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G = C23.34D28order 448 = 26·7

5th non-split extension by C23 of D28 acting via D28/D14=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.34D28, C8⋊Dic76C2, (C2×C28).40D4, (C2×C4).29D28, C22⋊C8.6D7, (C2×C8).107D14, C14.6(C2×SD16), C28.44D49C2, (C2×C14).13SD16, (C22×C4).75D14, (C22×C14).48D4, C28.279(C4○D4), (C2×C28).738C23, (C2×C56).118C22, C28.48D4.2C2, C22.101(C2×D28), C22.8(C56⋊C2), C14.7(C8.C22), C71(C23.47D4), C4.103(D42D7), C2.10(C8.D14), C4⋊Dic7.268C22, (C22×C28).48C22, (C2×Dic14).10C22, C14.14(C22.D4), C2.10(C22.D28), C2.9(C2×C56⋊C2), (C7×C22⋊C8).8C2, (C2×C14).121(C2×D4), (C2×C4⋊Dic7).11C2, (C2×C4).683(C22×D7), SmallGroup(448,255)

Series: Derived Chief Lower central Upper central

C1C2×C28 — C23.34D28
C1C7C14C28C2×C28C4⋊Dic7C2×C4⋊Dic7 — C23.34D28
C7C14C2×C28 — C23.34D28
C1C22C22×C4C22⋊C8

Generators and relations for C23.34D28
 G = < a,b,c,d,e | a2=b2=c2=1, d28=c, e2=cb=bc, dad-1=eae-1=ab=ba, ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d27 >

Subgroups: 508 in 104 conjugacy classes, 43 normal (25 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C7, C8 [×2], C2×C4 [×2], C2×C4 [×8], Q8 [×2], C23, C14 [×3], C14 [×2], C22⋊C4, C4⋊C4 [×5], C2×C8 [×2], C22×C4, C22×C4, C2×Q8, Dic7 [×4], C28 [×2], C28, C2×C14, C2×C14 [×2], C2×C14 [×2], C22⋊C8, Q8⋊C4 [×2], C4.Q8 [×2], C2×C4⋊C4, C22⋊Q8, C56 [×2], Dic14 [×2], C2×Dic7 [×6], C2×C28 [×2], C2×C28 [×2], C22×C14, C23.47D4, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7 [×2], C4⋊Dic7, C23.D7, C2×C56 [×2], C2×Dic14, C22×Dic7, C22×C28, C28.44D4 [×2], C8⋊Dic7 [×2], C7×C22⋊C8, C28.48D4, C2×C4⋊Dic7, C23.34D28
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D7, SD16 [×2], C2×D4, C4○D4 [×2], D14 [×3], C22.D4, C2×SD16, C8.C22, D28 [×2], C22×D7, C23.47D4, C56⋊C2 [×2], C2×D28, D42D7 [×2], C22.D28, C2×C56⋊C2, C8.D14, C23.34D28

Smallest permutation representation of C23.34D28
On 224 points
Generators in S224
(1 29)(2 95)(3 31)(4 97)(5 33)(6 99)(7 35)(8 101)(9 37)(10 103)(11 39)(12 105)(13 41)(14 107)(15 43)(16 109)(17 45)(18 111)(19 47)(20 57)(21 49)(22 59)(23 51)(24 61)(25 53)(26 63)(27 55)(28 65)(30 67)(32 69)(34 71)(36 73)(38 75)(40 77)(42 79)(44 81)(46 83)(48 85)(50 87)(52 89)(54 91)(56 93)(58 86)(60 88)(62 90)(64 92)(66 94)(68 96)(70 98)(72 100)(74 102)(76 104)(78 106)(80 108)(82 110)(84 112)(113 141)(114 208)(115 143)(116 210)(117 145)(118 212)(119 147)(120 214)(121 149)(122 216)(123 151)(124 218)(125 153)(126 220)(127 155)(128 222)(129 157)(130 224)(131 159)(132 170)(133 161)(134 172)(135 163)(136 174)(137 165)(138 176)(139 167)(140 178)(142 180)(144 182)(146 184)(148 186)(150 188)(152 190)(154 192)(156 194)(158 196)(160 198)(162 200)(164 202)(166 204)(168 206)(169 197)(171 199)(173 201)(175 203)(177 205)(179 207)(181 209)(183 211)(185 213)(187 215)(189 217)(191 219)(193 221)(195 223)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 81)(17 82)(18 83)(19 84)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 91)(27 92)(28 93)(29 94)(30 95)(31 96)(32 97)(33 98)(34 99)(35 100)(36 101)(37 102)(38 103)(39 104)(40 105)(41 106)(42 107)(43 108)(44 109)(45 110)(46 111)(47 112)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 65)(113 179)(114 180)(115 181)(116 182)(117 183)(118 184)(119 185)(120 186)(121 187)(122 188)(123 189)(124 190)(125 191)(126 192)(127 193)(128 194)(129 195)(130 196)(131 197)(132 198)(133 199)(134 200)(135 201)(136 202)(137 203)(138 204)(139 205)(140 206)(141 207)(142 208)(143 209)(144 210)(145 211)(146 212)(147 213)(148 214)(149 215)(150 216)(151 217)(152 218)(153 219)(154 220)(155 221)(156 222)(157 223)(158 224)(159 169)(160 170)(161 171)(162 172)(163 173)(164 174)(165 175)(166 176)(167 177)(168 178)
(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)
(1 204 94 166)(2 175 95 137)(3 202 96 164)(4 173 97 135)(5 200 98 162)(6 171 99 133)(7 198 100 160)(8 169 101 131)(9 196 102 158)(10 223 103 129)(11 194 104 156)(12 221 105 127)(13 192 106 154)(14 219 107 125)(15 190 108 152)(16 217 109 123)(17 188 110 150)(18 215 111 121)(19 186 112 148)(20 213 57 119)(21 184 58 146)(22 211 59 117)(23 182 60 144)(24 209 61 115)(25 180 62 142)(26 207 63 113)(27 178 64 140)(28 205 65 167)(29 176 66 138)(30 203 67 165)(31 174 68 136)(32 201 69 163)(33 172 70 134)(34 199 71 161)(35 170 72 132)(36 197 73 159)(37 224 74 130)(38 195 75 157)(39 222 76 128)(40 193 77 155)(41 220 78 126)(42 191 79 153)(43 218 80 124)(44 189 81 151)(45 216 82 122)(46 187 83 149)(47 214 84 120)(48 185 85 147)(49 212 86 118)(50 183 87 145)(51 210 88 116)(52 181 89 143)(53 208 90 114)(54 179 91 141)(55 206 92 168)(56 177 93 139)

G:=sub<Sym(224)| (1,29)(2,95)(3,31)(4,97)(5,33)(6,99)(7,35)(8,101)(9,37)(10,103)(11,39)(12,105)(13,41)(14,107)(15,43)(16,109)(17,45)(18,111)(19,47)(20,57)(21,49)(22,59)(23,51)(24,61)(25,53)(26,63)(27,55)(28,65)(30,67)(32,69)(34,71)(36,73)(38,75)(40,77)(42,79)(44,81)(46,83)(48,85)(50,87)(52,89)(54,91)(56,93)(58,86)(60,88)(62,90)(64,92)(66,94)(68,96)(70,98)(72,100)(74,102)(76,104)(78,106)(80,108)(82,110)(84,112)(113,141)(114,208)(115,143)(116,210)(117,145)(118,212)(119,147)(120,214)(121,149)(122,216)(123,151)(124,218)(125,153)(126,220)(127,155)(128,222)(129,157)(130,224)(131,159)(132,170)(133,161)(134,172)(135,163)(136,174)(137,165)(138,176)(139,167)(140,178)(142,180)(144,182)(146,184)(148,186)(150,188)(152,190)(154,192)(156,194)(158,196)(160,198)(162,200)(164,202)(166,204)(168,206)(169,197)(171,199)(173,201)(175,203)(177,205)(179,207)(181,209)(183,211)(185,213)(187,215)(189,217)(191,219)(193,221)(195,223), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,81)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,109)(45,110)(46,111)(47,112)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,65)(113,179)(114,180)(115,181)(116,182)(117,183)(118,184)(119,185)(120,186)(121,187)(122,188)(123,189)(124,190)(125,191)(126,192)(127,193)(128,194)(129,195)(130,196)(131,197)(132,198)(133,199)(134,200)(135,201)(136,202)(137,203)(138,204)(139,205)(140,206)(141,207)(142,208)(143,209)(144,210)(145,211)(146,212)(147,213)(148,214)(149,215)(150,216)(151,217)(152,218)(153,219)(154,220)(155,221)(156,222)(157,223)(158,224)(159,169)(160,170)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178), (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), (1,204,94,166)(2,175,95,137)(3,202,96,164)(4,173,97,135)(5,200,98,162)(6,171,99,133)(7,198,100,160)(8,169,101,131)(9,196,102,158)(10,223,103,129)(11,194,104,156)(12,221,105,127)(13,192,106,154)(14,219,107,125)(15,190,108,152)(16,217,109,123)(17,188,110,150)(18,215,111,121)(19,186,112,148)(20,213,57,119)(21,184,58,146)(22,211,59,117)(23,182,60,144)(24,209,61,115)(25,180,62,142)(26,207,63,113)(27,178,64,140)(28,205,65,167)(29,176,66,138)(30,203,67,165)(31,174,68,136)(32,201,69,163)(33,172,70,134)(34,199,71,161)(35,170,72,132)(36,197,73,159)(37,224,74,130)(38,195,75,157)(39,222,76,128)(40,193,77,155)(41,220,78,126)(42,191,79,153)(43,218,80,124)(44,189,81,151)(45,216,82,122)(46,187,83,149)(47,214,84,120)(48,185,85,147)(49,212,86,118)(50,183,87,145)(51,210,88,116)(52,181,89,143)(53,208,90,114)(54,179,91,141)(55,206,92,168)(56,177,93,139)>;

