metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D22⋊1D4, Dic11⋊2D4, C23.5D22, (C2×D44)⋊3C2, C2.9(D4×D11), (C2×C4).7D22, D22⋊C4⋊11C2, C11⋊1(C4⋊D4), C22⋊C4⋊4D11, C22.20(C2×D4), Dic11⋊C4⋊5C2, C22.9(C4○D4), (C2×C22).25C23, (C2×C44).53C22, C2.11(D44⋊5C2), (C22×C22).14C22, (C2×Dic11).7C22, C22.43(C22×D11), (C22×D11).18C22, (C2×C4×D11)⋊11C2, (C2×C11⋊D4)⋊2C2, (C11×C22⋊C4)⋊6C2, SmallGroup(352,79)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D22⋊D4
G = < a,b,c,d | a22=b2=c4=d2=1, bab=cac-1=dad=a-1, cbc-1=a20b, dbd=a9b, dcd=c-1 >
Subgroups: 658 in 94 conjugacy classes, 33 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C11, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D11, C22, C22, C4⋊D4, Dic11, Dic11, C44, D22, D22, C2×C22, C2×C22, C4×D11, D44, C2×Dic11, C11⋊D4, C2×C44, C22×D11, C22×C22, Dic11⋊C4, D22⋊C4, C11×C22⋊C4, C2×C4×D11, C2×D44, C2×C11⋊D4, D22⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D11, C4⋊D4, D22, C22×D11, D44⋊5C2, D4×D11, D22⋊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)
(1 135)(2 134)(3 133)(4 154)(5 153)(6 152)(7 151)(8 150)(9 149)(10 148)(11 147)(12 146)(13 145)(14 144)(15 143)(16 142)(17 141)(18 140)(19 139)(20 138)(21 137)(22 136)(23 115)(24 114)(25 113)(26 112)(27 111)(28 132)(29 131)(30 130)(31 129)(32 128)(33 127)(34 126)(35 125)(36 124)(37 123)(38 122)(39 121)(40 120)(41 119)(42 118)(43 117)(44 116)(45 99)(46 98)(47 97)(48 96)(49 95)(50 94)(51 93)(52 92)(53 91)(54 90)(55 89)(56 110)(57 109)(58 108)(59 107)(60 106)(61 105)(62 104)(63 103)(64 102)(65 101)(66 100)(67 165)(68 164)(69 163)(70 162)(71 161)(72 160)(73 159)(74 158)(75 157)(76 156)(77 155)(78 176)(79 175)(80 174)(81 173)(82 172)(83 171)(84 170)(85 169)(86 168)(87 167)(88 166)
(1 61 157 25)(2 60 158 24)(3 59 159 23)(4 58 160 44)(5 57 161 43)(6 56 162 42)(7 55 163 41)(8 54 164 40)(9 53 165 39)(10 52 166 38)(11 51 167 37)(12 50 168 36)(13 49 169 35)(14 48 170 34)(15 47 171 33)(16 46 172 32)(17 45 173 31)(18 66 174 30)(19 65 175 29)(20 64 176 28)(21 63 155 27)(22 62 156 26)(67 123 149 93)(68 122 150 92)(69 121 151 91)(70 120 152 90)(71 119 153 89)(72 118 154 110)(73 117 133 109)(74 116 134 108)(75 115 135 107)(76 114 136 106)(77 113 137 105)(78 112 138 104)(79 111 139 103)(80 132 140 102)(81 131 141 101)(82 130 142 100)(83 129 143 99)(84 128 144 98)(85 127 145 97)(86 126 146 96)(87 125 147 95)(88 124 148 94)
(1 168)(2 167)(3 166)(4 165)(5 164)(6 163)(7 162)(8 161)(9 160)(10 159)(11 158)(12 157)(13 156)(14 155)(15 176)(16 175)(17 174)(18 173)(19 172)(20 171)(21 170)(22 169)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(39 44)(40 43)(41 42)(45 66)(46 65)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)(67 145)(68 144)(69 143)(70 142)(71 141)(72 140)(73 139)(74 138)(75 137)(76 136)(77 135)(78 134)(79 133)(80 154)(81 153)(82 152)(83 151)(84 150)(85 149)(86 148)(87 147)(88 146)(89 101)(90 100)(91 99)(92 98)(93 97)(94 96)(102 110)(103 109)(104 108)(105 107)(111 117)(112 116)(113 115)(118 132)(119 131)(120 130)(121 129)(122 128)(123 127)(124 126)
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), (1,135)(2,134)(3,133)(4,154)(5,153)(6,152)(7,151)(8,150)(9,149)(10,148)(11,147)(12,146)(13,145)(14,144)(15,143)(16,142)(17,141)(18,140)(19,139)(20,138)(21,137)(22,136)(23,115)(24,114)(25,113)(26,112)(27,111)(28,132)(29,131)(30,130)(31,129)(32,128)(33,127)(34,126)(35,125)(36,124)(37,123)(38,122)(39,121)(40,120)(41,119)(42,118)(43,117)(44,116)(45,99)(46,98)(47,97)(48,96)(49,95)(50,94)(51,93)(52,92)(53,91)(54,90)(55,89)(56,110)(57,109)(58,108)(59,107)(60,106)(61,105)(62,104)(63,103)(64,102)(65,101)(66,100)(67,165)(68,164)(69,163)(70,162)(71,161)(72,160)(73,159)(74,158)(75,157)(76,156)(77,155)(78,176)(79,175)(80,174)(81,173)(82,172)(83,171)(84,170)(85,169)(86,168)(87,167)(88,166), (1,61,157,25)(2,60,158,24)(3,59,159,23)(4,58,160,44)(5,57,161,43)(6,56,162,42)(7,55,163,41)(8,54,164,40)(9,53,165,39)(10,52,166,38)(11,51,167,37)(12,50,168,36)(13,49,169,35)(14,48,170,34)(15,47,171,33)(16,46,172,32)(17,45,173,31)(18,66,174,30)(19,65,175,29)(20,64,176,28)(21,63,155,27)(22,62,156,26)(67,123,149,93)(68,122,150,92)(69,121,151,91)(70,120,152,90)(71,119,153,89)(72,118,154,110)(73,117,133,109)(74,116,134,108)(75,115,135,107)(76,114,136,106)(77,113,137,105)(78,112,138,104)(79,111,139,103)(80,132,140,102)(81,131,141,101)(82,130,142,100)(83,129,143,99)(84,128,144,98)(85,127,145,97)(86,126,146,96)(87,125,147,95)(88,124,148,94), (1,168)(2,167)(3,166)(4,165)(5,164)(6,163)(7,162)(8,161)(9,160)(10,159)(11,158)(12,157)(13,156)(14,155)(15,176)(16,175)(17,174)(18,173)(19,172)(20,171)(21,170)(22,169)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(39,44)(40,43)(41,42)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)(67,145)(68,144)(69,143)(70,142)(71,141)(72,140)(73,139)(74,138)(75,137)(76,136)(77,135)(78,134)(79,133)(80,154)(81,153)(82,152)(83,151)(84,150)(85,149)(86,148)(87,147)(88,146)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(102,110)(103,109)(104,108)(105,107)(111,117)(112,116)(113,115)(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,126)>;
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), (1,135)(2,134)(3,133)(4,154)(5,153)(6,152)(7,151)(8,150)(9,149)(10,148)(11,147)(12,146)(13,145)(14,144)(15,143)(16,142)(17,141)(18,140)(19,139)(20,138)(21,137)(22,136)(23,115)(24,114)(25,113)(26,112)(27,111)(28,132)(29,131)(30,130)(31,129)(32,128)(33,127)(34,126)(35,125)(36,124)(37,123)(38,122)(39,121)(40,120)(41,119)(42,118)(43,117)(44,116)(45,99)(46,98)(47,97)(48,96)(49,95)(50,94)(51,93)(52,92)(53,91)(54,90)(55,89)(56,110)(57,109)(58,108)(59,107)(60,106)(61,105)(62,104)(63,103)(64,102)(65,101)(66,100)(67,165)(68,164)(69,163)(70,162)(71,161)(72,160)(73,159)(74,158)(75,157)(76,156)(77,155)(78,176)(79,175)(80,174)(81,173)(82,172)(83,171)(84,170)(85,169)(86,168)(87,167)(88,166), (1,61,157,25)(2,60,158,24)(3,59,159,23)(4,58,160,44)(5,57,161,43)(6,56,162,42)(7,55,163,41)(8,54,164,40)(9,53,165,39)(10,52,166,38)(11,51,167,37)(12,50,168,36)(13,49,169,35)(14,48,170,34)(15,47,171,33)(16,46,172,32)(17,45,173,31)(18,66,174,30)(19,65,175,29)(20,64,176,28)(21,63,155,27)(22,62,156,26)(67,123,149,93)(68,122,150,92)(69,121,151,91)(70,120,152,90)(71,119,153,89)(72,118,154,110)(73,117,133,109)(74,116,134,108)(75,115,135,107)(76,114,136,106)(77,113,137,105)(78,112,138,104)(79,111,139,103)(80,132,140,102)(81,131,141,101)(82,130,142,100)(83,129,143,99)(84,128,144,98)(85,127,145,97)(86,126,146,96)(87,125,147,95)(88,124,148,94), (1,168)(2,167)(3,166)(4,165)(5,164)(6,163)(7,162)(8,161)(9,160)(10,159)(11,158)(12,157)(13,156)(14,155)(15,176)(16,175)(17,174)(18,173)(19,172)(20,171)(21,170)(22,169)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(39,44)(40,43)(41,42)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)(67,145)(68,144)(69,143)(70,142)(71,141)(72,140)(73,139)(74,138)(75,137)(76,136)(77,135)(78,134)(79,133)(80,154)(81,153)(82,152)(83,151)(84,150)(85,149)(86,148)(87,147)(88,146)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(102,110)(103,109)(104,108)(105,107)(111,117)(112,116)(113,115)(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,126) );
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)], [(1,135),(2,134),(3,133),(4,154),(5,153),(6,152),(7,151),(8,150),(9,149),(10,148),(11,147),(12,146),(13,145),(14,144),(15,143),(16,142),(17,141),(18,140),(19,139),(20,138),(21,137),(22,136),(23,115),(24,114),(25,113),(26,112),(27,111),(28,132),(29,131),(30,130),(31,129),(32,128),(33,127),(34,126),(35,125),(36,124),(37,123),(38,122),(39,121),(40,120),(41,119),(42,118),(43,117),(44,116),(45,99),(46,98),(47,97),(48,96),(49,95),(50,94),(51,93),(52,92),(53,91),(54,90),(55,89),(56,110),(57,109),(58,108),(59,107),(60,106),(61,105),(62,104),(63,103),(64,102),(65,101),(66,100),(67,165),(68,164),(69,163),(70,162),(71,161),(72,160),(73,159),(74,158),(75,157),(76,156),(77,155),(78,176),(79,175),(80,174),(81,173),(82,172),(83,171),(84,170),(85,169),(86,168),(87,167),(88,166)], [(1,61,157,25),(2,60,158,24),(3,59,159,23),(4,58,160,44),(5,57,161,43),(6,56,162,42),(7,55,163,41),(8,54,164,40),(9,53,165,39),(10,52,166,38),(11,51,167,37),(12,50,168,36),(13,49,169,35),(14,48,170,34),(15,47,171,33),(16,46,172,32),(17,45,173,31),(18,66,174,30),(19,65,175,29),(20,64,176,28),(21,63,155,27),(22,62,156,26),(67,123,149,93),(68,122,150,92),(69,121,151,91),(70,120,152,90),(71,119,153,89),(72,118,154,110),(73,117,133,109),(74,116,134,108),(75,115,135,107),(76,114,136,106),(77,113,137,105),(78,112,138,104),(79,111,139,103),(80,132,140,102),(81,131,141,101),(82,130,142,100),(83,129,143,99),(84,128,144,98),(85,127,145,97),(86,126,146,96),(87,125,147,95),(88,124,148,94)], [(1,168),(2,167),(3,166),(4,165),(5,164),(6,163),(7,162),(8,161),(9,160),(10,159),(11,158),(12,157),(13,156),(14,155),(15,176),(16,175),(17,174),(18,173),(19,172),(20,171),(21,170),(22,169),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(39,44),(40,43),(41,42),(45,66),(46,65),(47,64),(48,63),(49,62),(50,61),(51,60),(52,59),(53,58),(54,57),(55,56),(67,145),(68,144),(69,143),(70,142),(71,141),(72,140),(73,139),(74,138),(75,137),(76,136),(77,135),(78,134),(79,133),(80,154),(81,153),(82,152),(83,151),(84,150),(85,149),(86,148),(87,147),(88,146),(89,101),(90,100),(91,99),(92,98),(93,97),(94,96),(102,110),(103,109),(104,108),(105,107),(111,117),(112,116),(113,115),(118,132),(119,131),(120,130),(121,129),(122,128),(123,127),(124,126)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 11A | ··· | 11E | 22A | ··· | 22O | 22P | ··· | 22Y | 44A | ··· | 44T |
| 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 | 4 | 22 | 22 | 44 | 2 | 2 | 4 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D11 | D22 | D22 | D44⋊5C2 | D4×D11 |
| kernel | D22⋊D4 | Dic11⋊C4 | D22⋊C4 | C11×C22⋊C4 | C2×C4×D11 | C2×D44 | C2×C11⋊D4 | Dic11 | D22 | C22 | C22⋊C4 | C2×C4 | C23 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 5 | 10 | 5 | 20 | 10 |
Matrix representation of D22⋊D4 ►in GL4(𝔽89) generated by
| 62 | 55 | 0 | 0 |
| 38 | 61 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 59 | 50 | 0 | 0 |
| 55 | 30 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 5 | 54 | 0 | 0 |
| 16 | 84 | 0 | 0 |
| 0 | 0 | 38 | 87 |
| 0 | 0 | 55 | 51 |
| 69 | 21 | 0 | 0 |
| 70 | 20 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 51 | 1 |
G:=sub<GL(4,GF(89))| [62,38,0,0,55,61,0,0,0,0,1,0,0,0,0,1],[59,55,0,0,50,30,0,0,0,0,88,0,0,0,0,88],[5,16,0,0,54,84,0,0,0,0,38,55,0,0,87,51],[69,70,0,0,21,20,0,0,0,0,88,51,0,0,0,1] >;
D22⋊D4 in GAP, Magma, Sage, TeX
D_{22}\rtimes D_4 % in TeX
G:=Group("D22:D4"); // GroupNames label
G:=SmallGroup(352,79);
// by ID
G=gap.SmallGroup(352,79);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,55,506,188,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^22=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^20*b,d*b*d=a^9*b,d*c*d=c^-1>;
// generators/relations