metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8.D11, C22.9D8, C8.5D22, C44.4D4, C11⋊2SD32, Dic44⋊3C2, C88.3C22, C11⋊C16⋊2C2, (C11×D8).1C2, C2.5(D4⋊D11), C4.2(C11⋊D4), SmallGroup(352,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8.D11
G = < a,b,c,d | a8=b2=c11=1, d2=a4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c-1 >
Smallest permutation representation of D8.D11
►On 176 points
Generators in S176
(1 76 32 65 21 87 43 54)(2 77 33 66 22 88 44 55)(3 67 23 56 12 78 34 45)(4 68 24 57 13 79 35 46)(5 69 25 58 14 80 36 47)(6 70 26 59 15 81 37 48)(7 71 27 60 16 82 38 49)(8 72 28 61 17 83 39 50)(9 73 29 62 18 84 40 51)(10 74 30 63 19 85 41 52)(11 75 31 64 20 86 42 53)(89 133 122 166 100 144 111 155)(90 134 123 167 101 145 112 156)(91 135 124 168 102 146 113 157)(92 136 125 169 103 147 114 158)(93 137 126 170 104 148 115 159)(94 138 127 171 105 149 116 160)(95 139 128 172 106 150 117 161)(96 140 129 173 107 151 118 162)(97 141 130 174 108 152 119 163)(98 142 131 175 109 153 120 164)(99 143 132 176 110 154 121 165)
(1 54)(2 55)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)(43 76)(44 77)(89 111)(90 112)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)(97 119)(98 120)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)(105 127)(106 128)(107 129)(108 130)(109 131)(110 132)(133 144)(134 145)(135 146)(136 147)(137 148)(138 149)(139 150)(140 151)(141 152)(142 153)(143 154)
(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 21 99)(2 109 22 98)(3 108 12 97)(4 107 13 96)(5 106 14 95)(6 105 15 94)(7 104 16 93)(8 103 17 92)(9 102 18 91)(10 101 19 90)(11 100 20 89)(23 130 34 119)(24 129 35 118)(25 128 36 117)(26 127 37 116)(27 126 38 115)(28 125 39 114)(29 124 40 113)(30 123 41 112)(31 122 42 111)(32 132 43 121)(33 131 44 120)(45 152 56 141)(46 151 57 140)(47 150 58 139)(48 149 59 138)(49 148 60 137)(50 147 61 136)(51 146 62 135)(52 145 63 134)(53 144 64 133)(54 154 65 143)(55 153 66 142)(67 174 78 163)(68 173 79 162)(69 172 80 161)(70 171 81 160)(71 170 82 159)(72 169 83 158)(73 168 84 157)(74 167 85 156)(75 166 86 155)(76 176 87 165)(77 175 88 164)
G:=sub<Sym(176)| (1,76,32,65,21,87,43,54)(2,77,33,66,22,88,44,55)(3,67,23,56,12,78,34,45)(4,68,24,57,13,79,35,46)(5,69,25,58,14,80,36,47)(6,70,26,59,15,81,37,48)(7,71,27,60,16,82,38,49)(8,72,28,61,17,83,39,50)(9,73,29,62,18,84,40,51)(10,74,30,63,19,85,41,52)(11,75,31,64,20,86,42,53)(89,133,122,166,100,144,111,155)(90,134,123,167,101,145,112,156)(91,135,124,168,102,146,113,157)(92,136,125,169,103,147,114,158)(93,137,126,170,104,148,115,159)(94,138,127,171,105,149,116,160)(95,139,128,172,106,150,117,161)(96,140,129,173,107,151,118,162)(97,141,130,174,108,152,119,163)(98,142,131,175,109,153,120,164)(99,143,132,176,110,154,121,165), (1,54)(2,55)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77)(89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(133,144)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,151)(141,152)(142,153)(143,154), (