metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic11⋊3D4, C23.9D22, (C2×C22)⋊3D4, (D4×C22)⋊9C2, (C2×D4)⋊5D11, D22⋊C4⋊15C2, C11⋊5(C4⋊D4), C22.51(C2×D4), (C2×C4).19D22, C2.27(D4×D11), Dic11⋊C4⋊15C2, C22.32(C4○D4), C22⋊1(C11⋊D4), (C2×C22).54C23, (C2×C44).62C22, C23.D11⋊12C2, (C22×Dic11)⋊6C2, C2.18(D4⋊2D11), (C22×C22).21C22, C22.61(C22×D11), (C2×Dic11).38C22, (C22×D11).11C22, (C2×C11⋊D4)⋊6C2, C2.15(C2×C11⋊D4), SmallGroup(352,134)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic11⋊D4
G = < a,b,c,d | a22=c4=d2=1, b2=a11, bab-1=a-1, ac=ca, ad=da, cbc-1=a11b, bd=db, dcd=c-1 >
Subgroups: 538 in 94 conjugacy classes, 35 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C11, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, D11, C22, C22, C4⋊D4, Dic11, Dic11, C44, D22, C2×C22, C2×C22, C2×C22, C2×Dic11, C2×Dic11, C11⋊D4, C2×C44, D4×C11, C22×D11, C22×C22, Dic11⋊C4, D22⋊C4, C23.D11, C22×Dic11, C2×C11⋊D4, D4×C22, Dic11⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D11, C4⋊D4, D22, C11⋊D4, C22×D11, D4×D11, D4⋊2D11, C2×C11⋊D4, Dic11⋊D4
►On 176 points
Generators in S176
(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)
(1 109 12 98)(2 108 13 97)(3 107 14 96)(4 106 15 95)(5 105 16 94)(6 104 17 93)(7 103 18 92)(8 102 19 91)(9 101 20 90)(10 100 21 89)(11 99 22 110)(23 66 34 55)(24 65 35 54)(25 64 36 53)(26 63 37 52)(27 62 38 51)(28 61 39 50)(29 60 40 49)(30 59 41 48)(31 58 42 47)(32 57 43 46)(33 56 44 45)(67 129 78 118)(68 128 79 117)(69 127 80 116)(70 126 81 115)(71 125 82 114)(72 124 83 113)(73 123 84 112)(74 122 85 111)(75 121 86 132)(76 120 87 131)(77 119 88 130)(133 174 144 163)(134 173 145 162)(135 172 146 161)(136 171 147 160)(137 170 148 159)(138 169 149 158)(139 168 150 157)(140 167 151 156)(141 166 152 155)(142 165 153 176)(143 164 154 175)
(1 40 151 111)(2 41 152 112)(3 42 153 113)(4 43 154 114)(5 44 133 115)(6 23 134 116)(7 24 135 117)(8 25 136 118)(9 26 137 119)(10 27 138 120)(11 28 139 121)(12 29 140 122)(13 30 141 123)(14 31 142 124)(15 32 143 125)(16 33 144 126)(17 34 145 127)(18 35 146 128)(19 36 147 129)(20 37 148 130)(21 38 149 131)(22 39 150 132)(45 163 70 94)(46 164 71 95)(47 165 72 96)(48 166 73 97)(49 167 74 98)(50 168 75 99)(51 169 76 100)(52 170 77 101)(53 171 78 102)(54 172 79 103)(55 173 80 104)(56 174 81 105)(57 175 82 106)(58 176 83 107)(59 155 84 108)(60 156 85 109)(61 157 86 110)(62 158 87 89)(63 159 88 90)(64 160 67 91)(65 161 68 92)(66 162 69 93)
(1 60)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 92)(24 93)(25 94)(26 95)(27 96)(28 97)(29 98)(30 99)(31 100)(32 101)(33 102)(34 103)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(41 110)(42 89)(43 90)(44 91)(67 133)(68 134)(69 135)(70 136)(71 137)(72 138)(73 139)(74 140)(75 141)(76 142)(77 143)(78 144)(79 145)(80 146)(81 147)(82 148)(83 149)(84 150)(85 151)(86 152)(87 153)(88 154)(111 156)(112 157)(113 158)(114 159)(115 160)(116 161)(117 162)(118 163)(119 164)(120 165)(121 166)(122 167)(123 168)(124 169)(125 170)(126 171)(127 172)(128 173)(129 174)(130 175)(131 176)(132 155)
