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G = C24.8D14  order 448 = 26·7

8th non-split extension by C24 of D14 acting via D14/C7=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.8D14, C14.87(C4×D4), C2.5(D4×Dic7), C22⋊C4⋊5Dic7, C22.99(D4×D7), C2.5(D14⋊D4), C14.30(C4⋊D4), C23.7(C2×Dic7), (C2×Dic7).106D4, (C22×C4).307D14, C14.C42⋊31C2, C2.6(D14.D4), C14.32(C4.4D4), C22.52(C4○D28), (C23×C14).34C22, C7⋊6(C24.C22), C23.281(C22×D7), C14.44(C42⋊C2), C14.14(C42⋊2C2), C22.47(D4⋊2D7), (C22×C14).326C23, (C22×C28).343C22, C2.6(Dic7.D4), C2.7(C23.D14), C22.40(C22×Dic7), C14.31(C22.D4), C2.8(C23.21D14), (C22×Dic7).185C22, (C7×C22⋊C4)⋊8C4, (C2×C4×Dic7)⋊23C2, (C2×C4⋊Dic7)⋊11C2, (C2×C28).160(C2×C4), (C2×C14).320(C2×D4), (C2×C22⋊C4).14D7, (C2×C4).16(C2×Dic7), (C2×C14).79(C4○D4), (C14×C22⋊C4).19C2, (C22×C14).51(C2×C4), (C2×C23.D7).13C2, (C2×C14).179(C22×C4), SmallGroup(448,485)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C24.8D14
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C2×C4×Dic7 — C24.8D14
Lower central
C7 — C2×C14 — C24.8D14
Upper central
C1 — C23 — C2×C22⋊C4

Generators and relations for C24.8D14
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=c, f2=cb=bc, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >

Subgroups: 724 in 190 conjugacy classes, 75 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C42, C22⋊C4, 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, C4⋊Dic7, C23.D7, C7×C22⋊C4, C22×Dic7, C22×C28, C23×C14, C14.C42, C2×C4×Dic7, C2×C4⋊Dic7, C2×C23.D7, C14×C22⋊C4, C24.8D14
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, Dic7, D14, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C2×Dic7, C22×D7, C24.C22, C4○D28, D4×D7, D4⋊2D7, C22×Dic7, C23.D14, D14.D4, D14⋊D4, Dic7.D4, C23.21D14, D4×Dic7, C24.8D14

Smallest permutation representation of C24.8D14
►On 224 points
Generators in S224
(2 73)(4 75)(6 77)(8 79)(10 81)(12 83)(14 57)(16 59)(18 61)(20 63)(22 65)(24 67)(26 69)(28 71)(29 208)(30 44)(31 210)(32 46)(33 212)(34 48)(35 214)(36 50)(37 216)(38 52)(39 218)(40 54)(41 220)(42 56)(43 222)(45 224)(47 198)(49 200)(51 202)(53 204)(55 206)(85 172)(87 174)(89 176)(91 178)(93 180)(95 182)(97 184)(99 186)(101 188)(103 190)(105 192)(107 194)(109 196)(111 170)(113 167)(114 128)(115 141)(116 130)(117 143)(118 132)(119 145)(120 134)(121 147)(122 136)(123 149)(124 138)(125 151)(126 140)(127 153)(129 155)(131 157)(133 159)(135 161)(137 163)(139 165)(142 156)(144 158)(146 160)(148 162)(150 164)(152 166)(154 168)(197 211)(199 213)(201 215)(203 217)(205 219)(207 221)(209 223)

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

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

(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)

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

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

dim1111111222222244
type++++++++-+++-
imageC1C2C2C2C2C2C4D4D7C4○D4Dic7D14D14C4○D28D4×D7D4⋊2D7
kernelC24.8D14C14.C42C2×C4×Dic7C2×C4⋊Dic7C2×C23.D7C14×C22⋊C4C7×C22⋊C4C2×Dic7C2×C22⋊C4C2×C14C22⋊C4C22×C4C24C22C22C22
# reps121121843812632466

Matrix representation of C24.8D14 ►in GL5(𝔽29)

280000
01000
002800
00010
000528
,
280000
028000
002800
00010
00001
,
10000
028000
002800
000280
000028
,
10000
01000
00100
000280
000028
,
280000
015000
002700
0001728
000012
,
170000
002700
014000
000170
000017

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,1,5,0,0,0,0,28],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[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],[28,0,0,0,0,0,15,0,0,0,0,0,27,0,0,0,0,0,17,0,0,0,0,28,12],[17,0,0,0,0,0,0,14,0,0,0,27,0,0,0,0,0,0,17,0,0,0,0,0,17] >;
 

C24.8D14 in GAP, Magma, Sage, TeX 

C_2^4._8D_{14}
 
% in TeX
 
G:=Group("C2^4.8D14");
 
// GroupNames label
 
G:=SmallGroup(448,485);
 
// by ID
 
G=gap.SmallGroup(448,485);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,120,422,387,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=c,f^2=c*b=b*c,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,b*d=d*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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