(8,57,96,166),(9,167,97,58),(10,59,98,168),(11,141,99,60),(12,61,100,142),(13,143,101,62),(14,63,102,144),(15,145,103,64),(16,65,104,146),(17,147,105,66),(18,67,106,148),(19,149,107,68),(20,69,108,150),(21,151,109,70),(22,71,110,152),(23,153,111,72),(24,73,112,154),(25,155,85,74),(26,75,86,156),(27,157,87,76),(28,77,88,158),(29,177,123,198),(30,199,124,178),(31,179,125,200),(32,201,126,180),(33,181,127,202),(34,203,128,182),(35,183,129,204),(36,205,130,184),(37,185,131,206),(38,207,132,186),(39,187,133,208),(40,209,134,188),(41,189,135,210),(42,211,136,190),(43,191,137,212),(44,213,138,192),(45,193,139,214),(46,215,140,194),(47,195,113,216),(48,217,114,196),(49,169,115,218),(50,219,116,170),(51,171,117,220),(52,221,118,172),(53,173,119,222),(54,223,120,174),(55,175,121,224),(56,197,122,176)], [(1,31,103,139),(2,140,104,32),(3,33,105,113),(4,114,106,34),(5,35,107,115),(6,116,108,36),(7,37,109,117),(8,118,110,38),(9,39,111,119),(10,120,112,40),(11,41,85,121),(12,122,86,42),(13,43,87,123),(14,124,88,44),(15,45,89,125),(16,126,90,46),(17,47,91,127),(18,128,92,48),(19,49,93,129),(20,130,94,50),(21,51,95,131),(22,132,96,52),(23,53,97,133),(24,134,98,54),(25,55,99,135),(26,136,100,56),(27,29,101,137),(28,138,102,30),(57,172,152,207),(58,208,153,173),(59,174,154,209),(60,210,155,175),(61,176,156,211),(62,212,157,177),(63,178,158,213),(64,214,159,179),(65,180,160,215),(66,216,161,181),(67,182,162,217),(68,218,163,183),(69,184,164,219),(70,220,165,185),(71,186,166,221),(72,222,167,187),(73,188,168,223),(74,224,141,189),(75,190,142,197),(76,198,143,191),(77,192,144,199),(78,200,145,193),(79,194,146,201),(80,202,147,195),(81,196,148,203),(82,204,149,169),(83,170,150,205),(84,206,151,171)], [(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),(29,140,43,126),(30,125,44,139),(31,138,45,124),(32,123,46,137),(33,136,47,122),(34,121,48,135),(35,134,49,120),(36,119,50,133),(37,132,51,118),(38,117,52,131),(39,130,53,116),(40,115,54,129),(41,128,55,114),(42,113,56,127),(57,70,71,84),(58,83,72,69),(59,68,73,82),(60,81,74,67),(61,66,75,80),(62,79,76,65),(63,64,77,78),(85,106,99,92),(86,91,100,105),(87,104,101,90),(88,89,102,103),(93,98,107,112),(94,111,108,97),(95,96,109,110),(141,162,155,148),(142,147,156,161),(143,160,157,146),(144,145,158,159),(149,154,163,168),(150,167,164,153),(151,152,165,166),(169,223,183,209),(170,208,184,222),(171,221,185,207),(172,206,186,220),(173,219,187,205),(174,204,188,218),(175,217,189,203),(176,202,190,216),(177,215,191,201),(178,200,192,214),(179,213,193,199),(180,198,194,212),(181,211,195,197),(182,224,196,210)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 21 | 8 | 17 |
| 0 | 0 | 11 | 0 | 11 | 17 |
| 0 | 0 | 5 | 3 | 6 | 8 |
| 0 | 0 | 25 | 10 | 8 | 6 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 2 | 0 | 14 |
| 0 | 0 | 17 | 24 | 15 | 4 |
| 0 | 0 | 26 | 25 | 18 | 27 |
| 0 | 0 | 4 | 0 | 2 | 11 |
| 18 | 20 | 0 | 0 | 0 | 0 |
| 20 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 14 | 8 | 13 |
| 0 | 0 | 19 | 5 | 15 | 20 |
| 0 | 0 | 1 | 10 | 28 | 15 |
| 0 | 0 | 27 | 8 | 1 | 12 |
| 18 | 20 | 0 | 0 | 0 | 0 |
| 20 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 12 | 16 | 21 |
| 0 | 0 | 18 | 21 | 9 | 14 |
| 0 | 0 | 13 | 21 | 14 | 1 |
| 0 | 0 | 27 | 8 | 17 | 28 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,11,5,25,0,0,21,0,3,10,0,0,8,11,6,8,0,0,17,17,8,6],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,5,17,26,4,0,0,2,24,25,0,0,0,0,15,18,2,0,0,14,4,27,11],[18,20,0,0,0,0,20,11,0,0,0,0,0,0,13,19,1,27,0,0,14,5,10,8,0,0,8,15,28,1,0,0,13,20,15,12],[18,20,0,0,0,0,20,11,0,0,0,0,0,0,24,18,13,27,0,0,12,21,21,8,0,0,16,9,14,17,0,0,21,14,1,28] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | ··· | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 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 | 14 | 14 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D7 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | Q8×D7 | D4⋊6D14 | Q8.10D14 |
| kernel | C42.174D14 | C28.6Q8 | C42⋊D7 | C28⋊Q8 | Dic7.Q8 | C4.Dic14 | D7×C4⋊C4 | C4⋊C4⋊7D7 | D14⋊Q8 | Dic7⋊Q8 | D14⋊3Q8 | C7×C4⋊Q8 | C4×D7 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C14 | C14 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 3 | 3 | 12 | 6 | 1 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_4^2._{174}D_{14} % in TeX
G:=Group("C4^2.174D14"); // GroupNames label
G:=SmallGroup(448,1182);
// by ID
G=gap.SmallGroup(448,1182);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,268,675,297,136,18822]);
// 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=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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