metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic11⋊4D4, C23.14D22, C11⋊D4⋊C4, C11⋊2(C4×D4), D22⋊2(C2×C4), C2.2(D4×D11), D22⋊C4⋊10C2, C22⋊C4⋊7D11, (C2×C4).28D22, C22.18(C2×D4), C22⋊1(C4×D11), Dic11⋊C4⋊9C2, Dic11⋊1(C2×C4), C22.7(C22×C4), (C4×Dic11)⋊11C2, C22.22(C4○D4), (C2×C22).22C23, (C2×C44).51C22, C2.2(D4⋊2D11), (C22×Dic11)⋊1C2, (C22×C22).11C22, C22.14(C22×D11), (C2×Dic11).47C22, (C22×D11).16C22, (C2×C4×D11)⋊9C2, C2.9(C2×C4×D11), (C2×C22)⋊2(C2×C4), (C11×C22⋊C4)⋊9C2, (C2×C11⋊D4).2C2, SmallGroup(352,76)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic11⋊4D4
G = < a,b,c,d | a22=c4=d2=1, b2=a11, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 514 in 94 conjugacy classes, 43 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C11, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D11, C22, C22, C4×D4, Dic11, Dic11, C44, D22, D22, C2×C22, C2×C22, C2×C22, C4×D11, C2×Dic11, C2×Dic11, C11⋊D4, C2×C44, C22×D11, C22×C22, C4×Dic11, Dic11⋊C4, D22⋊C4, C11×C22⋊C4, C2×C4×D11, C22×Dic11, C2×C11⋊D4, Dic11⋊4D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, D11, C4×D4, D22, C4×D11, C22×D11, C2×C4×D11, D4×D11, D4⋊2D11, Dic11⋊4D4
►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)
(1 110 12 99)(2 109 13 98)(3 108 14 97)(4 107 15 96)(5 106 16 95)(6 105 17 94)(7 104 18 93)(8 103 19 92)(9 102 20 91)(10 101 21 90)(11 100 22 89)(23 174 34 163)(24 173 35 162)(25 172 36 161)(26 171 37 160)(27 170 38 159)(28 169 39 158)(29 168 40 157)(30 167 41 156)(31 166 42 155)(32 165 43 176)(33 164 44 175)(45 133 56 144)(46 154 57 143)(47 153 58 142)(48 152 59 141)(49 151 60 140)(50 150 61 139)(51 149 62 138)(52 148 63 137)(53 147 64 136)(54 146 65 135)(55 145 66 134)(67 125 78 114)(68 124 79 113)(69 123 80 112)(70 122 81 111)(71 121 82 132)(72 120 83 131)(73 119 84 130)(74 118 85 129)(75 117 86 128)(76 116 87 127)(77 115 88 126)
(1 45 39 87)(2 66 40 86)(3 65 41 85)(4 64 42 84)(5 63 43 83)(6 62 44 82)(7 61 23 81)(8 60 24 80)(9 59 25 79)(10 58 26 78)(11 57 27 77)(12 56 28 76)(13 55 29 75)(14 54 30 74)(15 53 31 73)(16 52 32 72)(17 51 33 71)(18 50 34 70)(19 49 35 69)(20 48 36 68)(21 47 37 67)(22 46 38 88)(89 154 159 126)(90 153 160 125)(91 152 161 124)(92 151 162 123)(93 150 163 122)(94 149 164 121)(95 148 165 120)(96 147 166 119)(97 146 167 118)(98 145 168 117)(99 144 169 116)(100 143 170 115)(101 142 171 114)(102 141 172 113)(103 140 173 112)(104 139 174 111)(105 138 175 132)(106 137 176 131)(107 136 155 130)(108 135 156 129)(109 134 157 128)(110 133 158 127)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 73)(21 74)(22 75)(23 62)(24 63)(25 64)(26 65)(27 66)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)(41 58)(42 59)(43 60)(44 61)(89 117)(90 118)(91 119)(92 120)(93 121)(94 122)(95 123)(96 124)(97 125)(98 126)(99 127)(100 128)(101 129)(102 130)(103 131)(104 132)(105 111)(106 112)(107 113)(108 114)(109 115)(110 116)(133 169)(134 170)(135 171)(136 172)(137 173)(138 174)(139 175)(140 176)(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)