G:=sub<Sym(176)| (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), (1,109,12,98)(2,108,13,97)(3,107,14,96)(4,106,15,95)(5,105,16,94)(6,104,17,93)(7,103,18,92)(8,102,19,91)(9,101,20,90)(10,100,21,89)(11,99,22,110)(23,66,34,55)(24,65,35,54)(25,64,36,53)(26,63,37,52)(27,62,38,51)(28,61,39,50)(29,60,40,49)(30,59,41,48)(31,58,42,47)(32,57,43,46)(33,56,44,45)(67,129,78,118)(68,128,79,117)(69,127,80,116)(70,126,81,115)(71,125,82,114)(72,124,83,113)(73,123,84,112)(74,122,85,111)(75,121,86,132)(76,120,87,131)(77,119,88,130)(133,174,144,163)(134,173,145,162)(135,172,146,161)(136,171,147,160)(137,170,148,159)(138,169,149,158)(139,168,150,157)(140,167,151,156)(141,166,152,155)(142,165,153,176)(143,164,154,175), (1,40,151,111)(2,41,152,112)(3,42,153,113)(4,43,154,114)(5,44,133,115)(6,23,134,116)(7,24,135,117)(8,25,136,118)(9,26,137,119)(10,27,138,120)(11,28,139,121)(12,29,140,122)(13,30,141,123)(14,31,142,124)(15,32,143,125)(16,33,144,126)(17,34,145,127)(18,35,146,128)(19,36,147,129)(20,37,148,130)(21,38,149,131)(22,39,150,132)(45,163,70,94)(46,164,71,95)(47,165,72,96)(48,166,73,97)(49,167,74,98)(50,168,75,99)(51,169,76,100)(52,170,77,101)(53,171,78,102)(54,172,79,103)(55,173,80,104)(56,174,81,105)(57,175,82,106)(58,176,83,107)(59,155,84,108)(60,156,85,109)(61,157,86,110)(62,158,87,89)(63,159,88,90)(64,160,67,91)(65,161,68,92)(66,162,69,93), (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,89)(43,90)(44,91)(67,133)(68,134)(69,135)(70,136)(71,137)(72,138)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154)(111,156)(112,157)(113,158)(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)(122,167)(123,168)(124,169)(125,170)(126,171)(127,172)(128,173)(129,174)(130,175)(131,176)(132,155)>;
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), (1,109,12,98)(2,108,13,97)(3,107,14,96)(4,106,15,95)(5,105,16,94)(6,104,17,93)(7,103,18,92)(8,102,19,91)(9,101,20,90)(10,100,21,89)(11,99,22,110)(23,66,34,55)(24,65,35,54)(25,64,36,53)(26,63,37,52)(27,62,38,51)(28,61,39,50)(29,60,40,49)(30,59,41,48)(31,58,42,47)(32,57,43,46)(33,56,44,45)(67,129,78,118)(68,128,79,117)(69,127,80,116)(70,126,81,115)(71,125,82,114)(72,124,83,113)(73,123,84,112)(74,122,85,111)(75,121,86,132)(76,120,87,131)(77,119,88,130)(133,174,144,163)(134,173,145,162)(135,172,146,161)(136,171,147,160)(137,170,148,159)(138,169,149,158)(139,168,150,157)(140,167,151,156)(141,166,152,155)(142,165,153,176)(143,164,154,175), (1,40,151,111)(2,41,152,112)(3,42,153,113)(4,43,154,114)(5,44,133,115)(6,23,134,116)(7,24,135,117)(8,25,136,118)(9,26,137,119)(10,27,138,120)(11,28,139,121)(12,29,140,122)(13,30,141,123)(14,31,142,124)(15,32,143,125)(16,33,144,126)(17,34,145,127)(18,35,146,128)(19,36,147,129)(20,37,148,130)(21,38,149,131)(22,39,150,132)(45,163,70,94)(46,164,71,95)(47,165,72,96)(48,166,73,97)(49,167,74,98)(50,168,75,99)(51,169,76,100)(52,170,77,101)(53,171,78,102)(54,172,79,103)(55,173,80,104)(56,174,81,105)(57,175,82,106)(58,176,83,107)(59,155,84,108)(60,156,85,109)(61,157,86,110)(62,158,87,89)(63,159,88,90)(64,160,67,91)(65,161,68,92)(66,162,69,93), (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,89)(43,90)(44,91)(67,133)(68,134)(69,135)(70,136)(71,137)(72,138)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154)(111,156)(112,157)(113,158)(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)(122,167)(123,168)(124,169)(125,170)(126,171)(127,172)(128,173)(129,174)(130,175)(131,176)(132,155) );
