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G = C7×C23.Q8  order 448 = 26·7

Direct product of C7 and C23.Q8

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

Aliases: C7×C23.Q8, (C2×C28).309D4, C23.3(C7×Q8), (C22×C14).3Q8, C24.10(C2×C14), C22.72(D4×C14), C22.22(Q8×C14), C2.C42⋊4C14, C14.89(C22⋊Q8), (C23×C14).7C22, C14.139(C4⋊D4), C23.79(C22×C14), (C22×C28).34C22, C14.34(C42⋊2C2), (C22×C14).460C23, (C2×C4⋊C4)⋊6C14, (C14×C4⋊C4)⋊33C2, (C2×C4).16(C7×D4), C2.8(C7×C4⋊D4), C2.8(C7×C22⋊Q8), (C2×C14).612(C2×D4), (C2×C22⋊C4).9C14, (C22×C4).7(C2×C14), C2.4(C7×C42⋊2C2), (C2×C14).110(C2×Q8), C22.39(C7×C4○D4), (C14×C22⋊C4).28C2, (C7×C2.C42)⋊6C2, (C2×C14).220(C4○D4), SmallGroup(448,804)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C7×C23.Q8
 G = < a,b,c,d,e,f | a7=b2=c2=d2=e4=1, f2=ce2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 330 in 186 conjugacy classes, 78 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C24, C28, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C23.Q8, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C23×C14, C7×C2.C42, C14×C22⋊C4, C14×C4⋊C4, C7×C23.Q8
Quotients: C1, C2, C22, C7, D4, Q8, C23, C14, C2×D4, C2×Q8, C4○D4, C2×C14, C4⋊D4, C22⋊Q8, C42⋊2C2, C7×D4, C7×Q8, C22×C14, C23.Q8, D4×C14, Q8×C14, C7×C4○D4, C7×C4⋊D4, C7×C22⋊Q8, C7×C42⋊2C2, C7×C23.Q8

Smallest permutation representation of C7×C23.Q8
►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 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 40)(16 41)(17 42)(18 36)(19 37)(20 38)(21 39)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)(105 120)(106 115)(107 116)(108 117)(109 118)(110 119)(111 113)(112 114)(127 152)(128 153)(129 154)(130 148)(131 149)(132 150)(133 151)(134 143)(135 144)(136 145)(137 146)(138 147)(139 141)(140 142)(155 173)(156 174)(157 175)(158 169)(159 170)(160 171)(161 172)(162 182)(163 176)(164 177)(165 178)(166 179)(167 180)(168 181)(183 196)(184 190)(185 191)(186 192)(187 193)(188 194)(189 195)(197 207)(198 208)(199 209)(200 210)(201 204)(202 205)(203 206)(211 220)(212 221)(213 222)(214 223)(215 224)(216 218)(217 219)

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

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

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

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

154 conjugacy classes

class 1 2A···2G2H2I4A···4L7A···7F14A···14AP14AQ···14BB28A···28BT
order12···2224···47···714···1414···1428···28
size11···1444···41···11···14···44···4

154 irreducible representations

dim11111111222222
type+++++-
imageC1C2C2C2C7C14C14C14D4Q8C4○D4C7×D4C7×Q8C7×C4○D4
kernelC7×C23.Q8C7×C2.C42C14×C22⋊C4C14×C4⋊C4C23.Q8C2.C42C2×C22⋊C4C2×C4⋊C4C2×C28C22×C14C2×C14C2×C4C23C22
# reps1133661818626361236

Matrix representation of C7×C23.Q8 ►in GL6(𝔽29)

2300000
0230000
0016000
0001600
0000230
0000023
,
100000
010000
001000
0002800
000010
0000028
,
100000
010000
0028000
0002800
0000280
0000028
,
100000
010000
0028000
0002800
000010
000001
,
010000
2800000
000100
001000
0000120
0000017
,
1480000
8150000
0001700
0017000
000001
000010

G:=sub<GL(6,GF(29))| [23,0,0,0,0,0,0,23,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,23,0,0,0,0,0,0,23],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,17],[14,8,0,0,0,0,8,15,0,0,0,0,0,0,0,17,0,0,0,0,17,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

C7×C23.Q8 in GAP, Magma, Sage, TeX 

C_7\times C_2^3.Q_8
 
% in TeX
 
G:=Group("C7xC2^3.Q8");
 
// GroupNames label
 
G:=SmallGroup(448,804);
 
// by ID
 
G=gap.SmallGroup(448,804);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1176,813,400,2438,2403]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^4=1,f^2=c*e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
 
// generators/relations
 

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