C2^2:Q8:25D7 - GroupNames

G = C₂²⋊Q₈⋊₂₅D₇ order 448 = 2⁶·7

1^st semidirect product of C₂²⋊Q₈ and D₇ acting through Inn(C₂²⋊Q₈)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂²⋊Q₈⋊₂₅D₇, C₄⋊C₄.₁₈₇D₁₄, (Q₈×Dic₇)⋊₁₁C₂, D₂₈⋊C₄⋊₂₃C₂, C₂₈⋊₇D₄.₁₅C₂, (C₂×Q₈).₁₂₂D₁₄, C₂²⋊C₄.₅₄D₁₄, C₂₈.₃Q₈⋊₂₂C₂, Dic₇⋊₄D₄⋊₁₃C₂, C₂₈.₂₀₈(C₄○D₄), C₂₈.₂₃D₄⋊₁₁C₂, C₄.₇₁(D₄⋊₂D₇), (C₂×C₂₈).₅₀₂C₂³, (C₂×C₁₄).₁₆₈C₂⁴, D₁₄⋊C₄.₁₉C₂², (C₂²×C₄).₃₇₀D₁₄, Dic₇.₄₂(C₄○D₄), (C₂×D₂₈).₁₄₇C₂², C₂².D₂₈⋊₁₅C₂, Dic₇⋊C₄.₂₃C₂², C₄⋊Dic₇.₂₁₂C₂², (Q₈×C₁₄).₁₀₃C₂², C₂².₃(Q₈⋊₂D₇), (C₂²×D₇).₇₃C₂³, C₂³.₁₈₆(C₂²×D₇), C₂².₁₈₉(C₂³×D₇), (C₂²×C₁₄).₁₉₆C₂³, (C₂²×C₂₈).₂₄₈C₂², C₇⋊₇(C₂³.₃₆C₂³), (C₄×Dic₇).₁₀₂C₂², (C₂×Dic₇).₂₃₂C₂³, (C₂²×Dic₇).₂₂₃C₂², (C₂×C₄×Dic₇)⋊₉C₂, C₂.₄₆(D₇×C₄○D₄), C₄⋊C₄⋊₇D₇⋊₂₃C₂, C₄⋊C₄⋊D₇⋊₁₆C₂, (C₇×C₂²⋊Q₈)⋊₅C₂, (C₂×C₄×D₇).₉₁C₂², C₁₄.₁₅₈(C₂×C₄○D₄), C₂.₄₄(C₂×D₄⋊₂D₇), C₂.₁₅(C₂×Q₈⋊₂D₇), (C₂×C₄).₄₄(C₂²×D₇), (C₂×C₁₄).₂₅(C₄○D₄), (C₇×C₄⋊C₄).₁₅₄C₂², (C₂×C₇⋊D₄).₃₇C₂², (C₇×C₂²⋊C₄).₂₃C₂², SmallGroup(448,1077)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — C₂²⋊Q₈⋊₂₅D₇

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×Dic₇ — C₂²×Dic₇ — C₂×C₄×Dic₇ — C₂²⋊Q₈⋊₂₅D₇

Lower central

C₇ — C₂×C₁₄ — C₂²⋊Q₈⋊₂₅D₇

Upper central

C₁ — C₂² — C₂²⋊Q₈

Generators and relations for C₂²⋊Q₈⋊₂₅D₇
G = < a,b,c,d,e,f | a²=b²=c⁴=e⁷=f²=1, d²=c², dad^-1=faf=ab=ba, ac=ca, ae=ea, bc=cb, bd=db, be=eb, bf=fb, dcd^-1=fcf=c^-1, ce=ec, de=ed, fdf=bc²d, fef=e^-1 >

