direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4○D4×C2×C14, C14.23C25, C28.89C24, (C23×C4)⋊10C14, (C23×C28)⋊17C2, (C2×C28)⋊18C23, D4⋊3(C22×C14), (C7×D4)⋊14C23, C2.3(C24×C14), Q8⋊3(C22×C14), (C7×Q8)⋊13C23, (D4×C14)⋊70C22, (C22×D4)⋊13C14, C4.12(C23×C14), C24.37(C2×C14), (Q8×C14)⋊59C22, (C22×Q8)⋊11C14, (C2×C14).386C24, (C22×C28)⋊67C22, C22.1(C23×C14), C23.46(C22×C14), (C23×C14).94C22, (C22×C14).269C23, (D4×C2×C14)⋊28C2, (Q8×C2×C14)⋊23C2, (C2×D4)⋊19(C2×C14), (C2×C4)⋊5(C22×C14), (C2×Q8)⋊19(C2×C14), (C22×C4)⋊20(C2×C14), SmallGroup(448,1388)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1010 in 890 conjugacy classes, 770 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×16], C22 [×19], C22 [×36], C7, C2×C4 [×72], D4 [×48], Q8 [×16], C23, C23 [×18], C23 [×12], C14, C14 [×6], C14 [×12], C22×C4, C22×C4 [×39], C2×D4 [×36], C2×Q8 [×12], C4○D4 [×64], C24 [×3], C28 [×16], C2×C14 [×19], C2×C14 [×36], C23×C4 [×3], C22×D4 [×3], C22×Q8, C2×C4○D4 [×24], C2×C28 [×72], C7×D4 [×48], C7×Q8 [×16], C22×C14, C22×C14 [×18], C22×C14 [×12], C22×C4○D4, C22×C28, C22×C28 [×39], D4×C14 [×36], Q8×C14 [×12], C7×C4○D4 [×64], C23×C14 [×3], C23×C28 [×3], D4×C2×C14 [×3], Q8×C2×C14, C14×C4○D4 [×24], C4○D4×C2×C14
Quotients:
C1, C2 [×31], C22 [×155], C7, C23 [×155], C14 [×31], C4○D4 [×4], C24 [×31], C2×C14 [×155], C2×C4○D4 [×6], C25, C22×C14 [×155], C22×C4○D4, C7×C4○D4 [×4], C23×C14 [×31], C14×C4○D4 [×6], C24×C14, C4○D4×C2×C14
Generators and relations
G = < a,b,c,d,e | a2=b14=c4=e2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >
►On 224 points
Generators in S224
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 43)(14 44)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 204)(30 205)(31 206)(32 207)(33 208)(34 209)(35 210)(36 197)(37 198)(38 199)(39 200)(40 201)(41 202)(42 203)(57 124)(58 125)(59 126)(60 113)(61 114)(62 115)(63 116)(64 117)(65 118)(66 119)(67 120)(68 121)(69 122)(70 123)(71 181)(72 182)(73 169)(74 170)(75 171)(76 172)(77 173)(78 174)(79 175)(80 176)(81 177)(82 178)(83 179)(84 180)(85 130)(86 131)(87 132)(88 133)(89 134)(90 135)(91 136)(92 137)(93 138)(94 139)(95 140)(96 127)(97 128)(98 129)(141 211)(142 212)(143 213)(144 214)(145 215)(146 216)(147 217)(148 218)(149 219)(150 220)(151 221)(152 222)(153 223)(154 224)(155 186)(156 187)(157 188)(158 189)(159 190)(160 191)(161 192)(162 193)(163 194)(164 195)(165 196)(166 183)(167 184)(168 185)
(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 70 177 97)(2 57 178 98)(3 58 179 85)(4 59 180 86)(5 60 181 87)(6 61 182 88)(7 62 169 89)(8 63 170 90)(9 64 171 91)(10 65 172 92)(11 66 173 93)(12 67 174 94)(13 68 175 95)(14 69 176 96)(15 205 160 154)(16 206 161 141)(17 207 162 142)(18 208 163 143)(19 209 164 144)(20 210 165 145)(21 197 166 146)(22 198 167 147)(23 199 168 148)(24 200 155 149)(25 201 156 150)(26 202 157 151)(27 203 158 152)(28 204 159 153)(29 190 223 112)(30 191 224 99)(31 192 211 100)(32 193 212 101)(33 194 213 102)(34 195 214 103)(35 196 215 104)(36 183 216 105)(37 184 217 106)(38 185 218 107)(39 186 219 108)(40 187 220 109)(41 188 221 110)(42 189 222 111)(43 121 79 140)(44 122 80 127)(45 123 81 128)(46 124 82 129)(47 125 83 130)(48 126 84 131)(49 113 71 132)(50 114 72 133)(51 115 73 134)(52 116 74 135)(53 117 75 136)(54 118 76 137)(55 119 77 138)(56 120 78 139)