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), (1,110,21,99)(2,109,22,98)(3,108,12,97)(4,107,13,96)(5,106,14,95)(6,105,15,94)(7,104,16,93)(8,103,17,92)(9,102,18,91)(10,101,19,90)(11,100,20,89)(23,130,34,119)(24,129,35,118)(25,128,36,117)(26,127,37,116)(27,126,38,115)(28,125,39,114)(29,124,40,113)(30,123,41,112)(31,122,42,111)(32,132,43,121)(33,131,44,120)(45,152,56,141)(46,151,57,140)(47,150,58,139)(48,149,59,138)(49,148,60,137)(50,147,61,136)(51,146,62,135)(52,145,63,134)(53,144,64,133)(54,154,65,143)(55,153,66,142)(67,174,78,163)(68,173,79,162)(69,172,80,161)(70,171,81,160)(71,170,82,159)(72,169,83,158)(73,168,84,157)(74,167,85,156)(75,166,86,155)(76,176,87,165)(77,175,88,164)>;
G:=Group( (1,76,32,65,21,87,43,54)(2,77,33,66,22,88,44,55)(3,67,23,56,12,78,34,45)(4,68,24,57,13,79,35,46)(5,69,25,58,14,80,36,47)(6,70,26,59,15,81,37,48)(7,71,27,60,16,82,38,49)(8,72,28,61,17,83,39,50)(9,73,29,62,18,84,40,51)(10,74,30,63,19,85,41,52)(11,75,31,64,20,86,42,53)(89,133,122,166,100,144,111,155)(90,134,123,167,101,145,112,156)(91,135,124,168,102,146,113,157)(92,136,125,169,103,147,114,158)(93,137,126,170,104,148,115,159)(94,138,127,171,105,149,116,160)(95,139,128,172,106,150,117,161)(96,140,129,173,107,151,118,162)(97,141,130,174,108,152,119,163)(98,142,131,175,109,153,120,164)(99,143,132,176,110,154,121,165), (1,54)(2,55)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77)(89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(133,144)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,151)(141,152)(142,153)(143,154), (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), (1,110,21,99)(2,109,22,98)(3,108,12,97)(4,107,13,96)(5,106,14,95)(6,105,15,94)(7,104,16,93)(8,103,17,92)(9,102,18,91)(10,101,19,90)(11,100,20,89)(23,130,34,119)(24,129,35,118)(25,128,36,117)(26,127,37,116)(27,126,38,115)(28,125,39,114)(29,124,40,113)(30,123,41,112)(31,122,42,111)(32,132,43,121)(33,131,44,120)(45,152,56,141)(46,151,57,140)(47,150,58,139)(48,149,59,138)(49,148,60,137)(50,147,61,136)(51,146,62,135)(52,145,63,134)(53,144,64,133)(54,154,65,143)(55,153,66,142)(67,174,78,163)(68,173,79,162)(69,172,80,161)(70,171,81,160)(71,170,82,159)(72,169,83,158)(73,168,84,157)(74,167,85,156)(75,166,86,155)(76,176,87,165)(77,175,88,164) );
G=PermutationGroup([[(1,76,32,65,21,87,43,54),(2,77,33,66,22,88,44,55),(3,67,23,56,12,78,34,45),(4,68,24,57,13,79,35,46),(5,69,25,58,14,80,36,47),(6,70,26,59,15,81,37,48),(7,71,27,60,16,82,38,49),(8,72,28,61,17,83,39,50),(9,73,29,62,18,84,40,51),(10,74,30,63,19,85,41,52),(11,75,31,64,20,86,42,53),(89,133,122,166,100,144,111,155),(90,134,123,167,101,145,112,156),(91,135,124,168,102,146,113,157),(92,136,125,169,103,147,114,158),(93,137,126,170,104,148,115,159),(94,138,127,171,105,149,116,160),(95,139,128,172,106,150,117,161),(96,140,129,173,107,151,118,162),(97,141,130,174,108,152,119,163),(98,142,131,175,109,153,120,164),(99,143,132,176,110,154,121,165)], [(1,54),(2,55),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