Subgroups: 988 in 234 conjugacy classes, 101 normal (43 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₇, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, D₇, C₁₄, C₁₄, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, Dic₇, Dic₇, C₂₈, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₄², C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄².C₂, C₄²⋊₂C₂, C₄×D₇, D₂₈, C₂×Dic₇, C₂×Dic₇, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₂×C₂₈, C₇×Q₈, C₂²×D₇, C₂²×C₁₄, C₂³.₃₆C₂³, C₄×Dic₇, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, C₄⋊Dic₇, D₁₄⋊C₄, C₇×C₂²⋊C₄, C₇×C₄⋊C₄, C₇×C₄⋊C₄, C₂×C₄×D₇, C₂×D₂₈, C₂²×Dic₇, C₂×C₇⋊D₄, C₂²×C₂₈, Q₈×C₁₄, Dic₇⋊₄D₄, C₂².D₂₈, C₂₈.₃Q₈, C₄⋊C₄⋊₇D₇, D₂₈⋊C₄, C₄⋊C₄⋊D₇, C₂×C₄×Dic₇, C₂₈⋊₇D₄, Q₈×Dic₇, C₂₈.₂₃D₄, C₇×C₂²⋊Q₈, C₂²⋊Q₈⋊₂₅D₇
Quotients: C₁, C₂, C₂², C₂³, D₇, C₄○D₄, C₂⁴, D₁₄, C₂×C₄○D₄, C₂²×D₇, C₂³.₃₆C₂³, D₄⋊₂D₇, Q₈⋊₂D₇, C₂³×D₇, C₂×D₄⋊₂D₇, C₂×Q₈⋊₂D₇, D₇×C₄○D₄, C₂²⋊Q₈⋊₂₅D₇

Smallest permutation representation of C₂²⋊Q₈⋊₂₅D₇
►On 224 points

Generators in S₂₂₄

(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(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 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 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)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(127 134)(128 135)(129 136)(130 137)(131 138)(132 139)(133 140)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(183 190)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)

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

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

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

(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)

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

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

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

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

70 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	···	4R	4S	4T	7A	7B	7C	14A	···	14I	14J	···	14O	28A	···	28L	28M	···	28X
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	4	4	4	···	4	4	4	7	7	7	14	···	14	14	···	14	28	···	28	28	···	28
size	1	1	1	1	2	2	28	28	2	2	2	2	4	4	4	4	7	7	7	7	14	···	14	28	28	2	2	2	2	···	2	4	···	4	4	···	4	8	···	8

70 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+				+	+	+	+	-	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₇	C₄○D₄	C₄○D₄	C₄○D₄	D₁₄	D₁₄	D₁₄	D₁₄	D₄⋊₂D₇	Q₈⋊₂D₇	D₇×C₄○D₄
kernel	C₂²⋊Q₈⋊₂₅D₇	Dic₇⋊₄D₄	C₂².D₂₈	C₂₈.₃Q₈	C₄⋊C₄⋊₇D₇	D₂₈⋊C₄	C₄⋊C₄⋊D₇	C₂×C₄×Dic₇	C₂₈⋊₇D₄	Q₈×Dic₇	C₂₈.₂₃D₄	C₇×C₂²⋊Q₈	C₂²⋊Q₈	Dic₇	C₂₈	C₂×C₁₄	C₂²⋊C₄	C₄⋊C₄	C₂²×C₄	C₂×Q₈	C₄	C₂²	C₂
# reps	1	2	2	1	2	1	2	1	1	1	1	1	3	4	4	4	6	9	3	3	6	6	6

Matrix representation of C₂²⋊Q₈⋊₂₅D₇ ►in GL₆(𝔽₂₉)

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	0
0	0	0	0	1	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	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	17	0
0	0	0	0	17	12

0	1	0	0	0	0
1	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	17	24
0	0	0	0	0	12

1	0	0	0	0	0
0	1	0	0	0	0
0	0	28	1	0	0
0	0	9	19	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	17	0	0	0	0
12	0	0	0	0	0
0	0	22	11	0	0
0	0	22	7	0	0
0	0	0	0	28	2
0	0	0	0	0	1

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,17,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,24,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,9,0,0,0,0,1,19,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,17,0,0,0,0,0,0,0,22,22,0,0,0,0,11,7,0,0,0,0,0,0,28,0,0,0,0,0,2,1] >;

C₂²⋊Q₈⋊₂₅D₇ in GAP, Magma, Sage, TeX

C_2^2\rtimes Q_8\rtimes_{25}D_7

% in TeX

G:=Group("C2^2:Q8:25D7");

// GroupNames label

G:=SmallGroup(448,1077);

// by ID

G=gap.SmallGroup(448,1077);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,100,794,297,136,18822]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=e^7=f^2=1,d^2=c^2,d*a*d^-1=f*a*f=a*b=b*a,a*c=c*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=f*c*f=c^-1,c*e=e*c,d*e=e*d,f*d*f=b*c^2*d,f*e*f=e^-1>;

// generators/relations

G = C22⋊Q8⋊25D7 order 448 = 26·7

1st semidirect product of C22⋊Q8 and D7 acting through Inn(C22⋊Q8)

G = C₂²⋊Q₈⋊₂₅D₇ order 448 = 2⁶·7

1^st semidirect product of C₂²⋊Q₈ and D₇ acting through Inn(C₂²⋊Q₈)