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G = C23.28D28  order 448 = 26·7

7th non-split extension by C23 of D28 acting via D28/C28=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.28D28, C24.65D14, (C2×C28)⋊35D4, (C23×C4)⋊1D7, (C23×C28)⋊1C2, C14.85(C4×D4), C23.38(C4×D7), C22⋊2(D14⋊C4), C14.70C22≀C2, C2.5(C28⋊7D4), C22.61(C2×D28), C14.80(C4⋊D4), (C22×C14).193D4, (C22×C4).408D14, C2.2(C24⋊D7), C7⋊4(C23.23D4), C23.83(C7⋊D4), C14.C42⋊25C2, C22.64(C4○D28), (C23×D7).24C22, C23.304(C22×D7), (C22×C28).485C22, (C22×C14).364C23, (C23×C14).100C22, C14.69(C22.D4), C2.5(C23.23D14), (C22×Dic7).67C22, (C2×C7⋊D4)⋊7C4, C2.29(C4×C7⋊D4), (C2×D14⋊C4)⋊11C2, (C2×C4)⋊15(C7⋊D4), C2.36(C2×D14⋊C4), (C2×Dic7)⋊9(C2×C4), (C22×D7)⋊5(C2×C4), (C2×C23.D7)⋊7C2, C22.150(C2×C4×D7), (C2×C14)⋊5(C22⋊C4), (C2×C14).550(C2×D4), C14.65(C2×C22⋊C4), (C22×C7⋊D4).7C2, C22.88(C2×C7⋊D4), (C2×C14).92(C4○D4), (C22×C14).100(C2×C4), (C2×C14).143(C22×C4), SmallGroup(448,747)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C23.28D28
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C23×D7 — C22×C7⋊D4 — C23.28D28
Lower central
C7 — C2×C14 — C23.28D28
Upper central
C1 — C23 — C23×C4

Generators and relations for C23.28D28
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=b, f2=db=bd, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de13 >

Subgroups: 1284 in 286 conjugacy classes, 87 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C14, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C23×C4, C22×D4, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C23.23D4, D14⋊C4, C23.D7, C22×Dic7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, C22×C28, C23×D7, C23×C14, C14.C42, C2×D14⋊C4, C2×C23.D7, C22×C7⋊D4, C23×C28, C23.28D28
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, C4○D4, D14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C4×D7, D28, C7⋊D4, C22×D7, C23.23D4, D14⋊C4, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C2×D14⋊C4, C4×C7⋊D4, C23.23D14, C28⋊7D4, C24⋊D7, C23.28D28

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

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

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

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

(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)

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

124 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4H4I···4N7A7B7C14A···14AS28A···28AV
order12···22222224···44···477714···1428···28
size11···1222228282···228···282222···22···2

124 irreducible representations

dim111111122222222222
type++++++++++++
imageC1C2C2C2C2C2C4D4D4D7C4○D4D14D14C7⋊D4C4×D7D28C7⋊D4C4○D28
kernelC23.28D28C14.C42C2×D14⋊C4C2×C23.D7C22×C7⋊D4C23×C28C2×C7⋊D4C2×C28C22×C14C23×C4C2×C14C22×C4C24C2×C4C23C23C23C22
# reps12211184434632412121224

Matrix representation of C23.28D28 ►in GL6(𝔽29)

100000
0280000
0028000
0025100
000010
000001
,
100000
010000
0028000
0002800
0000280
0000028
,
2800000
0280000
0028000
0002800
000010
000001
,
100000
010000
001000
000100
0000280
0000028
,
400000
0220000
0019000
003300
0000312
000079
,
0220000
400000
00261600
003300
0000928
00002220

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,25,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[4,0,0,0,0,0,0,22,0,0,0,0,0,0,19,3,0,0,0,0,0,3,0,0,0,0,0,0,3,7,0,0,0,0,12,9],[0,4,0,0,0,0,22,0,0,0,0,0,0,0,26,3,0,0,0,0,16,3,0,0,0,0,0,0,9,22,0,0,0,0,28,20] >;
 

C23.28D28 in GAP, Magma, Sage, TeX 

C_2^3._{28}D_{28}
 
% in TeX
 
G:=Group("C2^3.28D28");
 
// GroupNames label
 
G:=SmallGroup(448,747);
 
// by ID
 
G=gap.SmallGroup(448,747);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,758,58,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=b,f^2=d*b=b*d,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^13>;
 
// generators/relations
 

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