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)], [(1,109,12,98),(2,108,13,97),(3,107,14,96),(4,106,15,95),(5,105,16,94),(6,104,17,93),(7,103,18,92),(8,102,19,91),(9,101,20,90),(10,100,21,89),(11,99,22,110),(23,66,34,55),(24,65,35,54),(25,64,36,53),(26,63,37,52),(27,62,38,51),(28,61,39,50),(29,60,40,49),(30,59,41,48),(31,58,42,47),(32,57,43,46),(33,56,44,45),(67,129,78,118),(68,128,79,117),(69,127,80,116),(70,126,81,115),(71,125,82,114),(72,124,83,113),(73,123,84,112),(74,122,85,111),(75,121,86,132),(76,120,87,131),(77,119,88,130),(133,174,144,163),(134,173,145,162),(135,172,146,161),(136,171,147,160),(137,170,148,159),(138,169,149,158),(139,168,150,157),(140,167,151,156),(141,166,152,155),(142,165,153,176),(143,164,154,175)], [(1,40,151,111),(2,41,152,112),(3,42,153,113),(4,43,154,114),(5,44,133,115),(6,23,134,116),(7,24,135,117),(8,25,136,118),(9,26,137,119),(10,27,138,120),(11,28,139,121),(12,29,140,122),(13,30,141,123),(14,31,142,124),(15,32,143,125),(16,33,144,126),(17,34,145,127),(18,35,146,128),(19,36,147,129),(20,37,148,130),(21,38,149,131),(22,39,150,132),(45,163,70,94),(46,164,71,95),(47,165,72,96),(48,166,73,97),(49,167,74,98),(50,168,75,99),(51,169,76,100),(52,170,77,101),(53,171,78,102),(54,172,79,103),(55,173,80,104),(56,174,81,105),(57,175,82,106),(58,176,83,107),(59,155,84,108),(60,156,85,109),(61,157,86,110),(62,158,87,89),(63,159,88,90),(64,160,67,91),(65,161,68,92),(66,162,69,93)], [(1,60),(2,61),(3,62),(4,63),(5,64),(6,65),(7,66),(8,45),(9,46),(10,47),(11,48),(12,49),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,92),(24,93),(25,94),(26,95),(27,96),(28,97),(29,98),(30,99),(31,100),(32,101),(33,102),(34,103),(35,104),(36,105),(37,106),(38,107),(39,108),(40,109),(41,110),(42,89),(43,90),(44,91),(67,133),(68,134),(69,135),(70,136),(71,137),(72,138),(73,139),(74,140),(75,141),(76,142),(77,143),(78,144),(79,145),(80,146),(81,147),(82,148),(83,149),(84,150),(85,151),(86,152),(87,153),(88,154),(111,156),(112,157),(113,158),(114,159),(115,160),(116,161),(117,162),(118,163),(119,164),(120,165),(121,166),(122,167),(123,168),(124,169),(125,170),(126,171),(127,172),(128,173),(129,174),(130,175),(131,176),(132,155)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 11A | ··· | 11E | 22A | ··· | 22O | 22P | ··· | 22AI | 44A | ··· | 44J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 44 | 4 | 22 | 22 | 22 | 22 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D11 | D22 | D22 | C11⋊D4 | D4×D11 | D4⋊2D11 |
| kernel | Dic11⋊D4 | Dic11⋊C4 | D22⋊C4 | C23.D11 | C22×Dic11 | C2×C11⋊D4 | D4×C22 | Dic11 | C2×C22 | C22 | C2×D4 | C2×C4 | C23 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 20 | 5 | 5 |
Matrix representation of Dic11⋊D4 ►in GL4(𝔽89) generated by
| 0 | 88 | 0 | 0 |
| 1 | 34 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 14 | 27 | 0 | 0 |
| 85 | 75 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 14 | 27 | 0 | 0 |
| 62 | 75 | 0 | 0 |
| 0 | 0 | 42 | 50 |
| 0 | 0 | 27 | 47 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 47 | 39 |
| 0 | 0 | 5 | 42 |
G:=sub<GL(4,GF(89))| [0,1,0,0,88,34,0,0,0,0,1,0,0,0,0,1],[14,85,0,0,27,75,0,0,0,0,1,0,0,0,0,1],[14,62,0,0,27,75,0,0,0,0,42,27,0,0,50,47],[1,0,0,0,0,1,0,0,0,0,47,5,0,0,39,42] >;
Dic11⋊D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{11}\rtimes D_4 % in TeX
G:=Group("Dic11:D4"); // GroupNames label
G:=SmallGroup(352,134);
// by ID
G=gap.SmallGroup(352,134);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,218,188,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^22=c^4=d^2=1,b^2=a^11,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^11*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations