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G = C14×C22⋊Q8  order 448 = 26·7

Direct product of C14 and C22⋊Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C14×C22⋊Q8, C23⋊4(C7×Q8), C4.63(D4×C14), C22⋊1(Q8×C14), (C22×C14)⋊6Q8, (C2×C28).524D4, C28.470(C2×D4), (C22×Q8)⋊3C14, (C23×C4).12C14, (C23×C28).25C2, C24.34(C2×C14), (Q8×C14)⋊48C22, C22.60(D4×C14), C14.57(C22×Q8), (C2×C14).343C24, (C2×C28).656C23, C14.182(C22×D4), C23.70(C22×C14), C22.17(C23×C14), (C23×C14).91C22, (C22×C28).444C22, (C22×C14).258C23, C2.6(D4×C2×C14), C2.3(Q8×C2×C14), (C2×C4⋊C4)⋊15C14, (C14×C4⋊C4)⋊42C2, (Q8×C2×C14)⋊15C2, (C2×C14)⋊5(C2×Q8), C4⋊C4⋊10(C2×C14), (C2×Q8)⋊8(C2×C14), C2.6(C14×C4○D4), (C7×C4⋊C4)⋊66C22, (C2×C4).135(C7×D4), C14.225(C2×C4○D4), (C2×C14).682(C2×D4), C22.30(C7×C4○D4), C22⋊C4.10(C2×C14), (C2×C22⋊C4).11C14, (C14×C22⋊C4).31C2, (C22×C4).53(C2×C14), (C2×C4).12(C22×C14), (C2×C14).230(C4○D4), (C7×C22⋊C4).144C22, SmallGroup(448,1306)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C14×C22⋊Q8
Chief series
C1 — C2 — C22 — C2×C14 — C2×C28 — Q8×C14 — C7×C22⋊Q8 — C14×C22⋊Q8
Lower central
C1 — C22 — C14×C22⋊Q8
Upper central
C1 — C22×C14 — C14×C22⋊Q8

Subgroups: 450 in 322 conjugacy classes, 194 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×10], C22 [×12], C7, C2×C4 [×16], C2×C4 [×18], Q8 [×8], C23, C23 [×6], C23 [×4], C14 [×3], C14 [×4], C14 [×4], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×4], C24, C28 [×4], C28 [×10], C2×C14, C2×C14 [×10], C2×C14 [×12], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C22⋊Q8 [×8], C23×C4, C22×Q8, C2×C28 [×16], C2×C28 [×18], C7×Q8 [×8], C22×C14, C22×C14 [×6], C22×C14 [×4], C2×C22⋊Q8, C7×C22⋊C4 [×8], C7×C4⋊C4 [×12], C22×C28 [×2], C22×C28 [×8], C22×C28 [×4], Q8×C14 [×4], Q8×C14 [×4], C23×C14, C14×C22⋊C4 [×2], C14×C4⋊C4, C14×C4⋊C4 [×2], C7×C22⋊Q8 [×8], C23×C28, Q8×C2×C14, C14×C22⋊Q8

Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×4], Q8 [×4], C23 [×15], C14 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C2×C14 [×35], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C7×D4 [×4], C7×Q8 [×4], C22×C14 [×15], C2×C22⋊Q8, D4×C14 [×6], Q8×C14 [×6], C7×C4○D4 [×2], C23×C14, C7×C22⋊Q8 [×4], D4×C2×C14, Q8×C2×C14, C14×C4○D4, C14×C22⋊Q8

Generators and relations
 G = < a,b,c,d,e | a14=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽29)

280000
020000
002000
000130
000013
,
10000
028000
03100
000280
000028
,
10000
028000
002800
00010
00001
,
10000
017000
071200
000170
000412
,
280000
016100
041300
000102
0002219

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,20,0,0,0,0,0,20,0,0,0,0,0,13,0,0,0,0,0,13],[1,0,0,0,0,0,28,3,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,17,7,0,0,0,0,12,0,0,0,0,0,17,4,0,0,0,0,12],[28,0,0,0,0,0,16,4,0,0,0,1,13,0,0,0,0,0,10,22,0,0,0,2,19] >;
 

196 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P7A···7F14A···14AP14AQ···14BN28A···28AV28AW···28CR
order12···222224···44···47···714···1414···1428···2828···28
size11···122222···24···41···11···12···22···24···4

196 irreducible representations

dim111111111111222222
type+++++++-
imageC1C2C2C2C2C2C7C14C14C14C14C14D4Q8C4○D4C7×D4C7×Q8C7×C4○D4
kernelC14×C22⋊Q8C14×C22⋊C4C14×C4⋊C4C7×C22⋊Q8C23×C28Q8×C2×C14C2×C22⋊Q8C2×C22⋊C4C2×C4⋊C4C22⋊Q8C23×C4C22×Q8C2×C28C22×C14C2×C14C2×C4C23C22
# reps123811612184866444242424

In GAP, Magma, Sage, TeX 

C_{14}\times C_2^2\rtimes Q_8
 
% in TeX
 
G:=Group("C14xC2^2:Q8");
 
// GroupNames label
 
G:=SmallGroup(448,1306);
 
// by ID
 
G=gap.SmallGroup(448,1306);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,792,4790]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^14=b^2=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,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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ℤ
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