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G = (Q8×Dic7)⋊C2  order 448 = 26·7

10th semidirect product of Q8×Dic7 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C28⋊Q8⋊22C2, (C2×Dic7)⋊9Q8, C4⋊C4.186D14, (Q8×Dic7)⋊10C2, C22.5(Q8×D7), Dic7.3(C2×Q8), C22⋊Q8.10D7, Dic7.Q8⋊15C2, (C2×C28).48C23, (C2×Q8).121D14, C22⋊C4.52D14, Dic7⋊3Q8⋊23C2, Dic7⋊Q8⋊11C2, C28.207(C4○D4), C4.70(D4⋊2D7), C14.32(C22×Q8), (C2×C14).166C24, Dic7.6(C4○D4), (C22×C4).369D14, C28.48D4.15C2, Dic7⋊C4.22C22, C4⋊Dic7.211C22, C22⋊Dic14.2C2, (Q8×C14).101C22, (C2×Dic7).83C23, C22.187(C23×D7), C23.184(C22×D7), C23.D7.30C22, (C22×C28).247C22, (C22×C14).194C23, C7⋊4(C23.37C23), (C4×Dic7).255C22, C23.11D14.1C2, (C2×Dic14).157C22, (C22×Dic7).222C22, C2.15(C2×Q8×D7), C2.45(D7×C4○D4), (C2×C14).5(C2×Q8), C14.88(C2×C4○D4), (C2×C4×Dic7).15C2, (C7×C22⋊Q8).7C2, C2.43(C2×D4⋊2D7), (C7×C4⋊C4).152C22, (C2×C4).295(C22×D7), (C7×C22⋊C4).21C22, SmallGroup(448,1075)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — (Q8×Dic7)⋊C2
Chief series
C1 — C7 — C14 — C2×C14 — C2×Dic7 — C22×Dic7 — C2×C4×Dic7 — (Q8×Dic7)⋊C2
Lower central
C7 — C2×C14 — (Q8×Dic7)⋊C2

Subgroups: 796 in 222 conjugacy classes, 109 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×16], C22, C22 [×2], C22 [×2], C7, C2×C4 [×2], C2×C4 [×4], C2×C4 [×16], Q8 [×8], C23, C14 [×3], C14 [×2], C42 [×8], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×13], C22×C4, C22×C4 [×2], C2×Q8, C2×Q8 [×3], Dic7 [×6], Dic7 [×5], C28 [×2], C28 [×5], C2×C14, C2×C14 [×2], C2×C14 [×2], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8, C22⋊Q8 [×3], C42.C2 [×2], C4⋊Q8 [×2], Dic14 [×6], C2×Dic7 [×4], C2×Dic7 [×8], C2×Dic7 [×2], C2×C28 [×2], C2×C28 [×4], C2×C28 [×2], C7×Q8 [×2], C22×C14, C23.37C23, C4×Dic7 [×4], C4×Dic7 [×4], Dic7⋊C4 [×10], C4⋊Dic7, C4⋊Dic7 [×2], C23.D7 [×2], C7×C22⋊C4 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×Dic14, C2×Dic14 [×2], C22×Dic7 [×2], C22×C28, Q8×C14, C23.11D14 [×2], C22⋊Dic14 [×2], Dic7⋊3Q8, Dic7⋊3Q8 [×2], C28⋊Q8, Dic7.Q8 [×2], C2×C4×Dic7, C28.48D4, Dic7⋊Q8, Q8×Dic7, C7×C22⋊Q8, (Q8×Dic7)⋊C2

Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D7, C2×Q8 [×6], C4○D4 [×4], C24, D14 [×7], C22×Q8, C2×C4○D4 [×2], C22×D7 [×7], C23.37C23, D4⋊2D7 [×2], Q8×D7 [×2], C23×D7, C2×D4⋊2D7, C2×Q8×D7, D7×C4○D4, (Q8×Dic7)⋊C2

Generators and relations
 G = < a,b,c,d,e | a4=c14=e2=1, b2=a2, d2=c7, bab-1=a-1, ac=ca, ad=da, eae=ac7, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, de=ed >

Smallest permutation representation
►On 224 points
Generators in S224
(1 43 22 36)(2 44 23 37)(3 45 24 38)(4 46 25 39)(5 47 26 40)(6 48 27 41)(7 49 28 42)(8 50 15 29)(9 51 16 30)(10 52 17 31)(11 53 18 32)(12 54 19 33)(13 55 20 34)(14 56 21 35)(57 92 71 99)(58 93 72 100)(59 94 73 101)(60 95 74 102)(61 96 75 103)(62 97 76 104)(63 98 77 105)(64 85 78 106)(65 86 79 107)(66 87 80 108)(67 88 81 109)(68 89 82 110)(69 90 83 111)(70 91 84 112)(113 158 127 154)(114 159 128 141)(115 160 129 142)(116 161 130 143)(117 162 131 144)(118 163 132 145)(119 164 133 146)(120 165 134 147)(121 166 135 148)(122 167 136 149)(123 168 137 150)(124 155 138 151)(125 156 139 152)(126 157 140 153)(169 209 186 219)(170 210 187 220)(171 197 188 221)(172 198 189 222)(173 199 190 223)(174 200 191 224)(175 201 192 211)(176 202 193 212)(177 203 194 213)(178 204 195 214)(179 205 196 215)(180 206 183 216)(181 207 184 217)(182 208 185 218)

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

(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)

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

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

Matrix representation ►G ⊆ GL6(𝔽29)

2800000
0280000
00282400
000100
000001
0000280
,
100000
010000
0028000
0002800
0000170
0000012
,
7280000
100000
0028000
0002800
0000280
0000028
,
19220000
10100000
0017000
0001700
0000170
0000017
,
100000
010000
001000
00172800
0000280
000001

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,24,1,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,0,0,0,0,0,0,12],[7,1,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[19,10,0,0,0,0,22,10,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,17,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1] >;
 

70 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M···4R4S4T4U4V7A7B7C14A···14I14J···14O28A···28L28M···28X
order1222224444444444444···4444477714···1414···1428···2828···28
size11112222224444777714···14282828282222···24···44···48···8

70 irreducible representations

dim1111111111122222222444
type+++++++++++-+++++--
imageC1C2C2C2C2C2C2C2C2C2C2Q8D7C4○D4C4○D4D14D14D14D14D4⋊2D7Q8×D7D7×C4○D4
kernel(Q8×Dic7)⋊C2C23.11D14C22⋊Dic14Dic7⋊3Q8C28⋊Q8Dic7.Q8C2×C4×Dic7C28.48D4Dic7⋊Q8Q8×Dic7C7×C22⋊Q8C2×Dic7C22⋊Q8Dic7C28C22⋊C4C4⋊C4C22×C4C2×Q8C4C22C2
# reps1223121111143446933666

In GAP, Magma, Sage, TeX 

(Q_8\times Dic_7)\rtimes C_2
 
% in TeX
 
G:=Group("(Q8xDic7):C2");
 
// GroupNames label
 
G:=SmallGroup(448,1075);
 
// by ID
 
G=gap.SmallGroup(448,1075);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,100,570,185,80,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=c^14=e^2=1,b^2=a^2,d^2=c^7,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,e*a*e=a*c^7,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

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ℤ
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