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G = C7×D4⋊6D4  order 448 = 26·7

Direct product of C7 and D4⋊6D4

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

Aliases: C7×D4⋊6D4, C14.1172- (1+4), D4⋊6(C7×D4), (C7×D4)⋊24D4, C4⋊Q8⋊13C14, (C4×D4)⋊16C14, (D4×C28)⋊45C2, C4.41(D4×C14), C28⋊16(C4○D4), C4⋊D4⋊12C14, C28.402(C2×D4), C22⋊Q8⋊12C14, C22.6(D4×C14), C42.42(C2×C14), (C2×C14).367C24, (C4×C28).283C22, (C2×C28).675C23, C22.D4⋊9C14, C14.195(C22×D4), C2.9(C7×2- (1+4)), (D4×C14).321C22, C22.41(C23×C14), C23.41(C22×C14), (C22×C14).99C23, (Q8×C14).273C22, (C22×C28).453C22, C4⋊2(C7×C4○D4), (C7×C4⋊Q8)⋊34C2, (C14×C4⋊C4)⋊47C2, (C2×C4⋊C4)⋊20C14, C2.19(D4×C2×C14), (C2×C4○D4)⋊7C14, (C14×C4○D4)⋊23C2, C4⋊C4.31(C2×C14), (C7×C4⋊D4)⋊39C2, C2.21(C14×C4○D4), (C7×C22⋊Q8)⋊39C2, (C2×D4).66(C2×C14), C14.240(C2×C4○D4), (C2×C14).183(C2×D4), C22⋊C4.5(C2×C14), (C2×Q8).60(C2×C14), (C7×C4⋊C4).395C22, (C22×C4).65(C2×C14), (C2×C4).33(C22×C14), (C7×C22.D4)⋊28C2, (C7×C22⋊C4).87C22, SmallGroup(448,1330)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×D4⋊6D4
Chief series
C1 — C2 — C22 — C2×C14 — C2×C28 — D4×C14 — C7×C22.D4 — C7×D4⋊6D4
Lower central
C1 — C22 — C7×D4⋊6D4
Upper central
C1 — C2×C14 — C7×D4⋊6D4

Subgroups: 426 in 292 conjugacy classes, 166 normal (34 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×4], C22 [×10], C7, C2×C4 [×3], C2×C4 [×8], C2×C4 [×16], D4 [×4], D4 [×10], Q8 [×4], C23 [×4], C14 [×3], C14 [×6], C42, C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4 [×8], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×8], C28 [×4], C28 [×9], C2×C14, C2×C14 [×4], C2×C14 [×10], C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], C2×C28 [×3], C2×C28 [×8], C2×C28 [×16], C7×D4 [×4], C7×D4 [×10], C7×Q8 [×4], C22×C14 [×4], D4⋊6D4, C4×C28, C7×C22⋊C4 [×8], C7×C4⋊C4 [×2], C7×C4⋊C4 [×8], C22×C28 [×8], D4×C14 [×2], D4×C14 [×4], Q8×C14 [×2], C7×C4○D4 [×8], C14×C4⋊C4 [×2], D4×C28 [×2], C7×C4⋊D4 [×2], C7×C22⋊Q8 [×2], C7×C22.D4 [×4], C7×C4⋊Q8, C14×C4○D4 [×2], C7×D4⋊6D4

Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×4], C23 [×15], C14 [×15], C2×D4 [×6], C4○D4 [×2], C24, C2×C14 [×35], C22×D4, C2×C4○D4, 2- (1+4), C7×D4 [×4], C22×C14 [×15], D4⋊6D4, D4×C14 [×6], C7×C4○D4 [×2], C23×C14, D4×C2×C14, C14×C4○D4, C7×2- (1+4), C7×D4⋊6D4

Generators and relations
 G = < a,b,c,d,e | a7=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=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 63 39 75)(2 57 40 76)(3 58 41 77)(4 59 42 71)(5 60 36 72)(6 61 37 73)(7 62 38 74)(8 205 23 196)(9 206 24 190)(10 207 25 191)(11 208 26 192)(12 209 27 193)(13 210 28 194)(14 204 22 195)(15 199 221 211)(16 200 222 212)(17 201 223 213)(18 202 224 214)(19 203 218 215)(20 197 219 216)(21 198 220 217)(29 67 48 51)(30 68 49 52)(31 69 43 53)(32 70 44 54)(33 64 45 55)(34 65 46 56)(35 66 47 50)(78 131 94 119)(79 132 95 113)(80 133 96 114)(81 127 97 115)(82 128 98 116)(83 129 92 117)(84 130 93 118)(85 123 104 107)(86 124 105 108)(87 125 99 109)(88 126 100 110)(89 120 101 111)(90 121 102 112)(91 122 103 106)(134 175 150 187)(135 169 151 188)(136 170 152 189)(137 171 153 183)(138 172 154 184)(139 173 148 185)(140 174 149 186)(141 163 160 179)(142 164 161 180)(143 165 155 181)(144 166 156 182)(145 167 157 176)(146 168 158 177)(147 162 159 178)

(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 76)(58 77)(59 71)(60 72)(61 73)(62 74)(63 75)(106 122)(107 123)(108 124)(109 125)(110 126)(111 120)(112 121)(113 132)(114 133)(115 127)(116 128)(117 129)(118 130)(119 131)(134 150)(135 151)(136 152)(137 153)(138 154)(139 148)(140 149)(141 160)(142 161)(143 155)(144 156)(145 157)(146 158)(147 159)(190 206)(191 207)(192 208)(193 209)(194 210)(195 204)(196 205)(197 216)(198 217)(199 211)(200 212)(201 213)(202 214)(203 215)

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

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

Matrix representation ►G ⊆ GL4(𝔽29) generated by

20000
02000
0010
0001
,
02800
1000
0010
0001
,
1000
02800
0010
0001
,
28000
02800
00253
0044
,
01200
17000
002518
0044
G:=sub<GL(4,GF(29))| [20,0,0,0,0,20,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,28,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,0,28,0,0,0,0,25,4,0,0,3,4],[0,17,0,0,12,0,0,0,0,0,25,4,0,0,18,4] >;
 

175 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4H4I···4O7A···7F14A···14R14S···14AP14AQ···14BB28A···28AV28AW···28CL
order12222222224···44···47···714···1414···1414···1428···2828···28
size11112222442···24···41···11···12···24···42···24···4

175 irreducible representations

dim1111111111111111222244
type+++++++++-
imageC1C2C2C2C2C2C2C2C7C14C14C14C14C14C14C14D4C4○D4C7×D4C7×C4○D42- (1+4)C7×2- (1+4)
kernelC7×D4⋊6D4C14×C4⋊C4D4×C28C7×C4⋊D4C7×C22⋊Q8C7×C22.D4C7×C4⋊Q8C14×C4○D4D4⋊6D4C2×C4⋊C4C4×D4C4⋊D4C22⋊Q8C22.D4C4⋊Q8C2×C4○D4C7×D4C28D4C4C14C2
# reps122224126121212122461244242416

In GAP, Magma, Sage, TeX 

C_7\times D_4\rtimes_6D_4
 
% in TeX
 
G:=Group("C7xD4:6D4");
 
// GroupNames label
 
G:=SmallGroup(448,1330);
 
// by ID
 
G=gap.SmallGroup(448,1330);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,4790,604,1690]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^7=b^4=c^2=d^4=e^2=1,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,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
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