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), (1,110,12,99)(2,109,13,98)(3,108,14,97)(4,107,15,96)(5,106,16,95)(6,105,17,94)(7,104,18,93)(8,103,19,92)(9,102,20,91)(10,101,21,90)(11,100,22,89)(23,174,34,163)(24,173,35,162)(25,172,36,161)(26,171,37,160)(27,170,38,159)(28,169,39,158)(29,168,40,157)(30,167,41,156)(31,166,42,155)(32,165,43,176)(33,164,44,175)(45,133,56,144)(46,154,57,143)(47,153,58,142)(48,152,59,141)(49,151,60,140)(50,150,61,139)(51,149,62,138)(52,148,63,137)(53,147,64,136)(54,146,65,135)(55,145,66,134)(67,125,78,114)(68,124,79,113)(69,123,80,112)(70,122,81,111)(71,121,82,132)(72,120,83,131)(73,119,84,130)(74,118,85,129)(75,117,86,128)(76,116,87,127)(77,115,88,126), (1,45,39,87)(2,66,40,86)(3,65,41,85)(4,64,42,84)(5,63,43,83)(6,62,44,82)(7,61,23,81)(8,60,24,80)(9,59,25,79)(10,58,26,78)(11,57,27,77)(12,56,28,76)(13,55,29,75)(14,54,30,74)(15,53,31,73)(16,52,32,72)(17,51,33,71)(18,50,34,70)(19,49,35,69)(20,48,36,68)(21,47,37,67)(22,46,38,88)(89,154,159,126)(90,153,160,125)(91,152,161,124)(92,151,162,123)(93,150,163,122)(94,149,164,121)(95,148,165,120)(96,147,166,119)(97,146,167,118)(98,145,168,117)(99,144,169,116)(100,143,170,115)(101,142,171,114)(102,141,172,113)(103,140,173,112)(104,139,174,111)(105,138,175,132)(106,137,176,131)(107,136,155,130)(108,135,156,129)(109,134,157,128)(110,133,158,127), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,62)(24,63)(25,64)(26,65)(27,66)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(89,117)(90,118)(91,119)(92,120)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,111)(106,112)(107,113)(108,114)(109,115)(110,116)(133,169)(134,170)(135,171)(136,172)(137,173)(138,174)(139,175)(140,176)(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)>;
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), (1,110,12,99)(2,109,13,98)(3,108,14,97)(4,107,15,96)(5,106,16,95)(6,105,17,94)(7,104,18,93)(8,103,19,92)(9,102,20,91)(10,101,21,90)(11,100,22,89)(23,174,34,163)(24,173,35,162)(25,172,36,161)(26,171,37,160)(27,170,38,159)(28,169,39,158)(29,168,40,157)(30,167,41,156)(31,166,42,155)(32,165,43,176)(33,164,44,175)(45,133,56,144)(46,154,57,143)(47,153,58,142)(48,152,59,141)(49,151,60,140)(50,150,61,139)(51,149,62,138)(52,148,63,137)(53,147,64,136)(54,146,65,135)(55,145,66,134)(67,125,78,114)(68,124,79,113)(69,123,80,112)(70,122,81,111)(71,121,82,132)(72,120,83,131)(73,119,84,130)(74,118,85,129)(75,117,86,128)(76,116,87,127)(77,115,88,126), (1,45,39,87)(2,66,40,86)(3,65,41,85)(4,64,42,84)(5,63,43,83)(6,62,44,82)(7,61,23,81)(8,60,24,80)(9,59,25,79)(10,58,26,78)(11,57,27,77)(12,56,28,76)(13,55,29,75)(14,54,30,74)(15,53,31,73)(16,52,32,72)(17,51,33,71)(18,50,34,70)(19,49,35,69)(20,48,36,68)(21,47,37,67)(22,46,38,88)(89,154,159,126)(90,153,160,125)(91,152,161,124)(92,151,162,123)(93,150,163,122)(94,149,164,121)(95,148,165,120)(96,147,166,119)(97,146,167,118)(98,145,168,117)(99,144,169,116)(100,143,170,115)(101,142,171,114)(102,141,172,113)(103,140,173,112)(104,139,174,111)(105,138,175,132)(106,137,176,131)(107,136,155,130)(108,135,156,129)(109,134,157,128)(110,133,158,127), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,62)(24,63)(25,64)(26,65)(27,66)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(89,117)(90,118)(91,119)(92,120)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,111)(106,112)(107,113)(108,114)(109,115)(110,116)(133,169)(134,170)(135,171)(136,172)(137,173)(138,174)(139,175)(140,176)(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) );
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)], [(1,110,12,99),(2,109,13,98),(3,108,14,97),(4,107,15,96),(5,106,16,95),(6,105,17,94),(7,104,18,93),(8,103,19,92),(9,102,20,91),(10,101,21,90),(11,100,22,89),(23,174,34,163),(24,173,35,162),(25,172,36,161),(26,171,37,160),(27,170,38,159),(28,169,39,158),(29,168,40,157),(30,167,41,156),(31,166,42,155),(32,165,43,176),(33,164,44,175),(45,133,56,144),(46,154,57,143),(47,153,58,142),(48,152,59,141),(49,151,60,140),(50,150,61,139),(51,149,62,138),(52,148,63,137),(53,147,64,136),(54,146,65,135),(55,145,66,134),(67,125,78,114),(68,124,79,113),(69,123,80,112),(70,122,81,111),(71,121,82,132),(72,120,83,131),(73,119,84,130),(74,118,85,129),(75,117,86,128),(76,116,87,127),(77,115,88,126)], [(1,45,39,87),(2,66,40,86),(3,65,41,85),(4,64,42,84),(5,63,43,83),(6,62,44,82),(7,61,23,81),(8,60,24,80),(9,59,25,79),(10,58,26,78),(11,57,27,77),(12,56,28,76),(13,55,29,75),(14,54,30,74),(15,53,31,73),(16,52,32,72),(17,51,33,71),(18,50,34,70),(19,49,35,69),(20,48,36,68),(21,47,37,67),(22,46,38,88),(89,154,159,126),(90,153,160,125),(91,152,161,124),(92,151,162,123),(93,150,163,122),(94,149,164,121),(95,148,165,120),(96,147,166,119),(97,146,167,118),(98,145,168,117),(99,144,169,116),(100,143,170,115),(101,142,171,114),(102,141,172,113),(103,140,173,112),(104,139,174,111),(105,138,175,132),(106,137,176,131),(107,136,155,130),(108,135,156,129),(109,134,157,128),(110,133,158,127)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,81),(7,82),(8,83),(9,84),(10,85),(11,86),(12,87),(13,88),(14,67),(15,68),(16,69),(17,70),(18,71),(19,72),(20,73),(21,74),(22,75),(23,62),(24,63),(25,64),(26,65),(27,66),(28,45),(29,46),(30,47),(31,48),(32,49),(33,50),(34,51),(35,52),(36,53),(37,54),(38,55),(39,56),(40,57),(41,58),(42,59),(43,60),(44,61),(89,117),(90,118),(91,119),(92,120),(93,121),(94,122),(95,123),(96,124),(97,125),(98,126),(99,127),(100,128),(101,129),(102,130),(103,131),(104,132),(105,111),(106,112),(107,113),(108,114),(109,115),(110,116),(133,169),(134,170),(135,171),(136,172),(137,173),(138,174),(139,175),(140,176),(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)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 11A | ··· | 11E | 22A | ··· | 22O | 22P | ··· | 22Y | 44A | ··· | 44T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 22 | 22 | 2 | 2 | 2 | 2 | 11 | 11 | 11 | 11 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | C4○D4 | D11 | D22 | D22 | C4×D11 | D4×D11 | D4⋊2D11 |
| kernel | Dic11⋊4D4 | C4×Dic11 | Dic11⋊C4 | D22⋊C4 | C11×C22⋊C4 | C2×C4×D11 | C22×Dic11 | C2×C11⋊D4 | C11⋊D4 | Dic11 | C22 | C22⋊C4 | C2×C4 | C23 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 5 | 10 | 5 | 20 | 5 | 5 |
Matrix representation of Dic11⋊4D4 ►in GL4(𝔽89) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 88 |
| 0 | 0 | 57 | 33 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 54 | 34 |
| 0 | 0 | 32 | 35 |
| 0 | 1 | 0 | 0 |
| 88 | 0 | 0 | 0 |
| 0 | 0 | 56 | 88 |
| 0 | 0 | 20 | 33 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(89))| [1,0,0,0,0,1,0,0,0,0,1,57,0,0,88,33],[1,0,0,0,0,1,0,0,0,0,54,32,0,0,34,35],[0,88,0,0,1,0,0,0,0,0,56,20,0,0,88,33],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;
Dic11⋊4D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{11}\rtimes_4D_4 % in TeX
G:=Group("Dic11:4D4"); // GroupNames label
G:=SmallGroup(352,76);
// by ID
G=gap.SmallGroup(352,76);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,188,50,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^22=c^4=d^2=1,b^2=a^11,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations