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G = C2×D4.10D14  order 448 = 26·7

Direct product of C2 and D4.10D14

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D4.10D14, C28.48C24, C14.13C25, D14.7C24, D28.40C23, C14⋊22- (1+4), Dic7.8C24, Dic14.37C23, C4○D4⋊18D14, (C2×C14).4C24, (Q8×D7)⋊14C22, C2.14(D7×C24), C4.63(C23×D7), C7⋊D4.1C23, (C2×D4).254D14, C7⋊2(C2×2- (1+4)), C4○D28⋊26C22, (C2×Q8).212D14, (C4×D7).19C23, D4.29(C22×D7), (C7×D4).29C23, Q8.30(C22×D7), (C7×Q8).30C23, D4⋊2D7⋊13C22, (C2×C28).567C23, (C22×C4).292D14, C22.57(C23×D7), (C2×Dic14)⋊75C22, (C22×Dic14)⋊25C2, (C2×D28).290C22, (D4×C14).279C22, (Q8×C14).247C22, C23.215(C22×D7), (C22×C14).249C23, (C22×C28).303C22, (C2×Dic7).167C23, (C22×D7).250C23, (C22×Dic7).169C22, (C2×Q8×D7)⋊21C2, (C2×C4○D4)⋊14D7, (C14×C4○D4)⋊15C2, (C2×C4○D28)⋊38C2, (C2×D4⋊2D7)⋊30C2, (C7×C4○D4)⋊21C22, (C2×C4×D7).173C22, (C2×C4).253(C22×D7), (C2×C7⋊D4).144C22, SmallGroup(448,1377)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — C2×D4.10D14
Chief series
C1 — C7 — C14 — D14 — C22×D7 — C2×C4×D7 — C2×Q8×D7 — C2×D4.10D14
Lower central
C7 — C14 — C2×D4.10D14

Subgroups: 2772 in 794 conjugacy classes, 447 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×8], C4 [×12], C22, C22 [×6], C22 [×14], C7, C2×C4, C2×C4 [×15], C2×C4 [×54], D4 [×12], D4 [×28], Q8 [×4], Q8 [×36], C23 [×3], C23 [×2], D7 [×4], C14, C14 [×2], C14 [×6], C22×C4 [×3], C22×C4 [×12], C2×D4 [×3], C2×D4 [×7], C2×Q8, C2×Q8 [×49], C4○D4 [×8], C4○D4 [×72], Dic7 [×12], C28 [×8], D14 [×4], D14 [×4], C2×C14, C2×C14 [×6], C2×C14 [×6], C22×Q8 [×5], C2×C4○D4, C2×C4○D4 [×9], 2- (1+4) [×16], Dic14 [×36], C4×D7 [×24], D28 [×4], C2×Dic7 [×30], C7⋊D4 [×24], C2×C28, C2×C28 [×15], C7×D4 [×12], C7×Q8 [×4], C22×D7 [×2], C22×C14 [×3], C2×2- (1+4), C2×Dic14 [×33], C2×C4×D7 [×6], C2×D28, C4○D28 [×24], D4⋊2D7 [×48], Q8×D7 [×16], C22×Dic7 [×6], C2×C7⋊D4 [×6], C22×C28 [×3], D4×C14 [×3], Q8×C14, C7×C4○D4 [×8], C22×Dic14 [×3], C2×C4○D28 [×3], C2×D4⋊2D7 [×6], C2×Q8×D7 [×2], D4.10D14 [×16], C14×C4○D4, C2×D4.10D14

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], D7, C24 [×31], D14 [×15], 2- (1+4) [×2], C25, C22×D7 [×35], C2×2- (1+4), C23×D7 [×15], D4.10D14 [×2], D7×C24, C2×D4.10D14

Generators and relations
 G = < a,b,c,d,e | a2=b4=c2=1, d14=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=b2c, ce=ec, ede-1=d13 >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽29)

2800000
0280000
0028000
0002800
0000280
0000028
,
100000
010000
0072400
00102200
000885
0013101621
,
2800000
0280000
00280810
00028276
000010
000001
,
600000
050000
00525103
008172821
00192574
0091470
,
050000
600000
0027700
0020200
0000010
0000260

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,10,0,13,0,0,24,22,8,10,0,0,0,0,8,16,0,0,0,0,5,21],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,8,27,1,0,0,0,10,6,0,1],[6,0,0,0,0,0,0,5,0,0,0,0,0,0,5,8,19,9,0,0,25,17,25,14,0,0,10,28,7,7,0,0,3,21,4,0],[0,6,0,0,0,0,5,0,0,0,0,0,0,0,27,20,0,0,0,0,7,2,0,0,0,0,0,0,0,26,0,0,0,0,10,0] >;
 

94 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A···4H4I···4T7A7B7C14A···14I14J···14AA28A···28L28M···28AD
order12222···222224···44···477714···1414···1428···2828···28
size11112···2141414142···214···142222···24···42···24···4

94 irreducible representations

dim11111112222244
type++++++++++++--
imageC1C2C2C2C2C2C2D7D14D14D14D142- (1+4)D4.10D14
kernelC2×D4.10D14C22×Dic14C2×C4○D28C2×D4⋊2D7C2×Q8×D7D4.10D14C14×C4○D4C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C14C2
# reps13362161399324212

In GAP, Magma, Sage, TeX 

C_2\times D_4._{10}D_{14}
 
% in TeX
 
G:=Group("C2xD4.10D14");
 
// GroupNames label
 
G:=SmallGroup(448,1377);
 
// by ID
 
G=gap.SmallGroup(448,1377);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,297,136,1684,102,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^14=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b^2*c,c*e=e*c,e*d*e^-1=d^13>;
 
// generators/relations
 

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