metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.176D14, C14.382- (1+4), C4⋊Q8⋊14D7, C4⋊C4.125D14, (Q8×Dic7)⋊23C2, Dic7.Q8⋊42C2, (C2×Q8).148D14, C28.6Q8⋊25C2, C42⋊D7.9C2, C28.138(C4○D4), C4.42(D4⋊2D7), (C4×C28).216C22, (C2×C28).108C23, (C2×C14).275C24, D14⋊3Q8.13C2, D14⋊C4.154C22, Dic7⋊C4.63C22, C4⋊Dic7.254C22, (Q8×C14).142C22, C22.296(C23×D7), C7⋊7(C22.35C24), (C2×Dic7).273C23, (C4×Dic7).164C22, (C22×D7).120C23, C2.39(Q8.10D14), (C7×C4⋊Q8)⋊17C2, C4⋊C4⋊D7.4C2, C14.101(C2×C4○D4), C2.65(C2×D4⋊2D7), (C2×C4×D7).148C22, (C7×C4⋊C4).218C22, (C2×C4).221(C22×D7), SmallGroup(448,1184)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 716 in 192 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×13], C22, C22 [×3], C7, C2×C4, C2×C4 [×6], C2×C4 [×9], Q8 [×4], C23, D7, C14, C14 [×2], C42, C42 [×5], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×16], C22×C4, C2×Q8 [×2], Dic7 [×7], C28 [×2], C28 [×6], D14 [×3], C2×C14, C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×5], C42⋊2C2 [×4], C4⋊Q8, C4×D7 [×2], C2×Dic7, C2×Dic7 [×6], C2×C28, C2×C28 [×6], C7×Q8 [×4], C22×D7, C22.35C24, C4×Dic7, C4×Dic7 [×4], Dic7⋊C4 [×10], C4⋊Dic7 [×6], D14⋊C4 [×6], C4×C28, C7×C4⋊C4 [×4], C2×C4×D7, Q8×C14 [×2], C28.6Q8, C42⋊D7, Dic7.Q8 [×4], C4⋊C4⋊D7 [×4], Q8×Dic7 [×2], D14⋊3Q8 [×2], C7×C4⋊Q8, C42.176D14
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, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, Q8.10D14 [×2], C42.176D14
Generators and relations
G = < a,b,c,d | a4=b4=1, c14=a2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=b2c13 >
►On 224 points
Generators in S224
(1 111 15 97)(2 98 16 112)(3 85 17 99)(4 100 18 86)(5 87 19 101)(6 102 20 88)(7 89 21 103)(8 104 22 90)(9 91 23 105)(10 106 24 92)(11 93 25 107)(12 108 26 94)(13 95 27 109)(14 110 28 96)(29 214 43 200)(30 201 44 215)(31 216 45 202)(32 203 46 217)(33 218 47 204)(34 205 48 219)(35 220 49 206)(36 207 50 221)(37 222 51 208)(38 209 52 223)(39 224 53 210)(40 211 54 197)(41 198 55 212)(42 213 56 199)(57 152 71 166)(58 167 72 153)(59 154 73 168)(60 141 74 155)(61 156 75 142)(62 143 76 157)(63 158 77 144)(64 145 78 159)(65 160 79 146)(66 147 80 161)(67 162 81 148)(68 149 82 163)(69 164 83 150)(70 151 84 165)(113 189 127 175)(114 176 128 190)(115 191 129 177)(116 178 130 192)(117 193 131 179)(118 180 132 194)(119 195 133 181)(120 182 134 196)(121 169 135 183)(122 184 136 170)(123 171 137 185)(124 186 138 172)(125 173 139 187)(126 188 140 174)
(1 145 217 125)(2 126 218 146)(3 147 219 127)(4 128 220 148)(5 149 221 129)(6 130 222 150)(7 151 223 131)(8 132 224 152)(9 153 197 133)(10 134 198 154)(11 155 199 135)(12 136 200 156)(13 157 201 137)(14 138 202 158)(15 159 203 139)(16 140 204 160)(17 161 205 113)(18 114 206 162)(19 163 207 115)(20 116 208 164)(21 165 209 117)(22 118 210 166)(23 167 211 119)(24 120 212 168)(25 141 213 121)(26 122 214 142)(27 143 215 123)(28 124 216 144)(29 75 108 170)(30 171 109 76)(31 77 110 172)(32 173 111 78)(33 79 112 174)(34 175 85 80)(35 81 86 176)(36 177 87 82)(37 83 88 178)(38 179 89 84)(39 57 90 180)(40 181 91 58)(41 59 92 182)(42 183 93 60)(43 61 94 184)(44 185 95 62)(45 63 96 186)(46 187 97 64)(47 65 98 188)(48 189 99 66)(49 67 100 190)(50 191 101 68)(51 69 102 192)(52 193 103 70)(53 71 104 194)(54 195 105 72)(55 73 106 196)(56 169 107 74)
