C4^2.234D14 - GroupNames

G = C₄².₂₃₄D₁₄ order 448 = 2⁶·7

54^th non-split extension by C₄² of D₁₄ acting via D₁₄/D₇=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄².₂₃₄D₁₄, (D₄×Dic₇)⋊₃₀C₂, (Q₈×Dic₇)⋊₁₉C₂, C₄.₄D₄⋊₁₉D₇, (D₇×C₄²)⋊₁₀C₂, D₁₄⋊₃Q₈⋊₃₂C₂, D₁₄.₉(C₄○D₄), (C₂×D₄).₁₇₅D₁₄, C₂₈⋊₂D₄.₁₂C₂, (C₂×Q₈).₁₃₈D₁₄, C₂²⋊C₄.₇₄D₁₄, C₂₈.₆Q₈⋊₂₀C₂, Dic₇⋊₄D₄⋊₃₃C₂, D₁₄.D₄⋊₄₅C₂, C₂₈.₁₂₅(C₄○D₄), C₄.₃₈(D₄⋊₂D₇), (C₂×C₂₈).₅₀₄C₂³, (C₂×C₁₄).₂₂₄C₂⁴, (C₄×C₂₈).₁₈₇C₂², D₁₄⋊C₄.₃₆C₂², C₂³.₄₆(C₂²×D₇), Dic₇.₄₃(C₄○D₄), (D₄×C₁₄).₁₅₇C₂², C₂³.D₁₄⋊₄₁C₂, Dic₇⋊C₄.₇₀C₂², C₄⋊Dic₇.₂₃₄C₂², (C₂²×C₁₄).₅₄C₂³, (Q₈×C₁₄).₁₂₈C₂², C₂².₂₄₅(C₂³×D₇), C₂³.D₇.₅₇C₂², C₂³.₁₁D₁₄⋊₁₉C₂, C₇⋊₉(C₂³.₃₆C₂³), (C₄×Dic₇).₁₃₄C₂², (C₂×Dic₇).₃₁₀C₂³, (C₂²×D₇).₂₁₈C₂³, (C₂²×Dic₇).₁₄₄C₂², C₂.₈₀(D₇×C₄○D₄), C₁₄.₁₉₁(C₂×C₄○D₄), (C₇×C₄.₄D₄)⋊₁₆C₂, C₂.₅₆(C₂×D₄⋊₂D₇), (C₂×C₄×D₇).₂₉₈C₂², (C₂×C₄).₃₀₁(C₂²×D₇), (C₂×C₇⋊D₄).₆₂C₂², (C₇×C₂²⋊C₄).₆₆C₂², SmallGroup(448,1133)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — C₄².₂₃₄D₁₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×D₇ — C₂×C₄×D₇ — D₇×C₄² — C₄².₂₃₄D₁₄

Lower central

C₇ — C₂×C₁₄ — C₄².₂₃₄D₁₄

Upper central

C₁ — C₂² — C₄.₄D₄

Generators and relations for C₄².₂₃₄D₁₄
G = < a,b,c,d | a⁴=b⁴=c¹⁴=1, d²=a², ab=ba, cac^-1=dad^-1=ab², cbc^-1=dbd^-1=a²b, dcd^-1=a²c^-1 >

Subgroups: 940 in 234 conjugacy classes, 99 normal (43 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, D₇, C₁₄, C₁₄, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, Dic₇, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, 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, C₂×Dic₇, C₂×Dic₇, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₇×Q₈, C₂²×D₇, C₂²×C₁₄, C₂³.₃₆C₂³, C₄×Dic₇, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, C₄⋊Dic₇, D₁₄⋊C₄, C₂³.D₇, C₄×C₂₈, C₇×C₂²⋊C₄, C₂×C₄×D₇, C₂²×Dic₇, C₂×C₇⋊D₄, D₄×C₁₄, Q₈×C₁₄, C₂₈.₆Q₈, D₇×C₄², C₂³.₁₁D₁₄, C₂³.D₁₄, Dic₇⋊₄D₄, D₁₄.D₄, D₄×Dic₇, C₂₈⋊₂D₄, Q₈×Dic₇, D₁₄⋊₃Q₈, C₇×C₄.₄D₄, C₄².₂₃₄D₁₄
Quotients: C₁, C₂, C₂², C₂³, D₇, C₄○D₄, C₂⁴, D₁₄, C₂×C₄○D₄, C₂²×D₇, C₂³.₃₆C₂³, D₄⋊₂D₇, C₂³×D₇, C₂×D₄⋊₂D₇, D₇×C₄○D₄, C₄².₂₃₄D₁₄

Smallest permutation representation of C₄².₂₃₄D₁₄
►On 224 points

Generators in S₂₂₄

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

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

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

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

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

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

70 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	···	4F	4G	4H	4I	4J	4K	4L	4M	4N	4O	4P	4Q	4R	4S	4T	7A	7B	7C	14A	···	14I	14J	···	14O	28A	···	28R	28S	···	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	4	4	14	14	2	···	2	4	4	7	7	7	7	14	14	14	14	28	28	28	28	2	2	2	2	···	2	8	···	8	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
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	D₇×C₄○D₄
kernel	C₄².₂₃₄D₁₄	C₂₈.₆Q₈	D₇×C₄²	C₂³.₁₁D₁₄	C₂³.D₁₄	Dic₇⋊₄D₄	D₁₄.D₄	D₄×Dic₇	C₂₈⋊₂D₄	Q₈×Dic₇	D₁₄⋊₃Q₈	C₇×C₄.₄D₄	C₄.₄D₄	Dic₇	C₂₈	D₁₄	C₄²	C₂²⋊C₄	C₂×D₄	C₂×Q₈	C₄	C₂
# reps	1	1	1	2	2	2	2	1	1	1	1	1	3	4	4	4	3	12	3	3	6	12

Matrix representation of C₄².₂₃₄D₁₄ ►in GL₆(𝔽₂₉)

12	27	0	0	0	0
28	17	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	0	17

17	0	0	0	0	0
0	17	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	28

1	24	0	0	0	0
0	28	0	0	0	0
0	0	9	10	0	0
0	0	9	23	0	0
0	0	0	0	0	1
0	0	0	0	1	0

28	5	0	0	0	0
0	1	0	0	0	0
0	0	7	7	0	0
0	0	18	22	0	0
0	0	0	0	0	28
0	0	0	0	1	0

G:=sub<GL(6,GF(29))| [12,28,0,0,0,0,27,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,24,28,0,0,0,0,0,0,9,9,0,0,0,0,10,23,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[28,0,0,0,0,0,5,1,0,0,0,0,0,0,7,18,0,0,0,0,7,22,0,0,0,0,0,0,0,1,0,0,0,0,28,0] >;

C₄².₂₃₄D₁₄ in GAP, Magma, Sage, TeX

C_4^2._{234}D_{14}

% in TeX

G:=Group("C4^2.234D14");

// GroupNames label

G:=SmallGroup(448,1133);

// by ID

G=gap.SmallGroup(448,1133);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,100,1123,346,297,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^-1>;

// generators/relations

G = C42.234D14 order 448 = 26·7

54th non-split extension by C42 of D14 acting via D14/D7=C2

G = C₄².₂₃₄D₁₄ order 448 = 2⁶·7

54^th non-split extension by C₄² of D₁₄ acting via D₁₄/D₇=C₂