G:=Group( (1,29)(2,95)(3,31)(4,97)(5,33)(6,99)(7,35)(8,101)(9,37)(10,103)(11,39)(12,105)(13,41)(14,107)(15,43)(16,109)(17,45)(18,111)(19,47)(20,57)(21,49)(22,59)(23,51)(24,61)(25,53)(26,63)(27,55)(28,65)(30,67)(32,69)(34,71)(36,73)(38,75)(40,77)(42,79)(44,81)(46,83)(48,85)(50,87)(52,89)(54,91)(56,93)(58,86)(60,88)(62,90)(64,92)(66,94)(68,96)(70,98)(72,100)(74,102)(76,104)(78,106)(80,108)(82,110)(84,112)(113,141)(114,208)(115,143)(116,210)(117,145)(118,212)(119,147)(120,214)(121,149)(122,216)(123,151)(124,218)(125,153)(126,220)(127,155)(128,222)(129,157)(130,224)(131,159)(132,170)(133,161)(134,172)(135,163)(136,174)(137,165)(138,176)(139,167)(140,178)(142,180)(144,182)(146,184)(148,186)(150,188)(152,190)(154,192)(156,194)(158,196)(160,198)(162,200)(164,202)(166,204)(168,206)(169,197)(171,199)(173,201)(175,203)(177,205)(179,207)(181,209)(183,211)(185,213)(187,215)(189,217)(191,219)(193,221)(195,223), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,81)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,109)(45,110)(46,111)(47,112)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,65)(113,179)(114,180)(115,181)(116,182)(117,183)(118,184)(119,185)(120,186)(121,187)(122,188)(123,189)(124,190)(125,191)(126,192)(127,193)(128,194)(129,195)(130,196)(131,197)(132,198)(133,199)(134,200)(135,201)(136,202)(137,203)(138,204)(139,205)(140,206)(141,207)(142,208)(143,209)(144,210)(145,211)(146,212)(147,213)(148,214)(149,215)(150,216)(151,217)(152,218)(153,219)(154,220)(155,221)(156,222)(157,223)(158,224)(159,169)(160,170)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178), (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), (1,204,94,166)(2,175,95,137)(3,202,96,164)(4,173,97,135)(5,200,98,162)(6,171,99,133)(7,198,100,160)(8,169,101,131)(9,196,102,158)(10,223,103,129)(11,194,104,156)(12,221,105,127)(13,192,106,154)(14,219,107,125)(15,190,108,152)(16,217,109,123)(17,188,110,150)(18,215,111,121)(19,186,112,148)(20,213,57,119)(21,184,58,146)(22,211,59,117)(23,182,60,144)(24,209,61,115)(25,180,62,142)(26,207,63,113)(27,178,64,140)(28,205,65,167)(29,176,66,138)(30,203,67,165)(31,174,68,136)(32,201,69,163)(33,172,70,134)(34,199,71,161)(35,170,72,132)(36,197,73,159)(37,224,74,130)(38,195,75,157)(39,222,76,128)(40,193,77,155)(41,220,78,126)(42,191,79,153)(43,218,80,124)(44,189,81,151)(45,216,82,122)(46,187,83,149)(47,214,84,120)(48,185,85,147)(49,212,86,118)(50,183,87,145)(51,210,88,116)(52,181,89,143)(53,208,90,114)(54,179,91,141)(55,206,92,168)(56,177,93,139) );

G=PermutationGroup([(1,29),(2,95),(3,31),(4,97),(5,33),(6,99),(7,35),(8,101),(9,37),(10,103),(11,39),(12,105),(13,41),(14,107),(15,43),(16,109),(17,45),(18,111),(19,47),(20,57),(21,49),(22,59),(23,51),(24,61),(25,53),(26,63),(27,55),(28,65),(30,67),(32,69),(34,71),(36,73),(38,75),(40,77),(42,79),(44,81),(46,83),(48,85),(50,87),(52,89),(54,91),(56,93),(58,86),(60,88),(62,90),(64,92),(66,94),(68,96),(70,98),(72,100),(74,102),(76,104),(78,106),(80,108),(82,110),(84,112),(113,141),(114,208),(115,143),(116,210),(117,145),(118,212),(119,147),(120,214),(121,149),(122,216),(123,151),(124,218),(125,153),(126,220),(127,155),(128,222),(129,157),(130,224),(131,159),(132,170),(133,161),(134,172),(135,163),(136,174),(137,165),(138,176),(139,167),(140,178),(142,180),(144,182),(146,184),(148,186),(150,188),(152,190),(154,192),(156,194),(158,196),(160,198),(162,200),(164,202),(166,204),(168,206),(169,197),(171,199),(173,201),(175,203),(177,205),(179,207),(181,209),(183,211),(185,213),(187,215),(189,217),(191,219),(193,221),(195,223)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,73),(9,74),(10,75),(11,76),(12,77),(13,78),(14,79),(15,80),(16,81),(17,82),(18,83),(19,84),(20,85),(21,86),(22,87),(23,88),(24,89),(25,90),(26,91),(27,92),(28,93),(29,94),(30,95),(31,96),(32,97),(33,98),(34,99),(35,100),(36,101),(37,102),(38,103),(39,104),(40,105),(41,106),(42,107),(43,108),(44,109),(45,110),(46,111),(47,112),(48,57),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(55,64),(56,65),(113,179),(114,180),(115,181),(116,182),(117,183),(118,184),(119,185),(120,186),(121,187),(122,188),(123,189),(124,190),(125,191),(126,192),(127,193),(128,194),(129,195),(130,196),(131,197),(132,198),(133,199),(134,200),(135,201),(136,202),(137,203),(138,204),(139,205),(140,206),(141,207),(142,208),(143,209),(144,210),(145,211),(146,212),(147,213),(148,214),(149,215),(150,216),(151,217),(152,218),(153,219),(154,220),(155,221),(156,222),(157,223),(158,224),(159,169),(160,170),(161,171),(162,172),(163,173),(164,174),(165,175),(166,176),(167,177),(168,178)], [(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)], [(1,204,94,166),(2,175,95,137),(3,202,96,164),(4,173,97,135),(5,200,98,162),(6,171,99,133),(7,198,100,160),(8,169,101,131),(9,196,102,158),(10,223,103,129),(11,194,104,156),(12,221,105,127),(13,192,106,154),(14,219,107,125),(15,190,108,152),(16,217,109,123),(17,188,110,150),(18,215,111,121),(19,186,112,148),(20,213,57,119),(21,184,58,146),(22,211,59,117),(23,182,60,144),(24,209,61,115),(25,180,62,142),(26,207,63,113),(27,178,64,140),(28,205,65,167),(29,176,66,138),(30,203,67,165),(31,174,68,136),(32,201,69,163),(33,172,70,134),(34,199,71,161),(35,170,72,132),(36,197,73,159),(37,224,74,130),(38,195,75,157),(39,222,76,128),(40,193,77,155),(41,220,78,126),(42,191,79,153),(43,218,80,124),(44,189,81,151),(45,216,82,122),(46,187,83,149),(47,214,84,120),(48,185,85,147),(49,212,86,118),(50,183,87,145),(51,210,88,116),(52,181,89,143),(53,208,90,114),(54,179,91,141),(55,206,92,168),(56,177,93,139)])

79 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I7A7B7C8A8B8C8D14A···14I14J···14O28A···28L28M···28R56A···56X
order122222444444444777888814···1414···1428···2828···2856···56
size11112222428282828565622244442···24···42···24···44···4

79 irreducible representations

dim1111112222222222444
type+++++++++++++---
imageC1C2C2C2C2C2D4D4D7C4○D4SD16D14D14D28D28C56⋊C2C8.C22D42D7C8.D14
kernelC23.34D28C28.44D4C8⋊Dic7C7×C22⋊C8C28.48D4C2×C4⋊Dic7C2×C28C22×C14C22⋊C8C28C2×C14C2×C8C22×C4C2×C4C23C22C14C4C2
# reps12211111344636624166

Matrix representation of C23.34D28 in GL6(𝔽113)

11200000
01120000
001000
000100
000010
000087112
,
100000
010000
001000
000100
00001120
00000112
,
11200000
01120000
001000
000100
000010
000001
,
3490000
66230000
008910400
009910300
0000187
000087112
,
10250000
51030000
001067200
0037700
00009851
0000015

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,87,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,66,0,0,0,0,49,23,0,0,0,0,0,0,89,99,0,0,0,0,104,103,0,0,0,0,0,0,1,87,0,0,0,0,87,112],[10,5,0,0,0,0,25,103,0,0,0,0,0,0,106,37,0,0,0,0,72,7,0,0,0,0,0,0,98,0,0,0,0,0,51,15] >;

C23.34D28 in GAP, Magma, Sage, TeX

C_2^3._{34}D_{28}
% in TeX

G:=Group("C2^3.34D28");
// GroupNames label

G:=SmallGroup(448,255);
// by ID

G=gap.SmallGroup(448,255);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,254,219,58,1123,136,18822]);
// 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=e*a*e^-1=a*b=b*a,a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^27>;
// generators/relations

׿
×
𝔽