direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: D8×C22, C44.41D4, C88⋊12C22, C44.44C23, C8⋊2(C2×C22), (C2×C8)⋊3C22, (C2×C88)⋊11C2, (C2×D4)⋊4C22, D4⋊1(C2×C22), C4.6(D4×C11), (D4×C22)⋊13C2, C22.74(C2×D4), C2.11(D4×C22), (C2×C22).52D4, C4.1(C22×C22), (D4×C11)⋊10C22, C22.14(D4×C11), (C2×C44).129C22, (C2×C4).25(C2×C22), SmallGroup(352,167)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×C22
G = < a,b,c | a22=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 140 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C2×C4, D4, D4, C23, C11, C2×C8, D8, C2×D4, C22, C22, C22, C2×D8, C44, C2×C22, C2×C22, C88, C2×C44, D4×C11, D4×C11, C22×C22, C2×C88, C11×D8, D4×C22, D8×C22
Quotients: C1, C2, C22, D4, C23, C11, D8, C2×D4, C22, C2×D8, C2×C22, D4×C11, C22×C22, C11×D8, D4×C22, D8×C22
►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 83 100 27 50 127 150 156)(2 84 101 28 51 128 151 157)(3 85 102 29 52 129 152 158)(4 86 103 30 53 130 153 159)(5 87 104 31 54 131 154 160)(6 88 105 32 55 132 133 161)(7 67 106 33 56 111 134 162)(8 68 107 34 57 112 135 163)(9 69 108 35 58 113 136 164)(10 70 109 36 59 114 137 165)(11 71 110 37 60 115 138 166)(12 72 89 38 61 116 139 167)(13 73 90 39 62 117 140 168)(14 74 91 40 63 118 141 169)(15 75 92 41 64 119 142 170)(16 76 93 42 65 120 143 171)(17 77 94 43 66 121 144 172)(18 78 95 44 45 122 145 173)(19 79 96 23 46 123 146 174)(20 80 97 24 47 124 147 175)(21 81 98 25 48 125 148 176)(22 82 99 26 49 126 149 155)
(1 156)(2 157)(3 158)(4 159)(5 160)(6 161)(7 162)(8 163)(9 164)(10 165)(11 166)(12 167)(13 168)(14 169)(15 170)(16 171)(17 172)(18 173)(19 174)(20 175)(21 176)(22 155)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(41 64)(42 65)(43 66)(44 45)(67 134)(68 135)(69 136)(70 137)(71 138)(72 139)(73 140)(74 141)(75 142)(76 143)(77 144)(78 145)(79 146)(80 147)(81 148)(82 149)(83 150)(84 151)(85 152)(86 153)(87 154)(88 133)(89 116)(90 117)(91 118)(92 119)(93 120)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 127)(101 128)(102 129)(103 130)(104 131)(105 132)(106 111)(107 112)(108 113)(109 114)(110 115)
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), (1,83,100,27,50,127,150,156)(2,84,101,28,51,128,151,157)(3,85,102,29,52,129,152,158)(4,86,103,30,53,130,153,159)(5,87,104,31,54,131,154,160)(6,88,105,32,55,132,133,161)(7,67,106,33,56,111,134,162)(8,68,107,34,57,112,135,163)(9,69,108,35,58,113,136,164)(10,70,109,36,59,114,137,165)(11,71,110,37,60,115,138,166)(12,72,89,38,61,116,139,167)(13,73,90,39,62,117,140,168)(14,74,91,40,63,118,141,169)(15,75,92,41,64,119,142,170)(16,76,93,42,65,120,143,171)(17,77,94,43,66,121,144,172)(18,78,95,44,45,122,145,173)(19,79,96,23,46,123,146,174)(20,80,97,24,47,124,147,175)(21,81,98,25,48,125,148,176)(22,82,99,26,49,126,149,155), (1,156)(2,157)(3,158)(4,159)(5,160)(6,161)(7,162)(8,163)(9,164)(10,165)(11,166)(12,167)(13,168)(14,169)(15,170)(16,171)(17,172)(18,173)(19,174)(20,175)(21,176)(22,155)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,45)(67,134)(68,135)(69,136)(70,137)(71,138)(72,139)(73,140)(74,141)(75,142)(76,143)(77,144)(78,145)(79,146)(80,147)(81,148)(82,149)(83,150)(84,151)(85,152)(86,153)(87,154)(88,133)(89,116)(90,117)(91,118)(92,119)(93,120)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126)(100,127)(101,128)(102,129)(103,130)(104,131)(105,132)(106,111)(107,112)(108,113)(109,114)(110,115)>;
