Copied to
clipboard

?

G = C2×D14⋊3Q8  order 448 = 26·7

Direct product of C2 and D14⋊3Q8

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

Aliases: C2×D14⋊3Q8, D14⋊4(C2×Q8), (C2×Q8)⋊28D14, (C22×Q8)⋊5D7, (C22×D7)⋊7Q8, C28.258(C2×D4), (C2×C28).214D4, C14⋊5(C22⋊Q8), C22.37(Q8×D7), C4⋊Dic7⋊79C22, (Q8×C14)⋊35C22, C14.53(C22×Q8), (C2×C28).552C23, (C2×C14).305C24, Dic7⋊C4⋊75C22, C14.153(C22×D4), (C22×C4).384D14, D14⋊C4.157C22, C22.316(C23×D7), C23.341(C22×D7), (C22×C14).423C23, (C22×C28).285C22, C22.39(Q8⋊2D7), (C2×Dic7).157C23, (C23×D7).114C22, (C22×D7).241C23, (C22×Dic7).164C22, (Q8×C2×C14)⋊4C2, C7⋊6(C2×C22⋊Q8), C2.35(C2×Q8×D7), C4.98(C2×C7⋊D4), (D7×C22×C4).9C2, (C2×C4⋊Dic7)⋊46C2, (C2×C14).98(C2×Q8), (C2×Dic7⋊C4)⋊49C2, (C2×D14⋊C4).27C2, C14.127(C2×C4○D4), (C2×C14).588(C2×D4), C2.34(C2×Q8⋊2D7), (C2×C4×D7).262C22, C2.26(C22×C7⋊D4), (C2×C4).202(C7⋊D4), (C2×C4).242(C22×D7), C22.116(C2×C7⋊D4), (C2×C14).200(C4○D4), SmallGroup(448,1266)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C2×D14⋊3Q8
Chief series
C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — D7×C22×C4 — C2×D14⋊3Q8
Lower central
C7 — C2×C14 — C2×D14⋊3Q8

Subgroups: 1332 in 322 conjugacy classes, 135 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×6], C22 [×16], C7, C2×C4 [×10], C2×C4 [×24], Q8 [×8], C23, C23 [×10], D7 [×4], C14 [×3], C14 [×4], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4, C22×C4 [×2], C22×C4 [×11], C2×Q8 [×4], C2×Q8 [×4], C24, Dic7 [×6], C28 [×4], C28 [×4], D14 [×4], D14 [×12], C2×C14, C2×C14 [×6], C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C22⋊Q8 [×8], C23×C4, C22×Q8, C4×D7 [×8], C2×Dic7 [×6], C2×Dic7 [×6], C2×C28 [×10], C2×C28 [×4], C7×Q8 [×8], C22×D7 [×6], C22×D7 [×4], C22×C14, C2×C22⋊Q8, Dic7⋊C4 [×8], C4⋊Dic7 [×4], D14⋊C4 [×8], C2×C4×D7 [×4], C2×C4×D7 [×4], C22×Dic7, C22×Dic7 [×2], C22×C28, C22×C28 [×2], Q8×C14 [×4], Q8×C14 [×4], C23×D7, C2×Dic7⋊C4 [×2], C2×C4⋊Dic7, C2×D14⋊C4 [×2], D14⋊3Q8 [×8], D7×C22×C4, Q8×C2×C14, C2×D14⋊3Q8

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D7, C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, D14 [×7], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C7⋊D4 [×4], C22×D7 [×7], C2×C22⋊Q8, Q8×D7 [×2], Q8⋊2D7 [×2], C2×C7⋊D4 [×6], C23×D7, D14⋊3Q8 [×4], C2×Q8×D7, C2×Q8⋊2D7, C22×C7⋊D4, C2×D14⋊3Q8

Generators and relations
 G = < a,b,c,d,e | a2=b14=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b7c, ce=ec, ede-1=d-1 >

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

(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)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽29)

2800000
0280000
001000
000100
0000280
0000028
,
2800000
0280000
0028000
0002800
0000244
00002112
,
2800000
010000
001000
0002800
0000280
000031
,
010000
100000
000100
0028000
000010
000001
,
100000
010000
0012000
0001700
000010
000001

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,24,21,0,0,0,0,4,12],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,3,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

88 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P7A7B7C14A···14U28A···28AJ
order12···22222444444444444444477714···1428···28
size11···1141414142222444414141414282828282222···24···4

88 irreducible representations

dim1111111222222244
type++++++++-+++-+
imageC1C2C2C2C2C2C2D4Q8D7C4○D4D14D14C7⋊D4Q8×D7Q8⋊2D7
kernelC2×D14⋊3Q8C2×Dic7⋊C4C2×C4⋊Dic7C2×D14⋊C4D14⋊3Q8D7×C22×C4Q8×C2×C14C2×C28C22×D7C22×Q8C2×C14C22×C4C2×Q8C2×C4C22C22
# reps121281144349122466

In GAP, Magma, Sage, TeX 

C_2\times D_{14}\rtimes_3Q_8
 
% in TeX
 
G:=Group("C2xD14:3Q8");
 
// GroupNames label
 
G:=SmallGroup(448,1266);
 
// by ID
 
G=gap.SmallGroup(448,1266);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,675,297,136,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^14=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^7*c,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