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G = C24.64D14  order 448 = 26·7

4th non-split extension by C24 of D14 acting via D14/C14=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.64D14, C23.27D28, C23.16Dic14, (C23×C4).8D7, C28⋊7(C22⋊C4), (C22×C28)⋊14C4, (C2×C28).475D4, C4⋊2(C23.D7), (C22×C4)⋊8Dic7, C2.4(C28⋊7D4), (C23×C28).11C2, C22⋊2(C4⋊Dic7), C22.60(C2×D28), (C22×C14).26Q8, C14.79(C4⋊D4), C7⋊4(C23.7Q8), (C22×C14).143D4, (C22×C4).433D14, C14.68(C22⋊Q8), C23.31(C2×Dic7), C14.C42⋊24C2, C2.5(C28.48D4), C22.63(C4○D28), (C23×C14).99C22, C22.32(C2×Dic14), C23.303(C22×D7), C14.49(C42⋊C2), (C22×C14).363C23, (C22×C28).484C22, C22.50(C22×Dic7), (C22×Dic7).66C22, C2.12(C23.21D14), (C2×C14)⋊7(C4⋊C4), C14.56(C2×C4⋊C4), (C2×C4⋊Dic7)⋊16C2, C2.16(C2×C4⋊Dic7), (C2×C14).44(C2×Q8), (C2×C28).282(C2×C4), C2.6(C2×C23.D7), (C2×C14).549(C2×D4), C14.70(C2×C22⋊C4), (C2×C4).85(C2×Dic7), C22.87(C2×C7⋊D4), (C2×C14).91(C4○D4), (C2×C4).260(C7⋊D4), (C2×C23.D7).18C2, (C22×C14).136(C2×C4), (C2×C14).193(C22×C4), SmallGroup(448,746)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C24.64D14
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C2×C4⋊Dic7 — C24.64D14
Lower central
C7 — C2×C14 — C24.64D14
Upper central
C1 — C23 — C23×C4

Generators and relations for C24.64D14
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=d, f2=b, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de13 >

Subgroups: 772 in 234 conjugacy classes, 103 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×8], C22 [×12], C7, C2×C4 [×8], C2×C4 [×22], C23, C23 [×6], C23 [×4], C14 [×3], C14 [×4], C14 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C24, Dic7 [×4], C28 [×4], C28 [×2], C2×C14 [×3], C2×C14 [×8], C2×C14 [×12], C2.C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C23×C4, C2×Dic7 [×12], C2×C28 [×8], C2×C28 [×10], C22×C14, C22×C14 [×6], C22×C14 [×4], C23.7Q8, C4⋊Dic7 [×4], C23.D7 [×4], C22×Dic7 [×4], C22×C28 [×2], C22×C28 [×4], C22×C28 [×4], C23×C14, C14.C42 [×2], C2×C4⋊Dic7 [×2], C2×C23.D7 [×2], C23×C28, C24.64D14
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, D7, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], Dic7 [×4], D14 [×3], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], Dic14 [×2], D28 [×2], C2×Dic7 [×6], C7⋊D4 [×4], C22×D7, C23.7Q8, C4⋊Dic7 [×4], C23.D7 [×4], C2×Dic14, C2×D28, C4○D28 [×2], C22×Dic7, C2×C7⋊D4 [×2], C28.48D4 [×2], C2×C4⋊Dic7, C23.21D14, C28⋊7D4 [×2], C2×C23.D7, C24.64D14

Smallest permutation representation of C24.64D14
►On 224 points
Generators in S224
(85 132)(86 133)(87 134)(88 135)(89 136)(90 137)(91 138)(92 139)(93 140)(94 113)(95 114)(96 115)(97 116)(98 117)(99 118)(100 119)(101 120)(102 121)(103 122)(104 123)(105 124)(106 125)(107 126)(108 127)(109 128)(110 129)(111 130)(112 131)(141 203)(142 204)(143 205)(144 206)(145 207)(146 208)(147 209)(148 210)(149 211)(150 212)(151 213)(152 214)(153 215)(154 216)(155 217)(156 218)(157 219)(158 220)(159 221)(160 222)(161 223)(162 224)(163 197)(164 198)(165 199)(166 200)(167 201)(168 202)

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

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

(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 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)

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

124 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P7A7B7C14A···14AS28A···28AV
order12···222224···44···477714···1428···28
size11···122222···228···282222···22···2

124 irreducible representations

dim111111222222222222
type+++++++-+-++-+
imageC1C2C2C2C2C4D4D4Q8D7C4○D4Dic7D14D14C7⋊D4Dic14D28C4○D28
kernelC24.64D14C14.C42C2×C4⋊Dic7C2×C23.D7C23×C28C22×C28C2×C28C22×C14C22×C14C23×C4C2×C14C22×C4C22×C4C24C2×C4C23C23C22
# reps12221842234126324121224

Matrix representation of C24.64D14 ►in GL6(𝔽29)

100000
0280000
001000
0002800
000010
000001
,
2800000
0280000
001000
000100
000010
000001
,
2800000
0280000
0028000
0002800
000010
000001
,
100000
010000
001000
000100
0000280
0000028
,
700000
0250000
006000
000500
0000140
0000027
,
0250000
2200000
000500
006000
000002
0000150

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[7,0,0,0,0,0,0,25,0,0,0,0,0,0,6,0,0,0,0,0,0,5,0,0,0,0,0,0,14,0,0,0,0,0,0,27],[0,22,0,0,0,0,25,0,0,0,0,0,0,0,0,6,0,0,0,0,5,0,0,0,0,0,0,0,0,15,0,0,0,0,2,0] >;
 

C24.64D14 in GAP, Magma, Sage, TeX 

C_2^4._{64}D_{14}
 
% in TeX
 
G:=Group("C2^4.64D14");
 
// GroupNames label
 
G:=SmallGroup(448,746);
 
// by ID
 
G=gap.SmallGroup(448,746);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,477,232,422,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=d,f^2=b,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*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=d*e^13>;
 
// generators/relations
 

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