(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 14 203 216)(2 215 204 13)(3 12 205 214)(4 213 206 11)(5 10 207 212)(6 211 208 9)(7 8 209 210)(15 28 217 202)(16 201 218 27)(17 26 219 200)(18 199 220 25)(19 24 221 198)(20 197 222 23)(21 22 223 224)(29 85 94 48)(30 47 95 112)(31 111 96 46)(32 45 97 110)(33 109 98 44)(34 43 99 108)(35 107 100 42)(36 41 101 106)(37 105 102 40)(38 39 103 104)(49 93 86 56)(50 55 87 92)(51 91 88 54)(52 53 89 90)(57 179 194 70)(58 69 195 178)(59 177 196 68)(60 67 169 176)(61 175 170 66)(62 65 171 174)(63 173 172 64)(71 193 180 84)(72 83 181 192)(73 191 182 82)(74 81 183 190)(75 189 184 80)(76 79 185 188)(77 187 186 78)(113 136 147 142)(114 141 148 135)(115 134 149 168)(116 167 150 133)(117 132 151 166)(118 165 152 131)(119 130 153 164)(120 163 154 129)(121 128 155 162)(122 161 156 127)(123 126 157 160)(124 159 158 125)(137 140 143 146)(138 145 144 139)
G:=sub<Sym(224)| (1,111,15,97)(2,98,16,112)(3,85,17,99)(4,100,18,86)(5,87,19,101)(6,102,20,88)(7,89,21,103)(8,104,22,90)(9,91,23,105)(10,106,24,92)(11,93,25,107)(12,108,26,94)(13,95,27,109)(14,110,28,96)(29,214,43,200)(30,201,44,215)(31,216,45,202)(32,203,46,217)(33,218,47,204)(34,205,48,219)(35,220,49,206)(36,207,50,221)(37,222,51,208)(38,209,52,223)(39,224,53,210)(40,211,54,197)(41,198,55,212)(42,213,56,199)(57,152,71,166)(58,167,72,153)(59,154,73,168)(60,141,74,155)(61,156,75,142)(62,143,76,157)(63,158,77,144)(64,145,78,159)(65,160,79,146)(66,147,80,161)(67,162,81,148)(68,149,82,163)(69,164,83,150)(70,151,84,165)(113,189,127,175)(114,176,128,190)(115,191,129,177)(116,178,130,192)(117,193,131,179)(118,180,132,194)(119,195,133,181)(120,182,134,196)(121,169,135,183)(122,184,136,170)(123,171,137,185)(124,186,138,172)(125,173,139,187)(126,188,140,174), (1,145,217,125)(2,126,218,146)(3,147,219,127)(4,128,220,148)(5,149,221,129)(6,130,222,150)(7,151,223,131)(8,132,224,152)(9,153,197,133)(10,134,198,154)(11,155,199,135)(12,136,200,156)(13,157,201,137)(14,138,202,158)(15,159,203,139)(16,140,204,160)(17,161,205,113)(18,114,206,162)(19,163,207,115)(20,116,208,164)(21,165,209,117)(22,118,210,166)(23,167,211,119)(24,120,212,168)(25,141,213,121)(26,122,214,142)(27,143,215,123)(28,124,216,144)(29,75,108,170)(30,171,109,76)(31,77,110,172)(32,173,111,78)(33,79,112,174)(34,175,85,80)(35,81,86,176)(36,177,87,82)(37,83,88,178)(38,179,89,84)(39,57,90,180)(40,181,91,58)(41,59,92,182)(42,183,93,60)(43,61,94,184)(44,185,95,62)(45,63,96,186)(46,187,97,64)(47,65,98,188)(48,189,99,66)(49,67,100,190)(50,191,101,68)(51,69,102,192)(52,193,103,70)(53,71,104,194