metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.38D28, C22⋊C8⋊9D7, C8⋊Dic7⋊8C2, (C2×C28).44D4, (C2×C4).33D28, (C2×C8).109D14, C28⋊7D4.2C2, C2.D56⋊10C2, C14.8(C2×SD16), (C2×C14).14SD16, (C22×C4).85D14, (C22×C14).55D4, C28.282(C4○D4), C2.14(C8⋊D14), C14.11(C8⋊C22), (C2×C56).120C22, (C2×C28).745C23, (C2×D28).10C22, C22.108(C2×D28), C22.3(C56⋊C2), C7⋊1(C23.46D4), C4.106(D4⋊2D7), C4⋊Dic7.270C22, (C22×C28).52C22, C14.17(C22.D4), C2.13(C22.D28), (C2×C4⋊Dic7)⋊5C2, (C7×C22⋊C8)⋊11C2, C2.11(C2×C56⋊C2), (C2×C14).128(C2×D4), (C2×C4).690(C22×D7), SmallGroup(448,269)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.38D28
G = < a,b,c,d,e | a2=b2=c2=1, d28=c, e2=cb=bc, dad-1=ab=ba, ac=ca, ae=ea, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd27 >
Subgroups: 700 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C4.Q8, C2×C4⋊C4, C4⋊D4, C56, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C23.46D4, C4⋊Dic7, C4⋊Dic7, C4⋊Dic7, D14⋊C4, C2×C56, C2×D28, C22×Dic7, C2×C7⋊D4, C22×C28, C8⋊Dic7, C2.D56, C7×C22⋊C8, C2×C4⋊Dic7, C28⋊7D4, C23.38D28
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, C4○D4, D14, C22.D4, C2×SD16, C8⋊C22, D28, C22×D7, C23.46D4, C56⋊C2, C2×D28, D4⋊2D7, C22.D28, C2×C56⋊C2, C8⋊D14, C23.38D28
►On 224 points
Generators in S224
(1 29)(2 96)(3 31)(4 98)(5 33)(6 100)(7 35)(8 102)(9 37)(10 104)(11 39)(12 106)(13 41)(14 108)(15 43)(16 110)(17 45)(18 112)(19 47)(20 58)(21 49)(22 60)(23 51)(24 62)(25 53)(26 64)(27 55)(28 66)(30 68)(32 70)(34 72)(36 74)(38 76)(40 78)(42 80)(44 82)(46 84)(48 86)(50 88)(52 90)(54 92)(56 94)(57 85)(59 87)(61 89)(63 91)(65 93)(67 95)(69 97)(71 99)(73 101)(75 103)(77 105)(79 107)(81 109)(83 111)(113 191)(114 142)(115 193)(116 144)(117 195)(118 146)(119 197)(120 148)(121 199)(122 150)(123 201)(124 152)(125 203)(126 154)(127 205)(128 156)(129 207)(130 158)(131 209)(132 160)(133 211)(134 162)(135 213)(136 164)(137 215)(138 166)(139 217)(140 168)(141 219)(143 221)(145 223)(147 169)(149 171)(151 173)(153 175)(155 177)(157 179)(159 181)(161 183)(163 185)(165 187)(167 189)(170 198)(172 200)(174 202)(176 204)(178 206)(180 208)(182 210)(184 212)(186 214)(188 216)(190 218)(192 220)(194 222)(196 224)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 81)(16 82)(17 83)(18 84)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 97)(32 98)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 105)(40 106)(41 107)(42 108)(43 109)(44 110)(45 111)(46 112)(47 57)(48 58)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(113 219)(114 220)(115 221)(116 222)(117 223)(118 224)(119 169)(120 170)(121 171)(122 172)(123 173)(124 174)(125 175)(126 176)(127 177)(128 178)(129 179)(130 180)(131 181)(132 182)(133 183)(134 184)(135 185)(136 186)(137 187)(138 188)(139 189)(140 190)(141 191)(142 192)(143 193)(144 194)(145 195)(146 196)(147 197)(148 198)(149 199)(150 200)(151 201)(152 202)(153 203)(154 204)(155 205)(156 206)(157 207)(158 208)(159 209)(160 210)(161 211)(162 212)(163 213)(164 214)(165 215)(166 216)(167 217)(168 218)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(169 197)(170 198)(171 199)(172 200)(173 201)(174 202)(175 203)(176 204)(177 205)(178 206)(179 207)(180 208)(181 209)(182 210)(183 211)(184 212)(185 213)(186 214)(187 215)(188 216)(189 217)(190 218)(191 219)(192 220)(193 221)(194 222)(195 223)(196 224)
(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)