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), (1,83,100,27,50,127,150,156)(2,84,101,28,51,128,151,157)(3,85,102,29,52,129,152,158)(4,86,103,30,53,130,153,159)(5,87,104,31,54,131,154,160)(6,88,105,32,55,132,133,161)(7,67,106,33,56,111,134,162)(8,68,107,34,57,112,135,163)(9,69,108,35,58,113,136,164)(10,70,109,36,59,114,137,165)(11,71,110,37,60,115,138,166)(12,72,89,38,61,116,139,167)(13,73,90,39,62,117,140,168)(14,74,91,40,63,118,141,169)(15,75,92,41,64,119,142,170)(16,76,93,42,65,120,143,171)(17,77,94,43,66,121,144,172)(18,78,95,44,45,122,145,173)(19,79,96,23,46,123,146,174)(20,80,97,24,47,124,147,175)(21,81,98,25,48,125,148,176)(22,82,99,26,49,126,149,155), (1,156)(2,157)(3,158)(4,159)(5,160)(6,161)(7,162)(8,163)(9,164)(10,165)(11,166)(12,167)(13,168)(14,169)(15,170)(16,171)(17,172)(18,173)(19,174)(20,175)(21,176)(22,155)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,45)(67,134)(68,135)(69,136)(70,137)(71,138)(72,139)(73,140)(74,141)(75,142)(76,143)(77,144)(78,145)(79,146)(80,147)(81,148)(82,149)(83,150)(84,151)(85,152)(86,153)(87,154)(88,133)(89,116)(90,117)(91,118)(92,119)(93,120)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126)(100,127)(101,128)(102,129)(103,130)(104,131)(105,132)(106,111)(107,112)(108,113)(109,114)(110,115) );
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)], [(1,83,100,27,50,127,150,156),(2,84,101,28,51,128,151,157),(3,85,102,29,52,129,152,158),(4,86,103,30,53,130,153,159),(5,87,104,31,54,131,154,160),(6,88,105,32,55,132,133,161),(7,67,106,33,56,111,134,162),(8,68,107,34,57,112,135,163),(9,69,108,35,58,113,136,164),(10,70,109,36,59,114,137,165),(11,71,110,37,60,115,138,166),(12,72,89,38,61,116,139,167),(13,73,90,39,62,117,140,168),(14,74,91,40,63,118,141,169),(15,75,92,41,64,119,142,170),(16,76,93,42,65,120,143,171),(17,77,94,43,66,121,144,172),(18,78,95,44,45,122,145,173),(19,79,96,23,46,123,146,174),(20,80,97,24,47,124,147,175),(21,81,98,25,48,125,148,176),(22,82,99,26,49,126,149,155)], [(1,156),(2,157),(3,158),(4,159),(5,160),(6,161),(7,162),(8,163),(9,164),(10,165),(11,166),(12,167),(13,168),(14,169),(15,170),(16,171),(17,172),(18,173),(19,174),(20,175),(21,176),(22,155),(23,46),(24,47),(25,48),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59),(37,60),(38,61),(39,62),(40,63),(41,64),(42,65),(43,66),(44,45),(67,134),(68,135),(69,136),(70,137),(71,138),(72,139),(73,140),(74,141),(75,142),(76,143),(77,144),(78,145),(79,146),(80,147),(81,148),(82,149),(83,150),(84,151),(85,152),(86,153),(87,154),(88,133),(89,116),(90,117),(91,118),(92,119),(93,120),(94,121),(95,122),(96,123),(97,124),(98,125),(99,126),(100,127),(101,128),(102,129),(103,130),(104,131),(105,132),(106,111),(107,112),(108,113),(109,114),(110,115)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 8A | 8B | 8C | 8D | 11A | ··· | 11J | 22A | ··· | 22AD | 22AE | ··· | 22BR | 44A | ··· | 44T | 88A | ··· | 88AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C11 | C22 | C22 | C22 | D4 | D4 | D8 | D4×C11 | D4×C11 | C11×D8 |
| kernel | D8×C22 | C2×C88 | C11×D8 | D4×C22 | C2×D8 | C2×C8 | D8 | C2×D4 | C44 | C2×C22 | C22 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 10 | 10 | 40 | 20 | 1 | 1 | 4 | 10 | 10 | 40 |
Matrix representation of D8×C22 ►in GL3(𝔽89) generated by
| 88 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 0 | 32 |
| 88 | 0 | 0 |
| 0 | 57 | 32 |
| 0 | 57 | 57 |
| 1 | 0 | 0 |
| 0 | 57 | 32 |
| 0 | 32 | 32 |
G:=sub<GL(3,GF(89))| [88,0,0,0,32,0,0,0,32],[88,0,0,0,57,57,0,32,57],[1,0,0,0,57,32,0,32,32] >;
D8×C22 in GAP, Magma, Sage, TeX
D_8\times C_{22} % in TeX
G:=Group("D8xC22"); // GroupNames label
G:=SmallGroup(352,167);
// by ID
G=gap.SmallGroup(352,167);
# by ID
G:=PCGroup([6,-2,-2,-2,-11,-2,-2,1081,7924,3970,88]);
// Polycyclic
G:=Group<a,b,c|a^22=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations