metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.117D14, C14.1062+ (1+4), (C4×D4)⋊25D7, (D4×C28)⋊27C2, (C4×D28)⋊34C2, C28⋊2D4⋊11C2, C28⋊7D4⋊20C2, C4⋊C4.320D14, C28⋊2Q8⋊26C2, (C2×D4).224D14, C4.66(C4○D28), C28.114(C4○D4), (C4×C28).161C22, (C2×C28).165C23, (C2×C14).107C24, D14⋊C4.55C22, C22⋊C4.119D14, (C22×C4).215D14, C4.118(D4⋊2D7), C2.19(D4⋊8D14), Dic7.D4⋊11C2, (C2×D28).214C22, (D4×C14).266C22, C23.21D14⋊9C2, C4⋊Dic7.302C22, (C22×D7).41C23, C22.132(C23×D7), C23.104(C22×D7), (C22×C28).111C22, (C22×C14).177C23, C7⋊2(C22.49C24), (C2×Dic14).28C22, (C2×Dic7).209C23, (C4×Dic7).206C22, C23.D7.108C22, C4⋊C4⋊7D7⋊16C2, C2.56(C2×C4○D28), C14.49(C2×C4○D4), (C2×C4×D7).68C22, C2.24(C2×D4⋊2D7), (C7×C4⋊C4).335C22, (C2×C4).163(C22×D7), (C2×C7⋊D4).20C22, (C7×C22⋊C4).130C22, SmallGroup(448,1016)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1076 in 236 conjugacy classes, 99 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×9], C22, C22 [×12], C7, C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×8], Q8 [×2], C23 [×2], C23 [×2], D7 [×2], C14 [×3], C14 [×2], C42, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×5], C2×Q8 [×2], Dic7 [×6], C28 [×4], C28 [×3], D14 [×6], C2×C14, C2×C14 [×6], C42⋊C2 [×4], C4×D4, C4×D4, C4⋊D4 [×4], C4.4D4 [×4], C4⋊Q8, Dic14 [×2], C4×D7 [×4], D28 [×2], C2×Dic7 [×6], C7⋊D4 [×4], C2×C28 [×3], C2×C28 [×2], C2×C28 [×4], C7×D4 [×2], C22×D7 [×2], C22×C14 [×2], C22.49C24, C4×Dic7 [×4], C4⋊Dic7, C4⋊Dic7 [×4], D14⋊C4 [×6], C23.D7 [×4], C4×C28, C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14 [×2], C2×C4×D7 [×2], C2×D28, C2×C7⋊D4 [×4], C22×C28 [×2], D4×C14, C28⋊2Q8, C4×D28, Dic7.D4 [×4], C4⋊C4⋊7D7 [×2], C23.21D14 [×2], C28⋊7D4 [×2], C28⋊2D4 [×2], D4×C28, C42.117D14
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, C4○D28 [×2], D4⋊2D7 [×2], C23×D7, C2×C4○D28, C2×D4⋊2D7, D4⋊8D14, C42.117D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=1, d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=a2b2c-1 >
►On 224 points
Generators in S224
(1 210 93 176)(2 177 94 197)(3 198 95 178)(4 179 96 199)(5 200 97 180)(6 181 98 201)(7 202 85 182)(8 169 86 203)(9 204 87 170)(10 171 88 205)(11 206 89 172)(12 173 90 207)(13 208 91 174)(14 175 92 209)(15 32 49 193)(16 194 50 33)(17 34 51 195)(18 196 52 35)(19 36 53 183)(20 184 54 37)(21 38 55 185)(22 186 56 39)(23 40 43 187)(24 188 44 41)(25 42 45 189)(26 190 46 29)(27 30 47 191)(28 192 48 31)(57 84 168 100)(58 101 155 71)(59 72 156 102)(60 103 157 73)(61 74 158 104)(62 105 159 75)(63 76 160 106)(64 107 161 77)(65 78 162 108)(66 109 163 79)(67 80 164 110)(68 111 165 81)(69 82 166 112)(70 99 167 83)(113 140 147 215)(114 216 148 127)(115 128 149 217)(116 218 150 129)(117 130 151 219)(118 220 152 131)(119 132 153 221)(120 222 154 133)(121 134 141 223)(122 224 142 135)(123 136 143 211)(124 212 144 137)(125 138 145 213)(126 214 146 139)
(1 189 131 101)(2 190 132 102)(3 191 133 103)(4 192 134 104)(5 193 135 105)(6 194 136 106)(7 195 137 107)(8 196 138 108)(9 183 139 109)(10 184 140 110)(11 185 127 111)(12 186 128 112)(13 187 129 99)(14 188 130 100)(15 122 159 200)(16 123 160 201)(17 124 161 202)(18 125 162 203)(19 126 163 204)(20 113 164 205)(21 114 165 206)(22 115 166 207)(23 116 167 208)(24 117 168 209)(25 118 155 210)(26 119 156 197)(27 120 157 198)(28 121 158 199)(29 221 72 94)(30 222 73 95)(31 223 74 96)(32 224 75 97)(33 211 76 98)(34 212 77 85)(35 213 78 86)(36 214 79 87)(37 215 80 88)(38 216 81 89)(39 217 82 90)(40 218 83 91)(41 219 84 92)(42 220 71 93)(43 150 70 174)(44 151 57 175)(45 152 58 176)(46 153 59 177)(47 154 60 178)(48 141 61 179)(49 142 62 180)(50 143 63 181)(51 144 64 182)(52 145 65 169)(53 146 66 170)(54 147 67 171)(55 148 68 172)(56 149 69 173)
(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)
(1 100 220 41)(2 40 221 99)(3 112 222 39)(4 38 223 111)(5 110 224 37)(6 36 211 109)(7 108 212 35)(8 34 213 107)(9 106 214 33)(10 32 215 105)(11 104 216 31)(12 30 217 103)(13 102 218 29)(14 42 219 101)(15 113 62 171)(16 170 63 126)(17 125 64 169)(18 182 65 124)(19 123 66 181)(20 180 67 122)(21 121 68 179)(22 178 69 120)(23 119 70 177)(24 176 57 118)(25 117 58 175)(26 174 59 116)(27 115 60 173)(28 172 61 114)(43 153 167 197)(44 210 168 152)(45 151 155 209)(46 208 156 150)(47 149 157 207)(48 206 158 148)(49 147 159 205)(50 204 160 146)(51 145 161 203)(52 202 162 144)(53 143 163 201)(54 200 164 142)(55 141 165 199)(56 198 166 154)(71 92 189 130)(72 129 190 91)(73 90 191 128)(74 127 192 89)(75 88 193 140)(76 139 194 87)(77 86 195 138)(78 137 196 85)(79 98 183 136)(80 135 184 97)(81 96 185 134)(82 133 186 95)(83 94 187 132)(84 131 188 93)
G:=sub<Sym(224)| (1,210,93,176)(2,177,94,197)(3,198,95,178)(4,179,96,199)(5,200,97,180)(6,181,98,201)(7,202,85,182)(8,169,86,203)(9,204,87,170)(10,171,88,205)(11,206,89,172)(12,173,90,207)(13,208,91,174)(14,175,92,209)(15,32,49,193)(16,194,50,33)(17,34,51,195)(18,196,52,35)(19,36,53,183)(20,184,54,37)(21,38,55,185)(22,186,56,39)(23,40,43,187)(24,188,44,41)(25,42,45,189)(26,190,46,29)(27,30,47,191)(28,192,48,31)(57,84,168,100)(58,101,155,71)(59,72,156,102)(60,103,157,73)(61,74,158,104)(62,105,159,75)(63,76,160,106)(64,107,161,77)(65,78,162,108)(66,109,163,79)(67,80,164,110)(68,111,165,81)(69,82,166,112)(70,99,167,83)(113,140,147,215)(114,216,148,127)(115,128,149,217)(116,218,150,129)(117,130,151,219)(118,220,152,131)(119,132,153,221)(120,222,154,133)(121,134,141,223)(122,224,142,135)(123,136,143,211)(124,212,144,137)(125,138,145,213)(126,214,146,139), (1,189,131,101)(2,190,132,102)(3,191,133,103)(4,192,134,104)(5,193,135,105)(6,194,136,106)(7,195,137,107)(8,196,138,108)(9,183,139,109)(10,184,140,110)(11,185,127,111)(12,186,128,112)(13,187,129,99)(14,188,130,100)(15,122,159,200)(16,123,160,201)(17,124,161,202)(18,125,162,203)(19,126,163,204)(20,113,164,205)(21,114,165,206)(22,115,166,207)(23,116,167,208)(24,117,168,209)(25,118,155,210)(26,119,156,197)(27,120,157,198)(28,121,158,199)(29,221,72,94)(30,222,73,95)(31,223,74,96)(32,224,75,97)(33,211,76,98)(34,212,77,85)(35,213,78,86)(36,214,79,87)(37,215,80,88)(38,216,81,89)(39,217,82,90)(40,218,83,91)(41,219,84,92)(42,220,71,93)(43,150,70,174)(44,151,57,175)(45,152,58,176)(46,153,59,177)(47,154,60,178)(48,141,61,179)(49,142,62,180)(50,143,63,181)(51,144,64,182)(52,145,65,169)(53,146,66,170)(54,147,67,171)(55,148,68,172)(56,149,69,173), (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), (1,100,220,41)(2,40,221,99)(3,112,222,39)(4,38,223,111)(5,110,224,37)(6,36,211,109)(7,108,212,35)(8,34,213,107)(9,106,214,33)(10,32,215,105)(11,104,216,31)(12,30,217,103)(13,102,218,29)(14,42,219,101)(15,113,62,171)(16,170,63,126)(17,125,64,169)(18,182,65,124)(19,123,66,181)(20,180,67,122)(21,121,68,179)(22,178,69,120)(23,119,70,177)(24,176,57,118)(25,117,58,175)(26,174,59,116)(27,115,60,173)(28,172,61,114)(43,153,167,197)(44,210,168,152)(45,151,155,209)(46,208,156,150)(47,149,157,207)(48,206,158,148)(49,147,159,205)(50,204,160,146)(51,145,161,203)(52,202,162,144)(53,143,163,201)(54,200,164,142)(55,141,165,199)(56,198,166,154)(71,92,189,130)(72,129,190,91)(73,90,191,128)(74,127,192,89)(75,88,193,140)(76,139,194,87)(77,86,195,138)(78,137,196,85)(79,98,183,136)(80,135,184,97)(81,96,185,134)(82,133,186,95)(83,94,187,132)(84,131,188,93)>;
G:=Group( (1,210,93,176)(2,177,94,197)(3,198,95,178)(4,179,96,199)(5,200,97,180)(6,181,98,201)(7,202,85,182)(8,169,86,203)(9,204,87,170)(10,171,88,205)(11,206,89,172)(12,173,90,207)(13,208,91,174)(14,175,92,209)(15,32,49,193)(16,194,50,33)(17,34,51,195)(18,196,52,35)(19,36,53,183)(20,184,54,37)(21,38,55,185)(22,186,56,39)(23,40,43,187)(24,188,44,41)(25,42,45,189)(26,190,46,29)(27,30,47,191)(28,192,48,31)(57,84,168,100)(58,101,155,71)(59,72,156,102)(60,103,157,73)(61,74,158,104)(62,105,159,75)(63,76,160,106)(64,107,161,77)(65,78,162,108)(66,109,163,79)(67,80,164,110)(68,111,165,81)(69,82,166,112)(70,99,167,83)(113,140,147,215)(114,216,148,127)(115,128,149,217)(116,218,150,129)(117,130,151,219)(118,220,152,131)(119,132,153,221)(120,222,154,133)(121,134,141,223)(122,224,142,135)(123,136,143,211)(124,212,144,137)(125,138,145,213)(126,214,146,139), (1,189,131,101)(2,190,132,102)(3,191,133,103)(4,192,134,104)(5,193,135,105)(6,194,136,106)(7,195,137,107)(8,196,138,108)(9,183,139,109)(10,184,140,110)(11,185,127,111)(12,186,128,112)(13,187,129,99)(14,188,130,100)(15,122,159,200)(16,123,160,201)(17,124,161,202)(18,125,162,203)(19,126,163,204)(20,113,164,205)(21,114,165,206)(22,115,166,207)(23,116,167,208)(24,117,168,209)(25,118,155,210)(26,119,156,197)(27,120,157,198)(28,121,158,199)(29,221,72,94)(30,222,73,95)(31,223,74,96)(32,224,75,97)(33,211,76,98)(34,212,77,85)(35,213,78,86)(36,214,79,87)(37,215,80,88)(38,216,81,89)(39,217,82,90)(40,218,83,91)(41,219,84,92)(42,220,71,93)(43,150,70,174)(44,151,57,175)(45,152,58,176)(46,153,59,177)(47,154,60,178)(48,141,61,179)(49,142,62,180)(50,143,63,181)(51,144,64,182)(52,145,65,169)(53,146,66,170)(54,147,67,171)(55,148,68,172)(56,149,69,173), (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), (1,100,220,41)(2,40,221,99)(3,112,222,39)(4,38,223,111)(5,110,224,37)(6,36,211,109)(7,108,212,35)(8,34,213,107)(9,106,214,33)(10,32,215,105)(11,104,216,31)(12,30,217,103)(13,102,218,29)(14,42,219,101)(15,113,62,171)(16,170,63,126)(17,125,64,169)(18,182,65,124)(19,123,66,181)(20,180,67,122)(21,121,68,179)(22,178,69,120)(23,119,70,177)(24,176,57,118)(25,117,58,175)(26,174,59,116)(27,115,60,173)(28,172,61,114)(43,153,167,197)(44,210,168,152)(45,151,155,209)(46,208,156,150)(47,149,157,207)(48,206,158,148)(49,147,159,205)(50,204,160,146)(51,145,161,203)(52,202,162,144)(53,143,163,201)(54,200,164,142)(55,141,165,199)(56,198,166,154)(71,92,189,130)(72,129,190,91)(73,90,191,128)(74,127,192,89)(75,88,193,140)(76,139,194,87)(77,86,195,138)(78,137,196,85)(79,98,183,136)(80,135,184,97)(81,96,185,134)(82,133,186,95)(83,94,187,132)(84,131,188,93) );
G=PermutationGroup([(1,210,93,176),(2,177,94,197),(3,198,95,178),(4,179,96,199),(5,200,97,180),(6,181,98,201),(7,202,85,182),(8,169,86,203),(9,204,87,170),(10,171,88,205),(11,206,89,172),(12,173,90,207),(13,208,91,174),(14,175,92,209),(15,32,49,193),(16,194,50,33),(17,34,51,195),(18,196,52,35),(19,36,53,183),(20,184,54,37),(21,38,55,185),(22,186,56,39),(23,40,43,187),(24,188,44,41),(25,42,45,189),(26,190,46,29),(27,30,47,191),(28,192,48,31),(57,84,168,100),(58,101,155,71),(59,72,156,102),(60,103,157,73),(61,74,158,104),(62,105,159,75),(63,76,160,106),(64,107,161,77),(65,78,162,108),(66,109,163,79),(67,80,164,110),(68,111,165,81),(69,82,166,112),(70,99,167,83),(113,140,147,215),(114,216,148,127),(115,128,149,217),(116,218,150,129),(117,130,151,219),(118,220,152,131),(119,132,153,221),(120,222,154,133),(121,134,141,223),(122,224,142,135),(123,136,143,211),(124,212,144,137),(125,138,145,213),(126,214,146,139)], [(1,189,131,101),(2,190,132,102),(3,191,133,103),(4,192,134,104),(5,193,135,105),(6,194,136,106),(7,195,137,107),(8,196,138,108),(9,183,139,109),(10,184,140,110),(11,185,127,111),(12,186,128,112),(13,187,129,99),(14,188,130,100),(15,122,159,200),(16,123,160,201),(17,124,161,202),(18,125,162,203),(19,126,163,204),(20,113,164,205),(21,114,165,206),(22,115,166,207),(23,116,167,208),(24,117,168,209),(25,118,155,210),(26,119,156,197),(27,120,157,198),(28,121,158,199),(29,221,72,94),(30,222,73,95),(31,223,74,96),(32,224,75,97),(33,211,76,98),(34,212,77,85),(35,213,78,86),(36,214,79,87),(37,215,80,88),(38,216,81,89),(39,217,82,90),(40,218,83,91),(41,219,84,92),(42,220,71,93),(43,150,70,174),(44,151,57,175),(45,152,58,176),(46,153,59,177),(47,154,60,178),(48,141,61,179),(49,142,62,180),(50,143,63,181),(51,144,64,182),(52,145,65,169),(53,146,66,170),(54,147,67,171),(55,148,68,172),(56,149,69,173)], [(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)], [(1,100,220,41),(2,40,221,99),(3,112,222,39),(4,38,223,111),(5,110,224,37),(6,36,211,109),(7,108,212,35),(8,34,213,107),(9,106,214,33),(10,32,215,105),(11,104,216,31),(12,30,217,103),(13,102,218,29),(14,42,219,101),(15,113,62,171),(16,170,63,126),(17,125,64,169),(18,182,65,124),(19,123,66,181),(20,180,67,122),(21,121,68,179),(22,178,69,120),(23,119,70,177),(24,176,57,118),(25,117,58,175),(26,174,59,116),(27,115,60,173),(28,172,61,114),(43,153,167,197),(44,210,168,152),(45,151,155,209),(46,208,156,150),(47,149,157,207),(48,206,158,148),(49,147,159,205),(50,204,160,146),(51,145,161,203),(52,202,162,144),(53,143,163,201),(54,200,164,142),(55,141,165,199),(56,198,166,154),(71,92,189,130),(72,129,190,91),(73,90,191,128),(74,127,192,89),(75,88,193,140),(76,139,194,87),(77,86,195,138),(78,137,196,85),(79,98,183,136),(80,135,184,97),(81,96,185,134),(82,133,186,95),(83,94,187,132),(84,131,188,93)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 9 | 12 |
| 28 | 12 | 0 | 0 | 0 | 0 |
| 24 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 2 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 | 0 | 0 |
| 0 | 0 | 21 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 24 |
| 0 | 0 | 0 | 0 | 16 | 20 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 27 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 26 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 24 |
| 0 | 0 | 0 | 0 | 28 | 20 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,9,0,0,0,0,0,12],[28,24,0,0,0,0,12,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[12,2,0,0,0,0,1,17,0,0,0,0,0,0,8,21,0,0,0,0,8,3,0,0,0,0,0,0,9,16,0,0,0,0,24,20],[17,27,0,0,0,0,0,12,0,0,0,0,0,0,21,26,0,0,0,0,21,8,0,0,0,0,0,0,9,28,0,0,0,0,24,20] >;
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| 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 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | D14 | C4○D28 | 2+ (1+4) | D4⋊2D7 | D4⋊8D14 |
| kernel | C42.117D14 | C28⋊2Q8 | C4×D28 | Dic7.D4 | C4⋊C4⋊7D7 | C23.21D14 | C28⋊7D4 | C28⋊2D4 | D4×C28 | C4×D4 | C28 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C14 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 3 | 8 | 3 | 6 | 3 | 6 | 3 | 24 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_4^2._{117}D_{14} % in TeX
G:=Group("C4^2.117D14"); // GroupNames label
G:=SmallGroup(448,1016);
// by ID
G=gap.SmallGroup(448,1016);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,758,219,1571,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^-1>;
// generators/relations
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