(1 222 95 144)(2 143 96 221)(3 220 97 142)(4 141 98 219)(5 218 99 140)(6 139 100 217)(7 216 101 138)(8 137 102 215)(9 214 103 136)(10 135 104 213)(11 212 105 134)(12 133 106 211)(13 210 107 132)(14 131 108 209)(15 208 109 130)(16 129 110 207)(17 206 111 128)(18 127 112 205)(19 204 57 126)(20 125 58 203)(21 202 59 124)(22 123 60 201)(23 200 61 122)(24 121 62 199)(25 198 63 120)(26 119 64 197)(27 196 65 118)(28 117 66 195)(29 194 67 116)(30 115 68 193)(31 192 69 114)(32 113 70 191)(33 190 71 168)(34 167 72 189)(35 188 73 166)(36 165 74 187)(37 186 75 164)(38 163 76 185)(39 184 77 162)(40 161 78 183)(41 182 79 160)(42 159 80 181)(43 180 81 158)(44 157 82 179)(45 178 83 156)(46 155 84 177)(47 176 85 154)(48 153 86 175)(49 174 87 152)(50 151 88 173)(51 172 89 150)(52 149 90 171)(53 170 91 148)(54 147 92 169)(55 224 93 146)(56 145 94 223)
G:=sub<Sym(224)| (1,29)(2,96)(3,31)(4,98)(5,33)(6,100)(7,35)(8,102)(9,37)(10,104)(11,39)(12,106)(13,41)(14,108)(15,43)(16,110)(17,45)(18,112)(19,47)(20,58)(21,49)(22,60)(23,51)(24,62)(25,53)(26,64)(27,55)(28,66)(30,68)(32,70)(34,72)(36,74)(38,76)(40,78)(42,80)(44,82)(46,84)(48,86)(50,88)(52,90)(54,92)(56,94)(57,85)(59,87)(61,89)(63,91)(65,93)(67,95)(69,97)(71,99)(73,101)(75,103)(77,105)(79,107)(81,109)(83,111)(113,191)(114,142)(115,193)(116,144)(117,195)(118,146)(119,197)(120,148)(121,199)(122,150)(123,201)(124,152)(125,203)(126,154)(127,205)(128,156)(129,207)(130,158)(131,209)(132,160)(133,211)(134,162)(135,213)(136,164)(137,215)(138,166)(139,217)(140,168)(141,219)(143,221)(145,223)(147,169)(149,171)(151,173)(153,175)(155,177)(157,179)(159,181)(161,183)(163,185)(165,187)(167,189)(170,198)(172,200)(174,202)(176,204)(178,206)(180,208)(182,210)(184,212)(186,214)(188,216)(190,218)(192,220)(194,222)(196,224), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,81)(16,82)(17,83)(18,84)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,97)(32,98)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,105)(40,106)(41,107)(42,108)(43,109)(44,110)(45,111)(46,112)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(113,219)(114,220)(115,221)(116,222)(117,223)(118,224)(119,169)(120,170)(121,171)(122,172)(123,173)(124,174)(125,175)(126,176)(127,177)(128,178)(129,179)(130,180)(131,181)(132,182)(133,183)(134,184)(135,185)(136,186)(137,187)(138,188)(139,189)(140,190)(141,191)(142,192)(143,193)(144,194)(145,195)(146,196)(147,197)(148,198)(149,199)(150,200)(151,201)(152,202)(153,203)(154,204)(155,205)(156,206)(157,207)(158,208)(159,209)(160,210)(161,211)(162,212)(163,213)(164,214)(165,215)(166,216)(167,217)(168,218), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (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), (1,222,95,144)(2,143,96,221)(3,220,97,142)(4,141,98,219)(5,218,99,140)(6,139,100,217)(7,216,101,138)(8,137,102,215)(9,214,103,136)(10,135,104,213)(11,212,105,134)(12,133,106,211)(13,210,107,132)(14,131,108,209)(15,208,109,130)(16,129,110,207)(17,206,111,128)(18,127,112,205)(19,204,57,126)(20,125,58,203)(21,202,59,124)(22,123,60,201)(23,200,61,122)(24,121,62,199)(25,198,63,120)(26,119,64,197)(27,196,65,118)(28,117,66,195)(29,194,67,116)(30,115,68,193)(31,192,69,114)(32,113,70,191)(33,190,71,168)(34,167,72,189)(35,188,73,166)(36,165,74,187)(37,186,75,164)(38,163,76,185)(39,184,77,162)(40,161,78,183)(41,182,79,160)(42,159,80,181)(43,180,81,158)(44,157,82,179)(45,178,83,156)(46,155,84,177)(47,176,85,154)(48,153,86,175)(49,174,87,152)(50,151,88,173)(51,172,89,150)(52,149,90,171)(53,170,91,148)(54,147,92,169)(55,224,93,146)(56,145,94,223)>;
G:=Group( (1,29)(2,96)(3,31)(4,98)(5,33)(6,100)(7,35)(8,102)(9,37)(10,104)(11,39)(12,106)(13,41)(14,108)(15,43)(16,110)(17,45)(18,112)(19,47)(20,58)(21,49)(22,60)(23,51)(24,62)(25,53)(26,64)(27,55)(28,66)(30,68)(32,70)(34,72)(36,74)(38,76)(40,78)(42,80)(44,82)(46,84)(48,86)(50,88)(52,90)(54,92)(56,94)(57,85)(59,87)(61,89)(63,91)(65,93)(67,95)(69,97)(71,99)(73,101)(75,103)(77,105)(79,107)(81,109)(83,111)(113,191)(114,142)(115,193)(116,144)(117,195)(118,146)(119,197)(120,148)(121,199)(122,150)(123,201)(124,152)(125,203)(126,154)(127,205)(128,156)(129,207)(130,158)(131,209)(132,160)(133,211)(134,162)(135,213)(136,164)(137,215)(138,166)(139,217)(140,168)(141,219)(143,221)(145,223)(147,169)(149,171)(151,173)(153,175)(155,177)(157,179)(159,181)(161,183)(163,185)(165,187)(167,189)(170,198)(172,200)(174,202)(176,204)(178,206)(180,208)(182,210)(184,212)(186,214)(188,216)(190,218)(192,220)(194,222)(196,224), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,81)(16,82)(17,83)(18,84)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,97)(32,98)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,105)(40,106)(41,107)(42,108)(43,109)(44,110)(45,111)(46,112)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(113,219)(114,220)(115,221)(116,222)(117,223)(118,224)(119,169)(120,170)(121,171)(122,172)(123,173)(124,174)(125,175)(126,176)(127,177)(128,178)(129,179)(130,180)(131,181)(132,182)(133,183)(134,184)(135,185)(136,186)(137,187)(138,188)(139,189)(140,190)(141,191)(142,192)(143,193)(144,194)(145,195)(146,196)(147,197)(148,198)(149,199)(150,200)(151,201)(152,202)(153,203)(154,204)(155,205)(156,206)(157,207)(158,208)(159,209)(160,210)(161,211)(162,212)(163,213)(164,214)(165,215)(166,216)(167,217)(168,218), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (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), (1,222,95,144)(2,143,96,221)(3,220,97,142)(4,141,98,219)(5,218,99,140)(6,139,100,217)(7,216,101,138)(8,137,102,215)(9,214,103,136)(10,135,104,213)(11,212,105,134)(12,133,106,211)(13,210,107,132)(14,131,108,209)(15,208,109,130)(16,129,110,207)(17,206,111,128)(18,127,112,205)(19,204,57,126)(20,125,58,203)(21,202,59,124)(22,123,60,201)(23,200,61,122)(24,121,62,199)(25,198,63,120)(26,119,64,197)(27,196,65,118)(28,117,66,195)(29,194,67,116)(30,115,68,193)(31,192,69,114)(32,113,70,191)(33,190,71,168)(34,167,72,189)(35,188,73,166)(36,165,74,187)(37,186,75,164)(38,163,76,185)(39,184,77,162)(40,161,78,183)(41,182,79,160)(42,159,80,181)(43,180,81,158)(44,157,82,179)(45,178,83,156)(46,155,84,177)(47,176,85,154)(48,153,86,175)(49,174,87,152)(50,151,88,173)(51,172,89,150)(52,149,90,171)(53,170,91,148)(54,147,92,169)(55,224,93,146)(56,145,94,223) );
G=PermutationGroup([[(1,29),(2,96),(3,31),(4,98),(5,33),(6,100),(7,35),(8,102),(9,37),(10,104),(11,39),(12,106),(13,41),(14,108),(15,43),(16,110),(17,45),(18,112),(19,47),(20,58),(21,49),(22,60),(23,51),(24,62),(25,53),(26,64),(27,55),(28,66),(30,68),(32,70),(34,72),(36,74),(38,76),(40,78),(42,80),(44,82),(46,84),(48,86),(50,88),(52,90),(54,92),(56,94),(57,85),(59,87),(61,89),(63,91),(65,93),(67,95),(69,97),(71,99),(73,101),(75,103),(77,105),(79,107),(81,109),(83,111),(113,191),(114,142),(115,193),(116,144),(117,195),(118,146),(119,197),(120,148),(121,199),(122,150),(123,201),(124,152),(125,203),(126,154),(127,205),(128,156),(129,207),(130,158),(131,209),(132,160),(133,211),(134,162),(135,213),(136,164),(137,215),(138,166),(139,217),(140,168),(141,219),(143,221),(145,223),(147,169),(149,171),(151,173),(153,175),(155,177),(157,179),(159,181),(161,183),(163,185),(165,187),(167,189),(170,198),(172,200),(174,202),(176,204),(178,206),(180,208),(182,210),(184,212),(186,214),(188,216),(190,218),(192,220),(194,222),(196,224)], [(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,73),(8,74),(9,75),(10,76),(11,77),(12,78),(13,79),(14,80),(15,81),(16,82),(17,83),(18,84),(19,85),(20,86),(21,87),(22,88),(23,89),(24,90),(25,91),(26,92),(27,93),(28,94),(29,95),(30,96),(31,97),(32,98),(33,99),(34,100),(35,101),(36,102),(37,103),(38,104),(39,105),(40,106),(41,107),(42,108),(43,109),(44,110),(45,111),(46,112),(47,57),(48,58),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66),(113,219),(114,220),(115,221),(116,222),(117,223),(118,224),(119,169),(120,170),(121,171),(122,172),(123,173),(124,174),(125,175),(126,176),(127,177),(128,178),(129,179),(130,180),(131,181),(132,182),(133,183),(134,184),(135,185),(136,186),(137,187),(138,188),(139,189),(140,190),(141,191),(142,192),(143,193),(144,194),(145,195),(146,196),(147,197),(148,198),(149,199),(150,200),(151,201),(152,202),(153,203),(154,204),(155,205),(156,206),(157,207),(158,208),(159,209),(160,210),(161,211),(162,212),(163,213),(164,214),(165,215),(166,216),(167,217),(168,218)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112),(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148),(121,149),(122,150),(123,151),(124,152),(125,153),(126,154),(127,155),(128,156),(129,157),(130,158),(131,159),(132,160),(133,161),(134,162),(135,163),(136,164),(137,165),(138,166),(139,167),(140,168),(169,197),(170,198),(171,199),(172,200),(173,201),(174,202),(175,203),(176,204),(177,205),(178,206),(179,207),(180,208),(181,209),(182,210),(183,211),(184,212),(185,213),(186,214),(187,215),(188,216),(189,217),(190,218),(191,219),(192,220),(193,221),(194,222),(195,223),(196,224)], [(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)], [(1,222,95,144),(2,143,96,221),(3,220,97,142),(4,141,98,219),(5,218,99,140),(6,139,100,217),(7,216,101,138),(8,137,102,215),(9,214,103,136),(10,135,104,213),(11,212,105,134),(12,133,106,211),(13,210,107,132),(14,131,108,209),(15,208,109,130),(16,129,110,207),(17,206,111,128),(18,127,112,205),(19,204,57,126),(20,125,58,203),(21,202,59,124),(22,123,60,201),(23,200,61,122),(24,121,62,199),(25,198,63,120),(26,119,64,197),(27,196,65,118),(28,117,66,195),(29,194,67,116),(30,115,68,193),(31,192,69,114),(32,113,70,191),(33,190,71,168),(34,167,72,189),(35,188,73,166),(36,165,74,187),(37,186,75,164),(38,163,76,185),(39,184,77,162),(40,161,78,183),(41,182,79,160),(42,159,80,181),(43,180,81,158),(44,157,82,179),(45,178,83,156),(46,155,84,177),(47,176,85,154),(48,153,86,175),(49,174,87,152),(50,151,88,173),(51,172,89,150),(52,149,90,171),(53,170,91,148),(54,147,92,169),(55,224,93,146),(56,145,94,223)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 56 | 2 | 2 | 4 | 28 | 28 | 28 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | C4○D4 | SD16 | D14 | D14 | D28 | D28 | C56⋊C2 | C8⋊C22 | D4⋊2D7 | C8⋊D14 |
| kernel | C23.38D28 | C8⋊Dic7 | C2.D56 | C7×C22⋊C8 | C2×C4⋊Dic7 | C28⋊7D4 | C2×C28 | C22×C14 | C22⋊C8 | C28 | C2×C14 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C14 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 6 | 3 | 6 | 6 | 24 | 1 | 6 | 6 |
Matrix representation of C23.38D28 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 23 | 70 |
| 0 | 0 | 43 | 8 |
| 98 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 41 | 85 |
| 0 | 0 | 52 | 72 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[0,1,0,0,1,0,0,0,0,0,23,43,0,0,70,8],[98,0,0,0,0,15,0,0,0,0,41,52,0,0,85,72] >;
C23.38D28 in GAP, Magma, Sage, TeX
C_2^3._{38}D_{28} % in TeX
G:=Group("C2^3.38D28"); // GroupNames label
G:=SmallGroup(448,269);
// by ID
G=gap.SmallGroup(448,269);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,254,219,142,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^28=c,e^2=c*b=b*c,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^27>;
// generators/relations