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G = C24.45D14  order 448 = 26·7

3rd non-split extension by C24 of D14 acting via D14/D7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.45D14, C23.42D28, C23.50(C4×D7), (C22×Dic7)⋊8C4, (C22×C4).23D14, (C22×C14).61D4, C22.41(C2×D28), C23.50(C7⋊D4), C7⋊2(C23.34D4), (C23×Dic7).3C2, C14.C42⋊10C2, C22.22(D14⋊C4), (C23×C14).26C22, (C22×C28).22C22, C23.276(C22×D7), C14.25(C42⋊C2), C22.42(D4⋊2D7), (C22×C14).318C23, C2.3(C22.D28), C2.1(C23.18D14), C14.70(C22.D4), C2.12(C23.11D14), (C22×Dic7).183C22, C2.7(C2×D14⋊C4), (C2×C22⋊C4).5D7, C22.122(C2×C4×D7), (C2×C14).149(C2×D4), (C14×C22⋊C4).6C2, C14.34(C2×C22⋊C4), C22.46(C2×C7⋊D4), (C2×C23.D7).5C2, (C22×C14).46(C2×C4), (C2×Dic7).93(C2×C4), (C2×C14).139(C4○D4), (C2×C14).13(C22⋊C4), (C2×C14).104(C22×C4), SmallGroup(448,477)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C24.45D14
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C23×Dic7 — C24.45D14
Lower central
C7 — C2×C14 — C24.45D14
Upper central
C1 — C23 — C2×C22⋊C4

Generators and relations for C24.45D14
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=b, f2=bcd, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce13 >

Subgroups: 852 in 218 conjugacy classes, 79 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×8], C22 [×3], C22 [×8], C22 [×12], C7, C2×C4 [×28], C23, C23 [×6], C23 [×4], C14, C14 [×6], C14 [×4], C22⋊C4 [×4], C22×C4 [×2], C22×C4 [×12], C24, Dic7 [×6], C28 [×2], C2×C14 [×3], C2×C14 [×8], C2×C14 [×12], C2.C42 [×4], C2×C22⋊C4, C2×C22⋊C4, C23×C4, C2×Dic7 [×4], C2×Dic7 [×18], C2×C28 [×6], C22×C14, C22×C14 [×6], C22×C14 [×4], C23.34D4, C23.D7 [×2], C7×C22⋊C4 [×2], C22×Dic7 [×8], C22×Dic7 [×4], C22×C28 [×2], C23×C14, C14.C42 [×4], C2×C23.D7, C14×C22⋊C4, C23×Dic7, C24.45D14
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D7, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], D14 [×3], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C4×D7 [×2], D28 [×2], C7⋊D4 [×2], C22×D7, C23.34D4, D14⋊C4 [×4], C2×C4×D7, C2×D28, D4⋊2D7 [×4], C2×C7⋊D4, C23.11D14 [×2], C22.D28 [×2], C2×D14⋊C4, C23.18D14 [×2], C24.45D14

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

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

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

(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)

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

88 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4L4M4N4O4P7A7B7C14A···14U14V···14AG28A···28X
order12···2222244444···4444477714···1414···1428···28
size11···12222444414···14282828282222···24···44···4

88 irreducible representations

dim111111222222224
type++++++++++-
imageC1C2C2C2C2C4D4D7C4○D4D14D14C4×D7D28C7⋊D4D4⋊2D7
kernelC24.45D14C14.C42C2×C23.D7C14×C22⋊C4C23×Dic7C22×Dic7C22×C14C2×C22⋊C4C2×C14C22×C4C24C23C23C23C22
# reps1411184386312121212

Matrix representation of C24.45D14 ►in GL5(𝔽29)

10000
01200
002800
000280
000028
,
280000
01000
00100
00010
00001
,
10000
01000
00100
000280
000028
,
10000
028000
002800
00010
00001
,
120000
01000
0282800
000227
000223
,
120000
017000
0121200
000267
000113

G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,1,0,0,0,0,2,28,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[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],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,1,28,0,0,0,0,28,0,0,0,0,0,22,22,0,0,0,7,3],[12,0,0,0,0,0,17,12,0,0,0,0,12,0,0,0,0,0,26,11,0,0,0,7,3] >;
 

C24.45D14 in GAP, Magma, Sage, TeX 

C_2^4._{45}D_{14}
 
% in TeX
 
G:=Group("C2^4.45D14");
 
// GroupNames label
 
G:=SmallGroup(448,477);
 
// by ID
 
G=gap.SmallGroup(448,477);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,422,387,58,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=b,f^2=b*c*d,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,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=c*e^13>;
 
// generators/relations
 

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