,56),(13,57),(14,58),(15,59),(16,60),(17,61),(18,62),(19,63),(20,64),(21,65),(22,66),(23,78),(24,79),(25,80),(26,81),(27,82),(28,83),(29,84),(30,85),(31,86),(32,87),(33,88),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(40,73),(41,74),(42,75),(43,76),(44,77),(89,111),(90,112),(91,113),(92,114),(93,115),(94,116),(95,117),(96,118),(97,119),(98,120),(99,121),(100,122),(101,123),(102,124),(103,125),(104,126),(105,127),(106,128),(107,129),(108,130),(109,131),(110,132),(133,144),(134,145),(135,146),(136,147),(137,148),(138,149),(139,150),(140,151),(141,152),(142,153),(143,154)], [(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)], [(1,110,21,99),(2,109,22,98),(3,108,12,97),(4,107,13,96),(5,106,14,95),(6,105,15,94),(7,104,16,93),(8,103,17,92),(9,102,18,91),(10,101,19,90),(11,100,20,89),(23,130,34,119),(24,129,35,118),(25,128,36,117),(26,127,37,116),(27,126,38,115),(28,125,39,114),(29,124,40,113),(30,123,41,112),(31,122,42,111),(32,132,43,121),(33,131,44,120),(45,152,56,141),(46,151,57,140),(47,150,58,139),(48,149,59,138),(49,148,60,137),(50,147,61,136),(51,146,62,135),(52,145,63,134),(53,144,64,133),(54,154,65,143),(55,153,66,142),(67,174,78,163),(68,173,79,162),(69,172,80,161),(70,171,81,160),(71,170,82,159),(72,169,83,158),(73,168,84,157),(74,167,85,156),(75,166,86,155),(76,176,87,165),(77,175,88,164)]])
46 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 11A | ··· | 11E | 16A | 16B | 16C | 16D | 22A | ··· | 22E | 22F | ··· | 22O | 44A | ··· | 44E | 88A | ··· | 88J |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 11 | ··· | 11 | 16 | 16 | 16 | 16 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 8 | 2 | 88 | 2 | 2 | 2 | ··· | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 4 | ··· | 4 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | D4 | D8 | D11 | SD32 | D22 | C11⋊D4 | D4⋊D11 | D8.D11 |
| kernel | D8.D11 | C11⋊C16 | Dic44 | C11×D8 | C44 | C22 | D8 | C11 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 5 | 4 | 5 | 10 | 5 | 10 |
Matrix representation of D8.D11 ►in GL4(𝔽353) generated by
| 352 | 0 | 0 | 0 |
| 0 | 352 | 0 | 0 |
| 0 | 0 | 0 | 49 |
| 0 | 0 | 36 | 167 |
| 352 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 167 | 304 |
| 0 | 0 | 317 | 186 |
| 256 | 0 | 0 | 0 |
| 0 | 131 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 131 | 0 | 0 |
| 256 | 0 | 0 | 0 |
| 0 | 0 | 214 | 175 |
| 0 | 0 | 291 | 139 |
G:=sub<GL(4,GF(353))| [352,0,0,0,0,352,0,0,0,0,0,36,0,0,49,167],[352,0,0,0,0,1,0,0,0,0,167,317,0,0,304,186],[256,0,0,0,0,131,0,0,0,0,1,0,0,0,0,1],[0,256,0,0,131,0,0,0,0,0,214,291,0,0,175,139] >;
D8.D11 in GAP, Magma, Sage, TeX
D_8.D_{11} % in TeX
G:=Group("D8.D11"); // GroupNames label
G:=SmallGroup(352,33);
// by ID
G=gap.SmallGroup(352,33);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,96,73,218,116,122,579,297,69,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^11=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations
Export