metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.178D14, C14.842+ (1+4), C14.402- (1+4), C4⋊Q8⋊16D7, C4⋊C4.126D14, (C2×Q8).88D14, D28⋊C4⋊44C2, D14⋊2Q8⋊46C2, D14⋊3Q8⋊38C2, C28⋊1D4.13C2, C4.D28⋊27C2, C42⋊D7⋊26C2, Dic7⋊3Q8⋊43C2, D14.5D4⋊47C2, C28.140(C4○D4), C28.23D4⋊27C2, (C4×C28).218C22, (C2×C14).277C24, (C2×C28).639C23, C4.23(Q8⋊2D7), C2.88(D4⋊6D14), D14⋊C4.156C22, (C2×D28).173C22, C4⋊Dic7.255C22, (Q8×C14).144C22, C22.298(C23×D7), Dic7⋊C4.169C22, (C4×Dic7).166C22, (C2×Dic7).274C23, (C22×D7).122C23, C2.41(Q8.10D14), C7⋊11(C22.36C24), (C2×Dic14).192C22, (C7×C4⋊Q8)⋊19C2, C4⋊C4⋊D7⋊47C2, C14.124(C2×C4○D4), C2.32(C2×Q8⋊2D7), (C2×C4×D7).150C22, (C7×C4⋊C4).220C22, (C2×C4).602(C22×D7), SmallGroup(448,1186)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1036 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C7, C2×C4 [×3], C2×C4 [×4], C2×C4 [×9], D4 [×4], Q8 [×4], C23 [×3], D7 [×3], C14 [×3], C42, C42 [×3], C22⋊C4 [×12], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4 [×3], C2×D4 [×3], C2×Q8 [×2], C2×Q8, Dic7 [×5], C28 [×2], C28 [×6], D14 [×9], C2×C14, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×3], C42⋊2C2 [×2], C4⋊Q8, Dic14 [×2], C4×D7 [×4], D28 [×4], C2×Dic7 [×3], C2×Dic7 [×2], C2×C28 [×3], C2×C28 [×4], C7×Q8 [×2], C22×D7, C22×D7 [×2], C22.36C24, C4×Dic7, C4×Dic7 [×2], Dic7⋊C4 [×2], Dic7⋊C4 [×2], C4⋊Dic7 [×2], D14⋊C4 [×2], D14⋊C4 [×10], C4×C28, C7×C4⋊C4 [×2], C7×C4⋊C4 [×2], C2×Dic14, C2×C4×D7, C2×C4×D7 [×2], C2×D28, C2×D28 [×2], Q8×C14 [×2], C42⋊D7, C4.D28, Dic7⋊3Q8, D28⋊C4, D14.5D4 [×2], C28⋊1D4, D14⋊2Q8, C4⋊C4⋊D7 [×2], D14⋊3Q8 [×2], C28.23D4 [×2], C7×C4⋊Q8, C42.178D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×D7 [×7], C22.36C24, Q8⋊2D7 [×2], C23×D7, D4⋊6D14, C2×Q8⋊2D7, Q8.10D14, C42.178D14
Generators and relations
G = < a,b,c,d | a4=b4=1, c14=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c13 >
►On 224 points
Generators in S224
(1 42 15 56)(2 29 16 43)(3 44 17 30)(4 31 18 45)(5 46 19 32)(6 33 20 47)(7 48 21 34)(8 35 22 49)(9 50 23 36)(10 37 24 51)(11 52 25 38)(12 39 26 53)(13 54 27 40)(14 41 28 55)(57 104 71 90)(58 91 72 105)(59 106 73 92)(60 93 74 107)(61 108 75 94)(62 95 76 109)(63 110 77 96)(64 97 78 111)(65 112 79 98)(66 99 80 85)(67 86 81 100)(68 101 82 87)(69 88 83 102)(70 103 84 89)(113 151 127 165)(114 166 128 152)(115 153 129 167)(116 168 130 154)(117 155 131 141)(118 142 132 156)(119 157 133 143)(120 144 134 158)(121 159 135 145)(122 146 136 160)(123 161 137 147)(124 148 138 162)(125 163 139 149)(126 150 140 164)(169 204 183 218)(170 219 184 205)(171 206 185 220)(172 221 186 207)(173 208 187 222)(174 223 188 209)(175 210 189 224)(176 197 190 211)(177 212 191 198)(178 199 192 213)(179 214 193 200)(180 201 194 215)(181 216 195 202)(182 203 196 217)
(1 77 162 191)(2 192 163 78)(3 79 164 193)(4 194 165 80)(5 81 166 195)(6 196 167 82)(7 83 168 169)(8 170 141 84)(9 57 142 171)(10 172 143 58)(11 59 144 173)(12 174 145 60)(13 61 146 175)(14 176 147 62)(15 63 148 177)(16 178 149 64)(17 65 150 179)(18 180 151 66)(19 67 152 181)(20 182 153 68)(21 69 154 183)(22 184 155 70)(23 71 156 185)(24 186 157 72)(25 73 158 187)(26 188 159 74)(27 75 160 189)(28 190 161 76)(29 213 139 111)(30 112 140 214)(31 215 113 85)(32 86 114 216)(33 217 115 87)(34 88 116 218)(35 219 117 89)(36 90 118 220)(37 221 119 91)(38 92 120 222)(39 223 121 93)(40 94 122 224)(41 197 123 95)(42 96 124 198)(43 199 125 97)(44 98 126 200)(45 201 127 99)(46 100 128 202)(47 203 129 101)(48 102 130 204)(49 205 131 103)(50 104 132 206)(51 207 133 105)(52 106 134 208)(53 209 135 107)(54 108 136 210)(55 211 137 109)(56 110 138 212)
(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)
(1 197 15 211)(2 210 16 224)(3 223 17 209)(4 208 18 222)(5 221 19 207)(6 206 20 220)(7 219 21 205)(8 204 22 218)(9 217 23 203)(10 202 24 216)(11 215 25 201)(12 200 26 214)(13 213 27 199)(14 198 28 212)(29 61 43 75)(30 74 44 60)(31 59 45 73)(32 72 46 58)(33 57 47 71)(34 70 48 84)(35 83 49 69)(36 68 50 82)(37 81 51 67)(38 66 52 80)(39 79 53 65)(40 64 54 78)(41 77 55 63)(42 62 56 76)(85 158 99 144)(86 143 100 157)(87 156 101 142)(88 141 102 155)(89 154 103 168)(90 167 104 153)(91 152 105 166)(92 165 106 151)(93 150 107 164)(94 163 108 149)(95 148 109 162)(96 161 110 147)(97 146 111 160)(98 159 112 145)(113 173 127 187)(114 186 128 172)(115 171 129 185)(116 184 130 170)(117 169 131 183)(118 182 132 196)(119 195 133 181)(120 180 134 194)(121 193 135 179)(122 178 136 192)(123 191 137 177)(124 176 138 190)(125 189 139 175)(126 174 140 188)
G:=sub<Sym(224)| (1,42,15,56)(2,29,16,43)(3,44,17,30)(4,31,18,45)(5,46,19,32)(6,33,20,47)(7,48,21,34)(8,35,22,49)(9,50,23,36)(10,37,24,51)(11,52,25,38)(12,39,26,53)(13,54,27,40)(14,41,28,55)(57,104,71,90)(58,91,72,105)(59,106,73,92)(60,93,74,107)(61,108,75,94)(62,95,76,109)(63,110,77,96)(64,97,78,111)(65,112,79,98)(66,99,80,85)(67,86,81,100)(68,101,82,87)(69,88,83,102)(70,103,84,89)(113,151,127,165)(114,166,128,152)(115,153,129,167)(116,168,130,154)(117,155,131,141)(118,142,132,156)(119,157,133,143)(120,144,134,158)(121,159,135,145)(122,146,136,160)(123,161,137,147)(124,148,138,162)(125,163,139,149)(126,150,140,164)(169,204,183,218)(170,219,184,205)(171,206,185,220)(172,221,186,207)(173,208,187,222)(174,223,188,209)(175,210,189,224)(176,197,190,211)(177,212,191,198)(178,199,192,213)(179,214,193,200)(180,201,194,215)(181,216,195,202)(182,203,196,217), (1,77,162,191)(2,192,163,78)(3,79,164,193)(4,194,165,80)(5,81,166,195)(6,196,167,82)(7,83,168,169)(8,170,141,84)(9,57,142,171)(10,172,143,58)(11,59,144,173)(12,174,145,60)(13,61,146,175)(14,176,147,62)(15,63,148,177)(16,178,149,64)(17,65,150,179)(18,180,151,66)(19,67,152,181)(20,182,153,68)(21,69,154,183)(22,184,155,70)(23,71,156,185)(24,186,157,72)(25,73,158,187)(26,188,159,74)(27,75,160,189)(28,190,161,76)(29,213,139,111)(30,112,140,214)(31,215,113,85)(32,86,114,216)(33,217,115,87)(34,88,116,218)(35,219,117,89)(36,90,118,220)(37,221,119,91)(38,92,120,222)(39,223,121,93)(40,94,122,224)(41,197,123,95)(42,96,124,198)(43,199,125,97)(44,98,126,200)(45,201,127,99)(46,100,128,202)(47,203,129,101)(48,102,130,204)(49,205,131,103)(50,104,132,206)(51,207,133,105)(52,106,134,208)(53,209,135,107)(54,108,136,210)(55,211,137,109)(56,110,138,212), (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), (1,197,15,211)(2,210,16,224)(3,223,17,209)(4,208,18,222)(5,221,19,207)(6,206,20,220)(7,219,21,205)(8,204,22,218)(9,217,23,203)(10,202,24,216)(11,215,25,201)(12,200,26,214)(13,213,27,199)(14,198,28,212)(29,61,43,75)(30,74,44,60)(31,59,45,73)(32,72,46,58)(33,57,47,71)(34,70,48,84)(35,83,49,69)(36,68,50,82)(37,81,51,67)(38,66,52,80)(39,79,53,65)(40,64,54,78)(41,77,55,63)(42,62,56,76)(85,158,99,144)(86,143,100,157)(87,156,101,142)(88,141,102,155)(89,154,103,168)(90,167,104,153)(91,152,105,166)(92,165,106,151)(93,150,107,164)(94,163,108,149)(95,148,109,162)(96,161,110,147)(97,146,111,160)(98,159,112,145)(113,173,127,187)(114,186,128,172)(115,171,129,185)(116,184,130,170)(117,169,131,183)(118,182,132,196)(119,195,133,181)(120,180,134,194)(121,193,135,179)(122,178,136,192)(123,191,137,177)(124,176,138,190)(125,189,139,175)(126,174,140,188)>;
G:=Group( (1,42,15,56)(2,29,16,43)(3,44,17,30)(4,31,18,45)(5,46,19,32)(6,33,20,47)(7,48,21,34)(8,35,22,49)(9,50,23,36)(10,37,24,51)(11,52,25,38)(12,39,26,53)(13,54,27,40)(14,41,28,55)(57,104,71,90)(58,91,72,105)(59,106,73,92)(60,93,74,107)(61,108,75,94)(62,95,76,109)(63,110,77,96)(64,97,78,111)(65,112,79,98)(66,99,80,85)(67,86,81,100)(68,101,82,87)(69,88,83,102)(70,103,84,89)(113,151,127,165)(114,166,128,152)(115,153,129,167)(116,168,130,154)(117,155,131,141)(118,142,132,156)(119,157,133,143)(120,144,134,158)(121,159,135,145)(122,146,136,160)(123,161,137,147)(124,148,138,162)(125,163,139,149)(126,150,140,164)(169,204,183,218)(170,219,184,205)(171,206,185,220)(172,221,186,207)(173,208,187,222)(174,223,188,209)(175,210,189,224)(176,197,190,211)(177,212,191,198)(178,199,192,213)(179,214,193,200)(180,201,194,215)(181,216,195,202)(182,203,196,217), (1,77,162,191)(2,192,163,78)(3,79,164,193)(4,194,165,80)(5,81,166,195)(6,196,167,82)(7,83,168,169)(8,170,141,84)(9,57,142,171)(10,172,143,58)(11,59,144,173)(12,174,145,60)(13,61,146,175)(14,176,147,62)(15,63,148,177)(16,178,149,64)(17,65,150,179)(18,180,151,66)(19,67,152,181)(20,182,153,68)(21,69,154,183)(22,184,155,70)(23,71,156,185)(24,186,157,72)(25,73,158,187)(26,188,159,74)(27,75,160,189)(28,190,161,76)(29,213,139,111)(30,112,140,214)(31,215,113,85)(32,86,114,216)(33,217,115,87)(34,88,116,218)(35,219,117,89)(36,90,118,220)(37,221,119,91)(38,92,120,222)(39,223,121,93)(40,94,122,224)(41,197,123,95)(42,96,124,198)(43,199,125,97)(44,98,126,200)(45,201,127,99)(46,100,128,202)(47,203,129,101)(48,102,130,204)(49,205,131,103)(50,104,132,206)(51,207,133,105)(52,106,134,208)(53,209,135,107)(54,108,136,210)(55,211,137,109)(56,110,138,212), (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), (1,197,15,211)(2,210,16,224)(3,223,17,209)(4,208,18,222)(5,221,19,207)(6,206,20,220)(7,219,21,205)(8,204,22,218)(9,217,23,203)(10,202,24,216)(11,215,25,201)(12,200,26,214)(13,213,27,199)(14,198,28,212)(29,61,43,75)(30,74,44,60)(31,59,45,73)(32,72,46,58)(33,57,47,71)(34,70,48,84)(35,83,49,69)(36,68,50,82)(37,81,51,67)(38,66,52,80)(39,79,53,65)(40,64,54,78)(41,77,55,63)(42,62,56,76)(85,158,99,144)(86,143,100,157)(87,156,101,142)(88,141,102,155)(89,154,103,168)(90,167,104,153)(91,152,105,166)(92,165,106,151)(93,150,107,164)(94,163,108,149)(95,148,109,162)(96,161,110,147)(97,146,111,160)(98,159,112,145)(113,173,127,187)(114,186,128,172)(115,171,129,185)(116,184,130,170)(117,169,131,183)(118,182,132,196)(119,195,133,181)(120,180,134,194)(121,193,135,179)(122,178,136,192)(123,191,137,177)(124,176,138,190)(125,189,139,175)(126,174,140,188) );
G=PermutationGroup([(1,42,15,56),(2,29,16,43),(3,44,17,30),(4,31,18,45),(5,46,19,32),(6,33,20,47),(7,48,21,34),(8,35,22,49),(9,50,23,36),(10,37,24,51),(11,52,25,38),(12,39,26,53),(13,54,27,40),(14,41,28,55),(57,104,71,90),(58,91,72,105),(59,106,73,92),(60,93,74,107),(61,108,75,94),(62,95,76,109),(63,110,77,96),(64,97,78,111),(65,112,79,98),(66,99,80,85),(67,86,81,100),(68,101,82,87),(69,88,83,102),(70,103,84,89),(113,151,127,165),(114,166,128,152),(115,153,129,167),(116,168,130,154),(117,155,131,141),(118,142,132,156),(119,157,133,143),(120,144,134,158),(121,159,135,145),(122,146,136,160),(123,161,137,147),(124,148,138,162),(125,163,139,149),(126,150,140,164),(169,204,183,218),(170,219,184,205),(171,206,185,220),(172,221,186,207),(173,208,187,222),(174,223,188,209),(175,210,189,224),(176,197,190,211),(177,212,191,198),(178,199,192,213),(179,214,193,200),(180,201,194,215),(181,216,195,202),(182,203,196,217)], [(1,77,162,191),(2,192,163,78),(3,79,164,193),(4,194,165,80),(5,81,166,195),(6,196,167,82),(7,83,168,169),(8,170,141,84),(9,57,142,171),(10,172,143,58),(11,59,144,173),(12,174,145,60),(13,61,146,175),(14,176,147,62),(15,63,148,177),(16,178,149,64),(17,65,150,179),(18,180,151,66),(19,67,152,181),(20,182,153,68),(21,69,154,183),(22,184,155,70),(23,71,156,185),(24,186,157,72),(25,73,158,187),(26,188,159,74),(27,75,160,189),(28,190,161,76),(29,213,139,111),(30,112,140,214),(31,215,113,85),(32,86,114,216),(33,217,115,87),(34,88,116,218),(35,219,117,89),(36,90,118,220),(37,221,119,91),(38,92,120,222),(39,223,121,93),(40,94,122,224),(41,197,123,95),(42,96,124,198),(43,199,125,97),(44,98,126,200),(45,201,127,99),(46,100,128,202),(47,203,129,101),(48,102,130,204),(49,205,131,103),(50,104,132,206),(51,207,133,105),(52,106,134,208),(53,209,135,107),(54,108,136,210),(55,211,137,109),(56,110,138,212)], [(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)], [(1,197,15,211),(2,210,16,224),(3,223,17,209),(4,208,18,222),(5,221,19,207),(6,206,20,220),(7,219,21,205),(8,204,22,218),(9,217,23,203),(10,202,24,216),(11,215,25,201),(12,200,26,214),(13,213,27,199),(14,198,28,212),(29,61,43,75),(30,74,44,60),(31,59,45,73),(32,72,46,58),(33,57,47,71),(34,70,48,84),(35,83,49,69),(36,68,50,82),(37,81,51,67),(38,66,52,80),(39,79,53,65),(40,64,54,78),(41,77,55,63),(42,62,56,76),(85,158,99,144),(86,143,100,157),(87,156,101,142),(88,141,102,155),(89,154,103,168),(90,167,104,153),(91,152,105,166),(92,165,106,151),(93,150,107,164),(94,163,108,149),(95,148,109,162),(96,161,110,147),(97,146,111,160),(98,159,112,145),(113,173,127,187),(114,186,128,172),(115,171,129,185),(116,184,130,170),(117,169,131,183),(118,182,132,196),(119,195,133,181),(120,180,134,194),(121,193,135,179),(122,178,136,192),(123,191,137,177),(124,176,138,190),(125,189,139,175),(126,174,140,188)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 28 | 0 | 10 | 13 | 0 | 0 | 0 | 0 |
| 0 | 28 | 26 | 13 | 0 | 0 | 0 | 0 |
| 11 | 18 | 1 | 0 | 0 | 0 | 0 | 0 |
| 7 | 4 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 18 | 27 |
| 0 | 0 | 0 | 0 | 25 | 28 | 2 | 11 |
| 1 | 0 | 19 | 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 3 | 16 | 0 | 0 | 0 | 0 |
| 18 | 11 | 28 | 0 | 0 | 0 | 0 | 0 |
| 22 | 25 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 15 | 0 | 11 |
| 0 | 0 | 0 | 0 | 21 | 18 | 18 | 14 |
| 0 | 0 | 0 | 0 | 6 | 16 | 9 | 14 |
| 0 | 0 | 0 | 0 | 13 | 0 | 15 | 20 |
| 23 | 3 | 11 | 10 | 0 | 0 | 0 | 0 |
| 27 | 25 | 8 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 22 | 8 | 9 |
| 0 | 0 | 0 | 0 | 28 | 16 | 18 | 1 |
| 0 | 0 | 0 | 0 | 18 | 19 | 12 | 7 |
| 0 | 0 | 0 | 0 | 27 | 0 | 26 | 12 |
| 25 | 26 | 2 | 4 | 0 | 0 | 0 | 0 |
| 5 | 4 | 20 | 28 | 0 | 0 | 0 | 0 |
| 12 | 19 | 12 | 16 | 0 | 0 | 0 | 0 |
| 8 | 5 | 11 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 12 | 26 | 20 |
| 0 | 0 | 0 | 0 | 19 | 2 | 3 | 27 |
| 0 | 0 | 0 | 0 | 4 | 22 | 13 | 8 |
| 0 | 0 | 0 | 0 | 27 | 18 | 23 | 26 |
G:=sub<GL(8,GF(29))| [28,0,11,7,0,0,0,0,0,28,18,4,0,0,0,0,10,26,1,0,0,0,0,0,13,13,0,1,0,0,0,0,0,0,0,0,3,4,1,25,0,0,0,0,27,26,0,28,0,0,0,0,0,0,18,2,0,0,0,0,0,0,27,11],[1,0,18,22,0,0,0,0,0,1,11,25,0,0,0,0,19,3,28,0,0,0,0,0,16,16,0,28,0,0,0,0,0,0,0,0,11,21,6,13,0,0,0,0,15,18,16,0,0,0,0,0,0,18,9,15,0,0,0,0,11,14,14,20],[23,27,0,0,0,0,0,0,3,25,0,0,0,0,0,0,11,8,27,18,0,0,0,0,10,20,26,12,0,0,0,0,0,0,0,0,18,28,18,27,0,0,0,0,22,16,19,0,0,0,0,0,8,18,12,26,0,0,0,0,9,1,7,12],[25,5,12,8,0,0,0,0,26,4,19,5,0,0,0,0,2,20,12,11,0,0,0,0,4,28,16,17,0,0,0,0,0,0,0,0,17,19,4,27,0,0,0,0,12,2,22,18,0,0,0,0,26,3,13,23,0,0,0,0,20,27,8,26] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 28 | 28 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 14 | 14 | 28 | 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 | D7 | C4○D4 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | Q8⋊2D7 | D4⋊6D14 | Q8.10D14 |
| kernel | C42.178D14 | C42⋊D7 | C4.D28 | Dic7⋊3Q8 | D28⋊C4 | D14.5D4 | C28⋊1D4 | D14⋊2Q8 | C4⋊C4⋊D7 | D14⋊3Q8 | C28.23D4 | C7×C4⋊Q8 | C4⋊Q8 | C28 | C42 | C4⋊C4 | C2×Q8 | C14 | C14 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 3 | 4 | 3 | 12 | 6 | 1 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_4^2._{178}D_{14} % in TeX
G:=Group("C4^2.178D14"); // GroupNames label
G:=SmallGroup(448,1186);
// by ID
G=gap.SmallGroup(448,1186);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,758,219,100,675,570,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,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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