)(54,195,105,72)(55,73,106,196)(56,169,107,74), (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), (1,14,203,216)(2,215,204,13)(3,12,205,214)(4,213,206,11)(5,10,207,212)(6,211,208,9)(7,8,209,210)(15,28,217,202)(16,201,218,27)(17,26,219,200)(18,199,220,25)(19,24,221,198)(20,197,222,23)(21,22,223,224)(29,85,94,48)(30,47,95,112)(31,111,96,46)(32,45,97,110)(33,109,98,44)(34,43,99,108)(35,107,100,42)(36,41,101,106)(37,105,102,40)(38,39,103,104)(49,93,86,56)(50,55,87,92)(51,91,88,54)(52,53,89,90)(57,179,194,70)(58,69,195,178)(59,177,196,68)(60,67,169,176)(61,175,170,66)(62,65,171,174)(63,173,172,64)(71,193,180,84)(72,83,181,192)(73,191,182,82)(74,81,183,190)(75,189,184,80)(76,79,185,188)(77,187,186,78)(113,136,147,142)(114,141,148,135)(115,134,149,168)(116,167,150,133)(117,132,151,166)(118,165,152,131)(119,130,153,164)(120,163,154,129)(121,128,155,162)(122,161,156,127)(123,126,157,160)(124,159,158,125)(137,140,143,146)(138,145,144,139)>;
G:=Group( (1,111,15,97)(2,98,16,112)(3,85,17,99)(4,100,18,86)(5,87,19,101)(6,102,20,88)(7,89,21,103)(8,104,22,90)(9,91,23,105)(10,106,24,92)(11,93,25,107)(12,108,26,94)(13,95,27,109)(14,110,28,96)(29,214,43,200)(30,201,44,215)(31,216,45,202)(32,203,46,217)(33,218,47,204)(34,205,48,219)(35,220,49,206)(36,207,50,221)(37,222,51,208)(38,209,52,223)(39,224,53,210)(40,211,54,197)(41,198,55,212)(42,213,56,199)(57,152,71,166)(58,167,72,153)(59,154,73,168)(60,141,74,155)(61,156,75,142)(62,143,76,157)(63,158,77,144)(64,145,78,159)(65,160,79,146)(66,147,80,161)(67,162,81,148)(68,149,82,163)(69,164,83,150)(70,151,84,165)(113,189,127,175)(114,176,128,190)(115,191,129,177)(116,178,130,192)(117,193,131,179)(118,180,132,194)(119,195,133,181)(120,182,134,196)(121,169,135,183)(122,184,136,170)(123,171,137,185)(124,186,138,172)(125,173,139,187)(126,188,140,174), (1,145,217,125)(2,126,218,146)(3,147,219,127)(4,128,220,148)(5,149,221,129)(6,130,222,150)(7,151,223,131)(8,132,224,152)(9,153,197,133)(10,134,198,154)(11,155,199,135)(12,136,200,156)(13,157,201,137)(14,138,202,158)(15,159,203,139)(16,140,204,160)(17,161,205,113)(18,114,206,162)(19,163,207,115)(20,116,208,164)(21,165,209,117)(22,118,210,166)(23,167,211,119)(24,120,212,168)(25,141,213,121)(26,122,214,142)(27,143,215,123)(28,124,216,144)(29,75,108,170)(30,171,109,76)(31,77,110,172)(32,173,111,78)(33,79,112,174)(34,175,85,80)(35,81,86,176)(36,177,87,82)(37,83,88,178)(38,179,89,84)(39,57,90,180)(40,181,91,58)(41,59,92,182)(42,183,93,60)(43,61,94,184)(44,185,95,62)(45,63,96,186)(46,187,97,64)(47,65,98,188)(48,189,99,66)(49,67,100,190)(50,191,101,68)(51,69,102,192)(52,193,103,70)(53,71,104,194)(54,195,105,72)(55,73,106,196)(56,169,107,74), (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), (1,14,203,216)(2,215,204,13)(3,12,205,214)(4,213,206,11)(5,10,207,212)(6,211,208,9)(7,8,209,210)(15,28,217,202)(16,201,218,27)(17,26,219,200)(18,199,220,25)(19,24,221,198)(20,197,222,23)(21,22,223,224)(29,85,94,48)(30,47,95,112)(31,111,96,46)(32,45,97,110)(33,109,98,44)(34,43,99,108)(35,107,100,42)(36,41,101,106)(37,105,102,40)(38,39,103,104)(49,93,86,56)(50,55,87,92)(51,91,88,54)(52,53,89,90)(57,179,194,70)(58,69,195,178)(59,177,196,68)(60,67,169,176)(61,175,170,66)(62,65,171,174)(63,173,172,64)(71,193,180,84)(72,83,181,192)(73,191,182,82)(74,81,183,190)(75,189,184,80)(76,79,185,188)(77,187,186,78)(113,136,147,142)(114,141,148,135)(115,134,149,168)(116,167,150,133)(117,132,151,166)(118,165,152,131)(119,130,153,164)(120,163,154,129)(121,128,155,162)(122,161,156,127)(123,126,157,160)(124,159,158,125)(137,140,143,146)(138,145,144,139) );
G=PermutationGroup([(1,111,15,97),(2,98,16,112),(3,85,17,99),(4,100,18,86),(5,87,19,101),(6,102,20,88),(7,89,21,103),(8,104,22,90),(9,91,23,105),(10,106,24,92),(11,93,25,107),(12,108,26,94),(13,95,27,109),(14,110,28,96),(29,214,43,200),(30,201,44,215),(31,216,45,202),(32,203,46,217),(33,218,47,204),(34,205,48,219),(35,220,49,206),(36,207,50,221),(37,222,51,208),(38,209,52,223),(39,224,53,210),(40,211,54,197),(41,198,55,212),(42,213,56,199),(57,152,71,166),(58,167,72,153),(59,154,73,168),(60,141,74,155),(61,156,75,142),(62,143,76,157),(63,158,77,144),(64,145,78,159),(65,160,79,146),(66,147,80,161),(67,162,81,148),(68,149,82,163),(69,164,83,150),(70,151,84,165),(113,189,127,175),(114,176,128,190),(115,191,129,177),(116,178,130,192),(117,193,131,179),(118,180,132,194),(119,195,133,181),(120,182,134,196),(121,169,135,183),(122,184,136,170),(123,171,137,185),(124,186,138,172),(125,173,139,187),(126,188,140,174)], [(1,145,217,125),(2,126,218,146),(3,147,219,127),(4,128,220,148),(5,149,221,129),(6,130,222,150),(7,151,223,131),(8,132,224,152),(9,153,197,133),(10,134,198,154),(11,155,199,135),(12,136,200,156),(13,157,201,137),(14,138,202,158),(15,159,203,139),(16,140,204,160),(17,161,205,113),(18,114,206,162),(19,163,207,115),(20,116,208,164),(21,165,209,117),(22,118,210,166),(23,167,211,119),(24,120,212,168),(25,141,213,121),(26,122,214,142),(27,143,215,123),(28,124,216,144),(29,75,108,170),(30,171,109,76),(31,77,110,172),(32,173,111,78),(33,79,112,174),(34,175,85,80),(35,81,86,176),(36,177,87,82),(37,83,88,178),(38,179,89,84),(39,57,90,180),(40,181,91,58),(41,59,92,182),(42,183,93,60),(43,61,94,184),(44,185,95,62),(45,63,96,186),(46,187,97,64),(47,65,98,188),(48,189,99,66),(49,67,100,190),(50,191,101,68),(51,69,102,192),(52,193,103,70),(53,71,104,194),(54,195,105,72),(55,73,106,196),(56,169,107,74)], [(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)], [(1,14,203,216),(2,215,204,13),(3,12,205,214),(4,213,206,11),(5,10,207,212),(6,211,208,9),(7,8,209,210),(15,28,217,202),(16,201,218,27),(17,26,219,200),(18,199,220,25),(19,24,221,198),(20,197,222,23),(21,22,223,224),(29,85,94,48),(30,47,95,112),(31,111,96,46),(32,45,97,110),(33,109,98,44),(34,43,99,108),(35,107,100,42),(36,41,101,106),(37,105,102,40),(38,39,103,104),(49,93,86,56),(50,55,87,92),(51,91,88,54),(52,53,89,90),(57,179,194,70),(58,69,195,178),(59,177,196,68),(60,67,169,176),(61,175,170,66),(62,65,171,174),(63,173,172,64),(71,193,180,84),(72,83,181,192),(73,191,182,82),(74,81,183,190),(75,189,184,80),(76,79,185,188),(77,187,186,78),(113,136,147,142),(114,141,148,135),(115,134,149,168),(116,167,150,133),(117,132,151,166),(118,165,152,131),(119,130,153,164),(120,163,154,129),(121,128,155,162),(122,161,156,127),(123,126,157,160),(124,159,158,125),(137,140,143,146),(138,145,144,139)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 12 | 17 | 0 |
| 0 | 0 | 0 | 0 | 25 | 16 | 0 | 17 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 18 | 16 | 18 |
| 0 | 0 | 0 | 0 | 0 | 5 | 26 | 13 |
| 26 | 13 | 13 | 21 | 0 | 0 | 0 | 0 |
| 16 | 16 | 8 | 8 | 0 | 0 | 0 | 0 |
| 13 | 21 | 3 | 16 | 0 | 0 | 0 | 0 |
| 8 | 8 | 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 27 | 0 | 3 |
| 0 | 0 | 0 | 0 | 26 | 10 | 14 | 18 |
| 0 | 0 | 0 | 0 | 7 | 8 | 7 | 2 |
| 0 | 0 | 0 | 0 | 22 | 26 | 19 | 26 |
| 24 | 7 | 12 | 18 | 0 | 0 | 0 | 0 |
| 22 | 5 | 11 | 17 | 0 | 0 | 0 | 0 |
| 12 | 18 | 5 | 22 | 0 | 0 | 0 | 0 |
| 11 | 17 | 7 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 1 | 13 | 25 |
| 0 | 0 | 0 | 0 | 5 | 9 | 0 | 1 |
| 0 | 0 | 0 | 0 | 18 | 14 | 18 | 5 |
| 0 | 0 | 0 | 0 | 0 | 2 | 22 | 11 |
G:=sub<GL(8,GF(29))| [28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,12,0,2,25,0,0,0,0,0,12,12,16,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17],[0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,8,2,13,0,0,0,0,0,11,21,18,5,0,0,0,0,0,0,16,26,0,0,0,0,0,0,18,13],[26,16,13,8,0,0,0,0,13,16,21,8,0,0,0,0,13,8,3,13,0,0,0,0,21,8,16,13,0,0,0,0,0,0,0,0,15,26,7,22,0,0,0,0,27,10,8,26,0,0,0,0,0,14,7,19,0,0,0,0,3,18,2,26],[24,22,12,11,0,0,0,0,7,5,18,17,0,0,0,0,12,11,5,7,0,0,0,0,18,17,22,24,0,0,0,0,0,0,0,0,20,5,18,0,0,0,0,0,1,9,14,2,0,0,0,0,13,0,18,22,0,0,0,0,25,1,5,11] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4Q | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 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 | 28 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 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 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | 2- (1+4) | D4⋊2D7 | Q8.10D14 |
| kernel | C42.176D14 | C28.6Q8 | C42⋊D7 | Dic7.Q8 | C4⋊C4⋊D7 | Q8×Dic7 | D14⋊3Q8 | C7×C4⋊Q8 | C4⋊Q8 | C28 | C42 | C4⋊C4 | C2×Q8 | C14 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 3 | 4 | 3 | 12 | 6 | 2 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_4^2._{176}D_{14} % in TeX
G:=Group("C4^2.176D14"); // GroupNames label
G:=SmallGroup(448,1184);
// by ID
G=gap.SmallGroup(448,1184);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,758,219,100,675,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=a^2,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=b^2*c^13>;
// generators/relations
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