(1 193 177 101)(2 194 178 102)(3 195 179 103)(4 196 180 104)(5 183 181 105)(6 184 182 106)(7 185 169 107)(8 186 170 108)(9 187 171 109)(10 188 172 110)(11 189 173 111)(12 190 174 112)(13 191 175 99)(14 192 176 100)(15 43 160 79)(16 44 161 80)(17 45 162 81)(18 46 163 82)(19 47 164 83)(20 48 165 84)(21 49 166 71)(22 50 167 72)(23 51 168 73)(24 52 155 74)(25 53 156 75)(26 54 157 76)(27 55 158 77)(28 56 159 78)(29 67 223 94)(30 68 224 95)(31 69 211 96)(32 70 212 97)(33 57 213 98)(34 58 214 85)(35 59 215 86)(36 60 216 87)(37 61 217 88)(38 62 218 89)(39 63 219 90)(40 64 220 91)(41 65 221 92)(42 66 222 93)(113 146 132 197)(114 147 133 198)(115 148 134 199)(116 149 135 200)(117 150 136 201)(118 151 137 202)(119 152 138 203)(120 153 139 204)(121 154 140 205)(122 141 127 206)(123 142 128 207)(124 143 129 208)(125 144 130 209)(126 145 131 210)
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 71)(13 72)(14 73)(15 106)(16 107)(17 108)(18 109)(19 110)(20 111)(21 112)(22 99)(23 100)(24 101)(25 102)(26 103)(27 104)(28 105)(29 197)(30 198)(31 199)(32 200)(33 201)(34 202)(35 203)(36 204)(37 205)(38 206)(39 207)(40 208)(41 209)(42 210)(43 182)(44 169)(45 170)(46 171)(47 172)(48 173)(49 174)(50 175)(51 176)(52 177)(53 178)(54 179)(55 180)(56 181)(57 136)(58 137)(59 138)(60 139)(61 140)(62 127)(63 128)(64 129)(65 130)(66 131)(67 132)(68 133)(69 134)(70 135)(85 118)(86 119)(87 120)(88 121)(89 122)(90 123)(91 124)(92 125)(93 126)(94 113)(95 114)(96 115)(97 116)(98 117)(141 218)(142 219)(143 220)(144 221)(145 222)(146 223)(147 224)(148 211)(149 212)(150 213)(151 214)(152 215)(153 216)(154 217)(155 193)(156 194)(157 195)(158 196)(159 183)(160 184)(161 185)(162 186)(163 187)(164 188)(165 189)(166 190)(167 191)(168 192)
G:=sub<Sym(224)| (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,43)(14,44)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,204)(30,205)(31,206)(32,207)(33,208)(34,209)(35,210)(36,197)(37,198)(38,199)(39,200)(40,201)(41,202)(42,203)(57,124)(58,125)(59,126)(60,113)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,121)(69,122)(70,123)(71,181)(72,182)(73,169)(74,170)(75,171)(76,172)(77,173)(78,174)(79,175)(80,176)(81,177)(82,178)(83,179)(84,180)(85,130)(86,131)(87,132)(88,133)(89,134)(90,135)(91,136)(92,137)(93,138)(94,139)(95,140)(96,127)(97,128)(98,129)(141,211)(142,212)(143,213)(144,214)(145,215)(146,216)(147,217)(148,218)(149,219)(150,220)(151,221)(152,222)(153,223)(154,224)(155,186)(156,187)(157,188)(158,189)(159,190)(160,191)(161,192)(162,193)(163,194)(164,195)(165,196)(166,183)(167,184)(168,185), (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), (1,70,177,97)(2,57,178,98)(3,58,179,85)(4,59,180,86)(5,60,181,87)(6,61,182,88)(7,62,169,89)(8,63,170,90)(9,64,171,91)(10,65,172,92)(11,66,173,93)(12,67,174,94)(13,68,175,95)(14,69,176,96)(15,205,160,154)(16,206,161,141)(17,207,162,142)(18,208,163,143)(19,209,164,144)(20,210,165,145)(21,197,166,146)(22,198,167,147)(23,199,168,148)(24,200,155,149)(25,201,156,150)(26,202,157,151)(27,203,158,152)(28,204,159,153)(29,190,223,112)(30,191,224,99)(31,192,211,100)(32,193,212,101)(33,194,213,102)(34,195,214,103)(35,196,215,104)(36,183,216,105)(37,184,217,106)(38,185,218,107)(39,186,219,108)(40,187,220,109)(41,188,221,110)(42,189,222,111)(43,121,79,140)(44,122,80,127)(45,123,81,128)(46,124,82,129)(47,125,83,130)(48,126,84,131)(49,113,71,132)(50,114,72,133)(51,115,73,134)(52,116,74,135)(53,117,75,136)(54,118,76,137)(55,119,77,138)(56,120,78,139), (1,193,177,101)(2,194,178,102)(3,195,179,103)(4,196,180,104)(5,183,181,105)(6,184,182,106)(7,185,169,107)(8,186,170,108)(9,187,171,109)(10,188,172,110)(11,189,173,111)(12,190,174,112)(13,191,175,99)(14,192,176,100)(15,43,160,79)(16,44,161,80)(17,45,162,81)(18,46,163,82)(19,47,164,83)(20,48,165,84)(21,49,166,71)(22,50,167,72)(23,51,168,73)(24,52,155,74)(25,53,156,75)(26,54,157,76)(27,55,158,77)(28,56,159,78)(29,67,223,94)(30,68,224,95)(31,69,211,96)(32,70,212,97)(33,57,213,98)(34,58,214,85)(35,59,215,86)(36,60,216,87)(37,61,217,88)(38,62,218,89)(39,63,219,90)(40,64,220,91)(41,65,221,92)(42,66,222,93)(113,146,132,197)(114,147,133,198)(115,148,134,199)(116,149,135,200)(117,150,136,201)(118,151,137,202)(119,152,138,203)(120,153,139,204)(121,154,140,205)(122,141,127,206)(123,142,128,207)(124,143,129,208)(125,144,130,209)(126,145,131,210), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,71)(13,72)(14,73)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,197)(30,198)(31,199)(32,200)(33,201)(34,202)(35,203)(36,204)(37,205)(38,206)(39,207)(40,208)(41,209)(42,210)(43,182)(44,169)(45,170)(46,171)(47,172)(48,173)(49,174)(50,175)(51,176)(52,177)(53,178)(54,179)(55,180)(56,181)(57,136)(58,137)(59,138)(60,139)(61,140)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(85,118)(86,119)(87,120)(88,121)(89,122)(90,123)(91,124)(92,125)(93,126)(94,113)(95,114)(96,115)(97,116)(98,117)(141,218)(142,219)(143,220)(144,221)(145,222)(146,223)(147,224)(148,211)(149,212)(150,213)(151,214)(152,215)(153,216)(154,217)(155,193)(156,194)(157,195)(158,196)(159,183)(160,184)(161,185)(162,186)(163,187)(164,188)(165,189)(166,190)(167,191)(168,192)>;
G:=Group( (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,43)(14,44)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,204)(30,205)(31,206)(32,207)(33,208)(34,209)(35,210)(36,197)(37,198)(38,199)(39,200)(40,201)(41,202)(42,203)(57,124)(58,125)(59,126)(60,113)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,121)(69,122)(70,123)(71,181)(72,182)(73,169)(74,170)(75,171)(76,172)(77,173)(78,174)(79,175)(80,176)(81,177)(82,178)(83,179)(84,180)(85,130)(86,131)(87,132)(88,133)(89,134)(90,135)(91,136)(92,137)(93,138)(94,139)(95,140)(96,127)(97,128)(98,129)(141,211)(142,212)(143,213)(144,214)(145,215)(146,216)(147,217)(148,218)(149,219)(150,220)(151,221)(152,222)(153,223)(154,224)(155,186)(156,187)(157,188)(158,189)(159,190)(160,191)(161,192)(162,193)(163,194)(164,195)(165,196)(166,183)(167,184)(168,185), (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), (1,70,177,97)(2,57,178,98)(3,58,179,85)(4,59,180,86)(5,60,181,87)(6,61,182,88)(7,62,169,89)(8,63,170,90)(9,64,171,91)(10,65,172,92)(11,66,173,93)(12,67,174,94)(13,68,175,95)(14,69,176,96)(15,205,160,154)(16,206,161,141)(17,207,162,142)(18,208,163,143)(19,209,164,144)(20,210,165,145)(21,197,166,146)(22,198,167,147)(23,199,168,148)(24,200,155,149)(25,201,156,150)(26,202,157,151)(27,203,158,152)(28,204,159,153)(29,190,223,112)(30,191,224,99)(31,192,211,100)(32,193,212,101)(33,194,213,102)(34,195,214,103)(35,196,215,104)(36,183,216,105)(37,184,217,106)(38,185,218,107)(39,186,219,108)(40,187,220,109)(41,188,221,110)(42,189,222,111)(43,121,79,140)(44,122,80,127)(45,123,81,128)(46,124,82,129)(47,125,83,130)(48,126,84,131)(49,113,71,132)(50,114,72,133)(51,115,73,134)(52,116,74,135)(53,117,75,136)(54,118,76,137)(55,119,77,138)(56,120,78,139), (1,193,177,101)(2,194,178,102)(3,195,179,103)(4,196,180,104)(5,183,181,105)(6,184,182,106)(7,185,169,107)(8,186,170,108)(9,187,171,109)(10,188,172,110)(11,189,173,111)(12,190,174,112)(13,191,175,99)(14,192,176,100)(15,43,160,79)(16,44,161,80)(17,45,162,81)(18,46,163,82)(19,47,164,83)(20,48,165,84)(21,49,166,71)(22,50,167,72)(23,51,168,73)(24,52,155,74)(25,53,156,75)(26,54,157,76)(27,55,158,77)(28,56,159,78)(29,67,223,94)(30,68,224,95)(31,69,211,96)(32,70,212,97)(33,57,213,98)(34,58,214,85)(35,59,215,86)(36,60,216,87)(37,61,217,88)(38,62,218,89)(39,63,219,90)(40,64,220,91)(41,65,221,92)(42,66,222,93)(113,146,132,197)(114,147,133,198)(115,148,134,199)(116,149,135,200)(117,150,136,201)(118,151,137,202)(119,152,138,203)(120,153,139,204)(121,154,140,205)(122,141,127,206)(123,142,128,207)(124,143,129,208)(125,144,130,209)(126,145,131,210), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,71)(13,72)(14,73)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,197)(30,198)(31,199)(32,200)(33,201)(34,202)(35,203)(36,204)(37,205)(38,206)(39,207)(40,208)(41,209)(42,210)(43,182)(44,169)(45,170)(46,171)(47,172)(48,173)(49,174)(50,175)(51,176)(52,177)(53,178)(54,179)(55,180)(56,181)(57,136)(58,137)(59,138)(60,139)(61,140)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(85,118)(86,119)(87,120)(88,121)(89,122)(90,123)(91,124)(92,125)(93,126)(94,113)(95,114)(96,115)(97,116)(98,117)(141,218)(142,219)(143,220)(144,221)(145,222)(146,223)(147,224)(148,211)(149,212)(150,213)(151,214)(152,215)(153,216)(154,217)(155,193)(156,194)(157,195)(158,196)(159,183)(160,184)(161,185)(162,186)(163,187)(164,188)(165,189)(166,190)(167,191)(168,192) );
G=PermutationGroup([(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,43),(14,44),(15,99),(16,100),(17,101),(18,102),(19,103),(20,104),(21,105),(22,106),(23,107),(24,108),(25,109),(26,110),(27,111),(28,112),(29,204),(30,205),(31,206),(32,207),(33,208),(34,209),(35,210),(36,197),(37,198),(38,199),(39,200),(40,201),(41,202),(42,203),(57,124),(58,125),(59,126),(60,113),(61,114),(62,115),(63,116),(64,117),(65,118),(66,119),(67,120),(68,121),(69,122),(70,123),(71,181),(72,182),(73,169),(74,170),(75,171),(76,172),(77,173),(78,174),(79,175),(80,176),(81,177),(82,178),(83,179),(84,180),(85,130),(86,131),(87,132),(88,133),(89,134),(90,135),(91,136),(92,137),(93,138),(94,139),(95,140),(96,127),(97,128),(98,129),(141,211),(142,212),(143,213),(144,214),(145,215),(146,216),(147,217),(148,218),(149,219),(150,220),(151,221),(152,222),(153,223),(154,224),(155,186),(156,187),(157,188),(158,189),(159,190),(160,191),(161,192),(162,193),(163,194),(164,195),(165,196),(166,183),(167,184),(168,185)], [(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)], [(1,70,177,97),(2,57,178,98),(3,58,179,85),(4,59,180,86),(5,60,181,87),(6,61,182,88),(7,62,169,89),(8,63,170,90),(9,64,171,91),(10,65,172,92),(11,66,173,93),(12,67,174,94),(13,68,175,95),(14,69,176,96),(15,205,160,154),(16,206,161,141),(17,207,162,142),(18,208,163,143),(19,209,164,144),(20,210,165,145),(21,197,166,146),(22,198,167,147),(23,199,168,148),(24,200,155,149),(25,201,156,150),(26,202,157,151),(27,203,158,152),(28,204,159,153),(29,190,223,112),(30,191,224,99),(31,192,211,100),(32,193,212,101),(33,194,213,102),(34,195,214,103),(35,196,215,104),(36,183,216,105),(37,184,217,106),(38,185,218,107),(39,186,219,108),(40,187,220,109),(41,188,221,110),(42,189,222,111),(43,121,79,140),(44,122,80,127),(45,123,81,128),(46,124,82,129),(47,125,83,130),(48,126,84,131),(49,113,71,132),(50,114,72,133),(51,115,73,134),(52,116,74,135),(53,117,75,136),(54,118,76,137),(55,119,77,138),(56,120,78,139)], [(1,193,177,101),(2,194,178,102),(3,195,179,103),(4,196,180,104),(5,183,181,105),(6,184,182,106),(7,185,169,107),(8,186,170,108),(9,187,171,109),(10,188,172,110),(11,189,173,111),(12,190,174,112),(13,191,175,99),(14,192,176,100),(15,43,160,79),(16,44,161,80),(17,45,162,81),(18,46,163,82),(19,47,164,83),(20,48,165,84),(21,49,166,71),(22,50,167,72),(23,51,168,73),(24,52,155,74),(25,53,156,75),(26,54,157,76),(27,55,158,77),(28,56,159,78),(29,67,223,94),(30,68,224,95),(31,69,211,96),(32,70,212,97),(33,57,213,98),(34,58,214,85),(35,59,215,86),(36,60,216,87),(37,61,217,88),(38,62,218,89),(39,63,219,90),(40,64,220,91),(41,65,221,92),(42,66,222,93),(113,146,132,197),(114,147,133,198),(115,148,134,199),(116,149,135,200),(117,150,136,201),(118,151,137,202),(119,152,138,203),(120,153,139,204),(121,154,140,205),(122,141,127,206),(123,142,128,207),(124,143,129,208),(125,144,130,209),(126,145,131,210)], [(1,74),(2,75),(3,76),(4,77),(5,78),(6,79),(7,80),(8,81),(9,82),(10,83),(11,84),(12,71),(13,72),(14,73),(15,106),(16,107),(17,108),(18,109),(19,110),(20,111),(21,112),(22,99),(23,100),(24,101),(25,102),(26,103),(27,104),(28,105),(29,197),(30,198),(31,199),(32,200),(33,201),(34,202),(35,203),(36,204),(37,205),(38,206),(39,207),(40,208),(41,209),(42,210),(43,182),(44,169),(45,170),(46,171),(47,172),(48,173),(49,174),(50,175),(51,176),(52,177),(53,178),(54,179),(55,180),(56,181),(57,136),(58,137),(59,138),(60,139),(61,140),(62,127),(63,128),(64,129),(65,130),(66,131),(67,132),(68,133),(69,134),(70,135),(85,118),(86,119),(87,120),(88,121),(89,122),(90,123),(91,124),(92,125),(93,126),(94,113),(95,114),(96,115),(97,116),(98,117),(141,218),(142,219),(143,220),(144,221),(145,222),(146,223),(147,224),(148,211),(149,212),(150,213),(151,214),(152,215),(153,216),(154,217),(155,193),(156,194),(157,195),(158,196),(159,183),(160,184),(161,185),(162,186),(163,187),(164,188),(165,189),(166,190),(167,191),(168,192)])
Matrix representation ►G ⊆ GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 28 | 2 |
| 0 | 0 | 0 | 28 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 28 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,20,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,28,28,0,0,0,2,1],[1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,28] >;
280 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2S | 4A | ··· | 4H | 4I | ··· | 4T | 7A | ··· | 7F | 14A | ··· | 14AP | 14AQ | ··· | 14DJ | 28A | ··· | 28AV | 28AW | ··· | 28DP |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
280 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C4○D4 | C7×C4○D4 |
| kernel | C4○D4×C2×C14 | C23×C28 | D4×C2×C14 | Q8×C2×C14 | C14×C4○D4 | C22×C4○D4 | C23×C4 | C22×D4 | C22×Q8 | C2×C4○D4 | C2×C14 | C22 |
| # reps | 1 | 3 | 3 | 1 | 24 | 6 | 18 | 18 | 6 | 144 | 8 | 48 |
In GAP, Magma, Sage, TeX
C_4\circ D_4\times C_2\times C_{14} % in TeX
G:=Group("C4oD4xC2xC14"); // GroupNames label
G:=SmallGroup(448,1388);
// by ID
G=gap.SmallGroup(448,1388);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-7,-2,3165,1192]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^14=c^4=e^2=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