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G = C24.3D14  order 448 = 26·7

3rd non-split extension by C24 of D14 acting via D14/C7=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.3D14, C14.41(C4×D4), C23.D7⋊8C4, (C2×C28).247D4, C23.15(C4×D7), C22.95(D4×D7), C2.1(C28⋊2D4), (C22×C4).24D14, C14.82(C4⋊D4), (C2×Dic7).105D4, C14.C42⋊38C2, C14.30(C4.4D4), C22.50(C4○D28), (C23×C14).27C22, C7⋊4(C24.C22), C23.277(C22×D7), C14.11(C42⋊2C2), C14.26(C42⋊C2), C2.24(Dic7⋊4D4), C22.43(D4⋊2D7), (C22×C14).319C23, (C22×C28).341C22, C2.5(Dic7.D4), C2.5(C23.D14), C2.2(C23.18D14), C14.71(C22.D4), (C22×Dic7).35C22, C2.13(C23.11D14), C2.7(C4×C7⋊D4), (C2×C4×Dic7)⋊21C2, (C2×Dic7⋊C4)⋊7C2, (C2×C22⋊C4).6D7, C22.123(C2×C4×D7), (C2×C14).314(C2×D4), C22.47(C2×C7⋊D4), (C2×C23.D7).6C2, (C2×C4).167(C7⋊D4), (C14×C22⋊C4).23C2, (C22×C14).47(C2×C4), (C2×Dic7).58(C2×C4), (C2×C14).140(C4○D4), (C2×C14).105(C22×C4), SmallGroup(448,478)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C24.3D14
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C2×C23.D7 — C24.3D14
Lower central
C7 — C2×C14 — C24.3D14
Upper central
C1 — C23 — C2×C22⋊C4

Generators and relations for C24.3D14
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=b, f2=db=bd, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >

Subgroups: 724 in 190 conjugacy classes, 69 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic7, C28, C2×C14, C2×C14, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C24.C22, C4×Dic7, Dic7⋊C4, C23.D7, C23.D7, C7×C22⋊C4, C22×Dic7, C22×C28, C23×C14, C14.C42, C2×C4×Dic7, C2×Dic7⋊C4, C2×C23.D7, C14×C22⋊C4, C24.3D14
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C4×D7, C7⋊D4, C22×D7, C24.C22, C2×C4×D7, C4○D28, D4×D7, D4⋊2D7, C2×C7⋊D4, C23.11D14, C23.D14, Dic7⋊4D4, Dic7.D4, C4×C7⋊D4, C23.18D14, C28⋊2D4, C24.3D14

Smallest permutation representation of C24.3D14
►On 224 points
Generators in S224
(1 15)(2 221)(3 17)(4 223)(5 19)(6 197)(7 21)(8 199)(9 23)(10 201)(11 25)(12 203)(13 27)(14 205)(16 207)(18 209)(20 211)(22 213)(24 215)(26 217)(28 219)(29 43)(30 92)(31 45)(32 94)(33 47)(34 96)(35 49)(36 98)(37 51)(38 100)(39 53)(40 102)(41 55)(42 104)(44 106)(46 108)(48 110)(50 112)(52 86)(54 88)(56 90)(57 128)(58 170)(59 130)(60 172)(61 132)(62 174)(63 134)(64 176)(65 136)(66 178)(67 138)(68 180)(69 140)(70 182)(71 114)(72 184)(73 116)(74 186)(75 118)(76 188)(77 120)(78 190)(79 122)(80 192)(81 124)(82 194)(83 126)(84 196)(85 99)(87 101)(89 103)(91 105)(93 107)(95 109)(97 111)(113 153)(115 155)(117 157)(119 159)(121 161)(123 163)(125 165)(127 167)(129 141)(131 143)(133 145)(135 147)(137 149)(139 151)(142 171)(144 173)(146 175)(148 177)(150 179)(152 181)(154 183)(156 185)(158 187)(160 189)(162 191)(164 193)(166 195)(168 169)(198 212)(200 214)(202 216)(204 218)(206 220)(208 222)(210 224)

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

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

(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)

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

88 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G···4N4O4P4Q4R7A7B7C14A···14U14V···14AG28A···28X
order12···2224444444···4444477714···1414···1428···28
size11···14422224414···14282828282222···24···44···4

88 irreducible representations

dim111111122222222244
type++++++++++++-
imageC1C2C2C2C2C2C4D4D4D7C4○D4D14D14C7⋊D4C4×D7C4○D28D4×D7D4⋊2D7
kernelC24.3D14C14.C42C2×C4×Dic7C2×Dic7⋊C4C2×C23.D7C14×C22⋊C4C23.D7C2×Dic7C2×C28C2×C22⋊C4C2×C14C22×C4C24C2×C4C23C22C22C22
# reps121121822386312121239

Matrix representation of C24.3D14 ►in GL5(𝔽29)

10000
028000
025100
000280
00091
,
280000
028000
002800
000280
000028
,
10000
028000
002800
000280
000028
,
10000
01000
00100
000280
000028
,
120000
019000
03300
000417
0002825
,
170000
0261600
03300
00010
00001

G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,28,25,0,0,0,0,1,0,0,0,0,0,28,9,0,0,0,0,1],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[12,0,0,0,0,0,19,3,0,0,0,0,3,0,0,0,0,0,4,28,0,0,0,17,25],[17,0,0,0,0,0,26,3,0,0,0,16,3,0,0,0,0,0,1,0,0,0,0,0,1] >;
 

C24.3D14 in GAP, Magma, Sage, TeX 

C_2^4._3D_{14}
 
% in TeX
 
G:=Group("C2^4.3D14");
 
// GroupNames label
 
G:=SmallGroup(448,478);
 
// by ID
 
G=gap.SmallGroup(448,478);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,387,58,18822]);
 
// 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,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,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=e^13>;
 
// generators/relations
 

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