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G = C7×C23.41C23  order 448 = 26·7

Direct product of C7 and C23.41C23

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

Aliases: C7×C23.41C23, C14.1592+ (1+4), C14.1162- (1+4), (C2×C28)⋊9Q8, C4⋊Q8⋊12C14, C4.11(Q8×C14), C28.100(C2×Q8), C42.C2⋊7C14, C22⋊Q8.9C14, C22.5(Q8×C14), C42.41(C2×C14), C14.62(C22×Q8), (C4×C28).282C22, (C2×C14).364C24, (C2×C28).673C23, C42⋊C2.13C14, C2.8(C7×2- (1+4)), C23.40(C22×C14), C22.38(C23×C14), (Q8×C14).183C22, C2.11(C7×2+ (1+4)), (C22×C28).452C22, (C22×C14).263C23, (C2×C4)⋊2(C7×Q8), C2.8(Q8×C2×C14), (C7×C4⋊Q8)⋊33C2, (C2×C4⋊C4).20C14, (C14×C4⋊C4).49C2, C4⋊C4.30(C2×C14), (C2×C14).18(C2×Q8), (C2×Q8).27(C2×C14), (C7×C42.C2)⋊24C2, (C7×C22⋊Q8).19C2, C22⋊C4.18(C2×C14), (C7×C4⋊C4).394C22, (C2×C4).31(C22×C14), (C22×C4).64(C2×C14), (C7×C42⋊C2).27C2, (C7×C22⋊C4).152C22, SmallGroup(448,1327)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×C23.41C23
Chief series
C1 — C2 — C22 — C2×C14 — C2×C28 — C7×C4⋊C4 — C7×C4⋊Q8 — C7×C23.41C23
Lower central
C1 — C22 — C7×C23.41C23
Upper central
C1 — C2×C14 — C7×C23.41C23

Subgroups: 274 in 206 conjugacy classes, 162 normal (18 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×12], C22, C22 [×2], C22 [×2], C7, C2×C4 [×18], C2×C4 [×2], Q8 [×4], C23, C14 [×3], C14 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×20], C22×C4, C22×C4 [×2], C2×Q8 [×4], C28 [×4], C28 [×12], C2×C14, C2×C14 [×2], C2×C14 [×2], C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8 [×4], C2×C28 [×18], C2×C28 [×2], C7×Q8 [×4], C22×C14, C23.41C23, C4×C28 [×4], C7×C22⋊C4 [×4], C7×C4⋊C4 [×20], C22×C28, C22×C28 [×2], Q8×C14 [×4], C14×C4⋊C4, C7×C42⋊C2 [×2], C7×C22⋊Q8 [×4], C7×C42.C2 [×4], C7×C4⋊Q8 [×4], C7×C23.41C23

Quotients:
C1, C2 [×15], C22 [×35], C7, Q8 [×4], C23 [×15], C14 [×15], C2×Q8 [×6], C24, C2×C14 [×35], C22×Q8, 2+ (1+4), 2- (1+4), C7×Q8 [×4], C22×C14 [×15], C23.41C23, Q8×C14 [×6], C23×C14, Q8×C2×C14, C7×2+ (1+4), C7×2- (1+4), C7×C23.41C23

Generators and relations
 G = < a,b,c,d,e,f,g | a7=b2=c2=d2=1, e2=g2=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ebe-1=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, gfg-1=cf=fc, cg=gc, geg-1=de=ed, df=fd, dg=gd >

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)

(8 222)(9 223)(10 224)(11 218)(12 219)(13 220)(14 221)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(134 158)(135 159)(136 160)(137 161)(138 155)(139 156)(140 157)(141 152)(142 153)(143 154)(144 148)(145 149)(146 150)(147 151)(162 186)(163 187)(164 188)(165 189)(166 183)(167 184)(168 185)(169 180)(170 181)(171 182)(172 176)(173 177)(174 178)(175 179)(190 214)(191 215)(192 216)(193 217)(194 211)(195 212)(196 213)(197 208)(198 209)(199 210)(200 204)(201 205)(202 206)(203 207)

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽29)

2500000
0250000
001000
000100
000010
000001
,
2800000
0280000
001000
000100
00422280
00265028
,
100000
010000
0028000
0002800
0000280
0000028
,
2800000
0280000
001000
000100
000010
000001
,
21140000
1480000
0025720
001272727
001821117
008132124
,
100000
010000
0002800
001000
0026262828
000621
,
0280000
100000
000100
001000
002612828
0042501

G:=sub<GL(6,GF(29))| [25,0,0,0,0,0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,4,26,0,0,0,1,22,5,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[21,14,0,0,0,0,14,8,0,0,0,0,0,0,25,1,18,8,0,0,7,27,21,13,0,0,2,27,11,21,0,0,0,27,7,24],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,26,0,0,0,28,0,26,6,0,0,0,0,28,2,0,0,0,0,28,1],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,0,1,26,4,0,0,1,0,1,25,0,0,0,0,28,0,0,0,0,0,28,1] >;
 

154 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4P7A···7F14A···14R14S···14AD28A···28X28Y···28CR
order12222244444···47···714···1414···1428···2828···28
size11112222224···41···11···12···22···24···4

154 irreducible representations

dim111111111111224444
type++++++-+-
imageC1C2C2C2C2C2C7C14C14C14C14C14Q8C7×Q82+ (1+4)2- (1+4)C7×2+ (1+4)C7×2- (1+4)
kernelC7×C23.41C23C14×C4⋊C4C7×C42⋊C2C7×C22⋊Q8C7×C42.C2C7×C4⋊Q8C23.41C23C2×C4⋊C4C42⋊C2C22⋊Q8C42.C2C4⋊Q8C2×C28C2×C4C14C14C2C2
# reps11244466122424244241166

In GAP, Magma, Sage, TeX 

C_7\times C_2^3._{41}C_2^3
 
% in TeX
 
G:=Group("C7xC2^3.41C2^3");
 
// GroupNames label
 
G:=SmallGroup(448,1327);
 
// by ID
 
G=gap.SmallGroup(448,1327);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597,792,4790,1227,1192,3363]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=1,e^2=g^2=d,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*b*e^-1=b*c=c*b,b*d=d*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,g*f*g^-1=c*f=f*c,c*g=g*c,g*e*g^-1=d*e=e*d,d*f=f*d,d*g=g*d>;
 
// generators/relations
 

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