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G = C22×C4○D28  order 448 = 26·7

Direct product of C22 and C4○D28

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

Aliases: C22×C4○D28, C14.4C25, D28⋊14C23, C28.77C24, D14.1C24, C24.73D14, Dic7.2C24, Dic14⋊13C23, (C23×C4)⋊8D7, (C4×D7)⋊8C23, C7⋊D4⋊7C23, C2.5(D7×C24), (C23×C28)⋊11C2, (C2×C28)⋊15C23, (C22×C4)⋊46D14, C4.76(C23×D7), (C22×D28)⋊25C2, (C2×D28)⋊66C22, C22.7(C23×D7), (C2×C14).326C24, (C22×C28)⋊62C22, (C22×Dic14)⋊26C2, (C2×Dic14)⋊77C22, C23.347(C22×D7), (C23×C14).116C22, (C22×C14).433C23, (C2×Dic7).296C23, (C22×D7).245C23, (C23×D7).116C22, (C22×Dic7).239C22, C14⋊1(C2×C4○D4), C7⋊1(C22×C4○D4), (C2×C4×D7)⋊72C22, (D7×C22×C4)⋊26C2, (C2×C4)⋊12(C22×D7), (C2×C14)⋊13(C4○D4), (C2×C7⋊D4)⋊56C22, (C22×C7⋊D4)⋊22C2, SmallGroup(448,1368)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — C22×C4○D28
Chief series
C1 — C7 — C14 — D14 — C22×D7 — C23×D7 — D7×C22×C4 — C22×C4○D28
Lower central
C7 — C14 — C22×C4○D28

Subgroups: 3332 in 890 conjugacy classes, 463 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×8], C4 [×8], C22 [×11], C22 [×44], C7, C2×C4 [×28], C2×C4 [×44], D4 [×48], Q8 [×16], C23, C23 [×6], C23 [×24], D7 [×8], C14, C14 [×6], C14 [×4], C22×C4 [×2], C22×C4 [×12], C22×C4 [×26], C2×D4 [×36], C2×Q8 [×12], C4○D4 [×64], C24, C24 [×2], Dic7 [×8], C28 [×8], D14 [×8], D14 [×24], C2×C14 [×11], C2×C14 [×12], C23×C4, C23×C4 [×2], C22×D4 [×3], C22×Q8, C2×C4○D4 [×24], Dic14 [×16], C4×D7 [×32], D28 [×16], C2×Dic7 [×12], C7⋊D4 [×32], C2×C28 [×28], C22×D7 [×12], C22×D7 [×8], C22×C14, C22×C14 [×6], C22×C14 [×4], C22×C4○D4, C2×Dic14 [×12], C2×C4×D7 [×24], C2×D28 [×12], C4○D28 [×64], C22×Dic7 [×2], C2×C7⋊D4 [×24], C22×C28 [×2], C22×C28 [×12], C23×D7 [×2], C23×C14, C22×Dic14, D7×C22×C4 [×2], C22×D28, C2×C4○D28 [×24], C22×C7⋊D4 [×2], C23×C28, C22×C4○D28

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], D7, C4○D4 [×4], C24 [×31], D14 [×15], C2×C4○D4 [×6], C25, C22×D7 [×35], C22×C4○D4, C4○D28 [×4], C23×D7 [×15], C2×C4○D28 [×6], D7×C24, C22×C4○D28

Generators and relations
 G = < a,b,c,d,e | a2=b2=c4=e2=1, d14=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d13 >

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

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

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

(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)

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

Matrix representation ►G ⊆ GL5(𝔽29)

10000
01000
00100
000280
000028
,
280000
01000
00100
00010
00001
,
280000
012000
001200
00010
00001
,
10000
001200
012000
0002721
0001620
,
10000
001700
012000
000828
000521

G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,27,16,0,0,0,21,20],[1,0,0,0,0,0,0,12,0,0,0,17,0,0,0,0,0,0,8,5,0,0,0,28,21] >;
 

136 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2S4A···4H4I4J4K4L4M···4T7A7B7C14A···14AS28A···28AV
order12···222222···24···444444···477714···1428···28
size11···1222214···141···1222214···142222···22···2

136 irreducible representations

dim111111122222
type++++++++++
imageC1C2C2C2C2C2C2D7C4○D4D14D14C4○D28
kernelC22×C4○D28C22×Dic14D7×C22×C4C22×D28C2×C4○D28C22×C7⋊D4C23×C28C23×C4C2×C14C22×C4C24C22
# reps112124213842348

In GAP, Magma, Sage, TeX 

C_2^2\times C_4\circ D_{28}
 
% in TeX
 
G:=Group("C2^2xC4oD28");
 
// GroupNames label
 
G:=SmallGroup(448,1368);
 
// by ID
 
G=gap.SmallGroup(448,1368);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,136,1684,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^4=e^2=1,d^14=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d^13>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
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