D28:6Q8 - GroupNames

G = D₂₈⋊₆Q₈ order 448 = 2⁶·7

4^th semidirect product of D₂₈ and Q₈ acting via Q₈/C₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₂₈⋊₆Q₈, C₂₈.₁₈D₈, C₄².₈₁D₁₄, C₄⋊Q₈⋊₅D₇, C₄.₁₂(Q₈×D₇), C₄⋊C₄.₈₃D₁₄, C₇⋊₅(D₄⋊Q₈), C₂₈⋊C₈⋊₃₅C₂, C₁₄.₆₀(C₂×D₈), C₂₈.₃₉(C₂×Q₈), C₄.₁₆(D₄⋊D₇), (C₄×D₂₈).₁₉C₂, (C₂×C₂₈).₁₅₇D₄, C₂₈.₈₂(C₄○D₄), C₂₈.Q₈⋊₄₃C₂, C₁₄.D₈.₁₅C₂, (C₄×C₂₈).₁₃₄C₂², (C₂×C₂₈).₄₀₅C₂³, C₄.₃₅(Q₈⋊₂D₇), C₁₄.₇₆(C₂²⋊Q₈), C₂.₁₃(D₁₄⋊₃Q₈), (C₂×D₂₈).₂₄₉C₂², C₁₄.₉₇(C₈.C₂²), C₄⋊Dic₇.₃₄₈C₂², C₂.₁₈(C₂₈.C₂³), (C₇×C₄⋊Q₈)⋊₅C₂, C₂.₁₅(C₂×D₄⋊D₇), (C₂×C₁₄).₅₃₆(C₂×D₄), (C₂×C₇⋊C₈).₁₃₈C₂², (C₂×C₄).₁₈₉(C₇⋊D₄), (C₇×C₄⋊C₄).₁₃₀C₂², (C₂×C₄).₅₀₂(C₂²×D₇), C₂².₂₀₈(C₂×C₇⋊D₄), SmallGroup(448,621)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — D₂₈⋊₆Q₈

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂×C₂₈ — C₂×D₂₈ — C₄×D₂₈ — D₂₈⋊₆Q₈

Lower central

C₇ — C₁₄ — C₂×C₂₈ — D₂₈⋊₆Q₈

Upper central

C₁ — C₂² — C₄² — C₄⋊Q₈

Generators and relations for D₂₈⋊₆Q₈
G = < a,b,c,d | a²⁸=b²=c⁴=1, d²=c², bab=a^-1, cac^-1=a¹⁵, ad=da, cbc^-1=a⁷b, bd=db, dcd^-1=c^-1 >

Subgroups: 556 in 108 conjugacy classes, 45 normal (29 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, D₇, C₁₄, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂×D₄, C₂×Q₈, Dic₇, C₂₈, C₂₈, C₂₈, D₁₄, C₂×C₁₄, D₄⋊C₄, C₄⋊C₈, C₂.D₈, C₄×D₄, C₄⋊Q₈, C₇⋊C₈, C₄×D₇, D₂₈, D₂₈, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₇×Q₈, C₂²×D₇, D₄⋊Q₈, C₂×C₇⋊C₈, C₄⋊Dic₇, D₁₄⋊C₄, C₄×C₂₈, C₇×C₄⋊C₄, C₇×C₄⋊C₄, C₂×C₄×D₇, C₂×D₂₈, Q₈×C₁₄, C₂₈⋊C₈, C₂₈.Q₈, C₁₄.D₈, C₄×D₂₈, C₇×C₄⋊Q₈, D₂₈⋊₆Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₇, D₈, C₂×D₄, C₂×Q₈, C₄○D₄, D₁₄, C₂²⋊Q₈, C₂×D₈, C₈.C₂², C₇⋊D₄, C₂²×D₇, D₄⋊Q₈, D₄⋊D₇, Q₈×D₇, Q₈⋊₂D₇, C₂×C₇⋊D₄, C₂×D₄⋊D₇, C₂₈.C₂³, D₁₄⋊₃Q₈, D₂₈⋊₆Q₈

Smallest permutation representation of D₂₈⋊₆Q₈
►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)

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

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

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

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

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

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

61 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	7A	7B	7C	8A	8B	8C	8D	14A	···	14I	28A	···	28R	28S	···	28AD
order	1	2	2	2	2	2	4	4	4	4	4	4	4	4	4	7	7	7	8	8	8	8	14	···	14	28	···	28	28	···	28
size	1	1	1	1	28	28	2	2	2	2	4	8	8	28	28	2	2	2	28	28	28	28	2	···	2	4	···	4	8	···	8

61 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+	+	+	-	+	+	+		+	+		-	+	-	+
image	C₁	C₂	C₂	C₂	C₂	C₂	Q₈	D₄	D₇	D₈	C₄○D₄	D₁₄	D₁₄	C₇⋊D₄	C₈.C₂²	D₄⋊D₇	Q₈×D₇	Q₈⋊₂D₇	C₂₈.C₂³
kernel	D₂₈⋊₆Q₈	C₂₈⋊C₈	C₂₈.Q₈	C₁₄.D₈	C₄×D₂₈	C₇×C₄⋊Q₈	D₂₈	C₂×C₂₈	C₄⋊Q₈	C₂₈	C₂₈	C₄²	C₄⋊C₄	C₂×C₄	C₁₄	C₄	C₄	C₄	C₂
# reps	1	1	2	2	1	1	2	2	3	4	2	3	6	12	1	6	3	3	6

Matrix representation of D₂₈⋊₆Q₈ ►in GL₆(𝔽₁₁₃)

112	0	0	0	0	0
0	112	0	0	0	0
0	0	1	9	0	0
0	0	104	33	0	0
0	0	0	0	1	111
0	0	0	0	1	112

112	0	0	0	0	0
111	1	0	0	0	0
0	0	1	0	0	0
0	0	104	112	0	0
0	0	0	0	1	111
0	0	0	0	0	112

112	1	0	0	0	0
111	1	0	0	0	0
0	0	79	5	0	0
0	0	108	34	0	0
0	0	0	0	51	62
0	0	0	0	82	62

15	0	0	0	0	0
30	98	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	112	0
0	0	0	0	0	112

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,104,0,0,0,0,9,33,0,0,0,0,0,0,1,1,0,0,0,0,111,112],[112,111,0,0,0,0,0,1,0,0,0,0,0,0,1,104,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,111,112],[112,111,0,0,0,0,1,1,0,0,0,0,0,0,79,108,0,0,0,0,5,34,0,0,0,0,0,0,51,82,0,0,0,0,62,62],[15,30,0,0,0,0,0,98,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112] >;

D₂₈⋊₆Q₈ in GAP, Magma, Sage, TeX

D_{28}\rtimes_6Q_8

% in TeX

G:=Group("D28:6Q8");

// GroupNames label

G:=SmallGroup(448,621);

// by ID

G=gap.SmallGroup(448,621);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,254,219,100,1123,297,136,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^28=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d^-1=c^-1>;

// generators/relations

G = D28⋊6Q8 order 448 = 26·7

4th semidirect product of D28 and Q8 acting via Q8/C4=C2

G = D₂₈⋊₆Q₈ order 448 = 2⁶·7

4^th semidirect product of D₂₈ and Q₈ acting via Q₈/C₄=C₂