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G = C24.20D14  order 448 = 26·7

20th non-split extension by C24 of D14 acting via D14/C7=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.20D14, (C2×Dic7)⋊6D4, (C2×C28).302D4, (C22×D4).8D7, C14.73C22≀C2, C22.283(D4×D7), C2.25(C28⋊2D4), (C22×C14).110D4, (C22×C4).152D14, C2.6(C24⋊D7), C7⋊5(C23.10D4), C23.30(C7⋊D4), C14.130(C4⋊D4), C14.C42⋊45C2, C14.48(C4.4D4), (C23×C14).48C22, C23.384(C22×D7), C2.34(Dic7⋊D4), C2.14(C28.17D4), (C22×C14).367C23, (C22×C28).395C22, C22.106(D4⋊2D7), C14.84(C22.D4), (C22×Dic7).68C22, C2.17(C23.18D14), (D4×C2×C14).12C2, (C2×Dic7⋊C4)⋊43C2, (C2×C14).556(C2×D4), (C2×C4).85(C7⋊D4), (C2×C23.D7)⋊11C2, C22.218(C2×C7⋊D4), (C2×C14).163(C4○D4), SmallGroup(448,756)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C14 — C24.20D14
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C2×C23.D7 — C24.20D14
Lower central
C7 — C22×C14 — C24.20D14
Upper central
C1 — C23 — C22×D4

Generators and relations for C24.20D14
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=1, f2=dc=cd, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >

Subgroups: 884 in 238 conjugacy classes, 65 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, Dic7, C28, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C23.10D4, Dic7⋊C4, C23.D7, C22×Dic7, C22×Dic7, C22×C28, D4×C14, C23×C14, C14.C42, C2×Dic7⋊C4, C2×C23.D7, D4×C2×C14, C24.20D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C7⋊D4, C22×D7, C23.10D4, D4×D7, D4⋊2D7, C2×C7⋊D4, C23.18D14, C28.17D4, C28⋊2D4, Dic7⋊D4, C24⋊D7, C24.20D14

Smallest permutation representation of C24.20D14
►On 224 points
Generators in S224
(2 52)(4 54)(6 56)(8 44)(10 46)(12 48)(14 50)(16 34)(18 36)(20 38)(22 40)(24 42)(26 30)(28 32)(57 178)(58 143)(59 180)(60 145)(61 182)(62 147)(63 170)(64 149)(65 172)(66 151)(67 174)(68 153)(69 176)(70 141)(71 166)(72 205)(73 168)(74 207)(75 156)(76 209)(77 158)(78 197)(79 160)(80 199)(81 162)(82 201)(83 164)(84 203)(85 142)(86 179)(87 144)(88 181)(89 146)(90 169)(91 148)(92 171)(93 150)(94 173)(95 152)(96 175)(97 154)(98 177)(99 159)(100 198)(101 161)(102 200)(103 163)(104 202)(105 165)(106 204)(107 167)(108 206)(109 155)(110 208)(111 157)(112 210)(114 195)(116 183)(118 185)(120 187)(122 189)(124 191)(126 193)(127 223)(129 211)(131 213)(133 215)(135 217)(137 219)(139 221)

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

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

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

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

82 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C···4J7A7B7C14A···14U14V···14AS28A···28L
order12···22222444···477714···1414···1428···28
size11···144444428···282222···24···44···4

82 irreducible representations

dim1111122222222244
type++++++++++++-
imageC1C2C2C2C2D4D4D4D7C4○D4D14D14C7⋊D4C7⋊D4D4×D7D4⋊2D7
kernelC24.20D14C14.C42C2×Dic7⋊C4C2×C23.D7D4×C2×C14C2×Dic7C2×C28C22×C14C22×D4C2×C14C22×C4C24C2×C4C23C22C22
# reps111412243636122439

Matrix representation of C24.20D14 ►in GL6(𝔽29)

100000
3280000
001000
000100
000010
00002528
,
2800000
0280000
001000
000100
000010
000001
,
100000
010000
0028000
0002800
0000280
0000028
,
100000
010000
001000
000100
0000280
0000028
,
2400000
16230000
00251400
001400
0000105
00001519
,
190000
0280000
0014400
00161500
000042
0000725

G:=sub<GL(6,GF(29))| [1,3,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,25,0,0,0,0,0,28],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[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],[24,16,0,0,0,0,0,23,0,0,0,0,0,0,25,1,0,0,0,0,14,4,0,0,0,0,0,0,10,15,0,0,0,0,5,19],[1,0,0,0,0,0,9,28,0,0,0,0,0,0,14,16,0,0,0,0,4,15,0,0,0,0,0,0,4,7,0,0,0,0,2,25] >;
 

C24.20D14 in GAP, Magma, Sage, TeX 

C_2^4._{20}D_{14}
 
% in TeX
 
G:=Group("C2^4.20D14");
 
// GroupNames label
 
G:=SmallGroup(448,756);
 
// by ID
 
G=gap.SmallGroup(448,756);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,254,387,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^14=1,f^2=d*c=c*d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*b*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^-1>;
 
// generators/relations
 

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