metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D22.4D4, C23.4D22, D22⋊C4⋊5C2, C44⋊C4⋊4C2, (C2×C4).6D22, C2.8(D4×D11), C22⋊C4⋊3D11, C22.19(C2×D4), C22.8(C4○D4), C23.D11⋊4C2, Dic11⋊C4⋊10C2, (C2×C22).24C23, (C2×C44).52C22, C2.8(D4⋊2D11), C11⋊1(C22.D4), C2.10(D44⋊5C2), (C22×C22).13C22, (C2×Dic11).6C22, C22.42(C22×D11), (C22×D11).17C22, (C2×C4×D11)⋊10C2, (C11×C22⋊C4)⋊5C2, (C2×C11⋊D4).3C2, SmallGroup(352,78)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D22.D4
G = < a,b,c,d | a22=b2=c4=1, d2=a11, bab=a-1, ac=ca, ad=da, cbc-1=a11b, bd=db, dcd-1=a11c-1 >
Subgroups: 466 in 78 conjugacy classes, 31 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, C22.D4, Dic11, C44, D22, D22, C2×C22, C2×C22, C4×D11, C2×Dic11, C11⋊D4, C2×C44, C22×D11, C22×C22, Dic11⋊C4, C44⋊C4, D22⋊C4, C23.D11, C11×C22⋊C4, C2×C4×D11, C2×C11⋊D4, D22.D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D11, C22.D4, D22, C22×D11, D44⋊5C2, D4×D11, D4⋊2D11, 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)
(1 134)(2 133)(3 154)(4 153)(5 152)(6 151)(7 150)(8 149)(9 148)(10 147)(11 146)(12 145)(13 144)(14 143)(15 142)(16 141)(17 140)(18 139)(19 138)(20 137)(21 136)(22 135)(23 122)(24 121)(25 120)(26 119)(27 118)(28 117)(29 116)(30 115)(31 114)(32 113)(33 112)(34 111)(35 132)(36 131)(37 130)(38 129)(39 128)(40 127)(41 126)(42 125)(43 124)(44 123)(45 103)(46 102)(47 101)(48 100)(49 99)(50 98)(51 97)(52 96)(53 95)(54 94)(55 93)(56 92)(57 91)(58 90)(59 89)(60 110)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 164)(68 163)(69 162)(70 161)(71 160)(72 159)(73 158)(74 157)(75 156)(76 155)(77 176)(78 175)(79 174)(80 173)(81 172)(82 171)(83 170)(84 169)(85 168)(86 167)(87 166)(88 165)
(1 92 146 57)(2 93 147 58)(3 94 148 59)(4 95 149 60)(5 96 150 61)(6 97 151 62)(7 98 152 63)(8 99 153 64)(9 100 154 65)(10 101 133 66)(11 102 134 45)(12 103 135 46)(13 104 136 47)(14 105 137 48)(15 106 138 49)(16 107 139 50)(17 108 140 51)(18 109 141 52)(19 110 142 53)(20 89 143 54)(21 90 144 55)(22 91 145 56)(23 161 128 87)(24 162 129 88)(25 163 130 67)(26 164 131 68)(27 165 132 69)(28 166 111 70)(29 167 112 71)(30 168 113 72)(31 169 114 73)(32 170 115 74)(33 171 116 75)(34 172 117 76)(35 173 118 77)(36 174 119 78)(37 175 120 79)(38 176 121 80)(39 155 122 81)(40 156 123 82)(41 157 124 83)(42 158 125 84)(43 159 126 85)(44 160 127 86)
(1 120 12 131)(2 121 13 132)(3 122 14 111)(4 123 15 112)(5 124 16 113)(6 125 17 114)(7 126 18 115)(8 127 19 116)(9 128 20 117)(10 129 21 118)(11 130 22 119)(23 143 34 154)(24 144 35 133)(25 145 36 134)(26 146 37 135)(27 147 38 136)(28 148 39 137)(29 149 40 138)(30 150 41 139)(31 151 42 140)(32 152 43 141)(33 153 44 142)(45 78 56 67)(46 79 57 68)(47 80 58 69)(48 81 59 70)(49 82 60 71)(50 83 61 72)(51 84 62 73)(52 85 63 74)(53 86 64 75)(54 87 65 76)(55 88 66 77)(89 161 100 172)(90 162 101 173)(91 163 102 174)(92 164 103 175)(93 165 104 176)(94 166 105 155)(95 167 106 156)(96 168 107 157)(97 169 108 158)(98 170 109 159)(99 171 110 160)
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), (1,134)(2,133)(3,154)(4,153)(5,152)(6,151)(7,150)(8,149)(9,148)(10,147)(11,146)(12,145)(13,144)(14,143)(15,142)(16,141)(17,140)(18,139)(19,138)(20,137)(21,136)(22,135)(23,122)(24,121)(25,120)(26,119)(27,118)(28,117)(29,116)(30,115)(31,114)(32,113)(33,112)(34,111)(35,132)(36,131)(37,130)(38,129)(39,128)(40,127)(41,126)(42,125)(43,124)(44,123)(45,103)(46,102)(47,101)(48,100)(49,99)(50,98)(51,97)(52,96)(53,95)(54,94)(55,93)(56,92)(57,91)(58,90)(59,89)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,164)(68,163)(69,162)(70,161)(71,160)(72,159)(73,158)(74,157)(75,156)(76,155)(77,176)(78,175)(79,174)(80,173)(81,172)(82,171)(83,170)(84,169)(85,168)(86,167)(87,166)(88,165), (1,92,146,57)(2,93,147,58)(3,94,148,59)(4,95,149,60)(5,96,150,61)(6,97,151,62)(7,98,152,63)(8,99,153,64)(9,100,154,65)(10,101,133,66)(11,102,134,45)(12,103,135,46)(13,104,136,47)(14,105,137,48)(15,106,138,49)(16,107,139,50)(17,108,140,51)(18,109,141,52)(19,110,142,53)(20,89,143,54)(21,90,144,55)(22,91,145,56)(23,161,128,87)(24,162,129,88)(25,163,130,67)(26,164,131,68)(27,165,132,69)(28,166,111,70)(29,167,112,71)(30,168,113,72)(31,169,114,73)(32,170,115,74)(33,171,116,75)(34,172,117,76)(35,173,118,77)(36,174,119,78)(37,175,120,79)(38,176,121,80)(39,155,122,81)(40,156,123,82)(41,157,124,83)(42,158,125,84)(43,159,126,85)(44,160,127,86), (1,120,12,131)(2,121,13,132)(3,122,14,111)(4,123,15,112)(5,124,16,113)(6,125,17,114)(7,126,18,115)(8,127,19,116)(9,128,20,117)(10,129,21,118)(11,130,22,119)(23,143,34,154)(24,144,35,133)(25,145,36,134)(26,146,37,135)(27,147,38,136)(28,148,39,137)(29,149,40,138)(30,150,41,139)(31,151,42,140)(32,152,43,141)(33,153,44,142)(45,78,56,67)(46,79,57,68)(47,80,58,69)(48,81,59,70)(49,82,60,71)(50,83,61,72)(51,84,62,73)(52,85,63,74)(53,86,64,75)(54,87,65,76)(55,88,66,77)(89,161,100,172)(90,162,101,173)(91,163,102,174)(92,164,103,175)(93,165,104,176)(94,166,105,155)(95,167,106,156)(96,168,107,157)(97,169,108,158)(98,170,109,159)(99,171,110,160)>;
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), (1,134)(2,133)(3,154)(4,153)(5,152)(6,151)(7,150)(8,149)(9,148)(10,147)(11,146)(12,145)(13,144)(14,143)(15,142)(16,141)(17,140)(18,139)(19,138)(20,137)(21,136)(22,135)(23,122)(24,121)(25,120)(26,119)(27,118)(28,117)(29,116)(30,115)(31,114)(32,113)(33,112)(34,111)(35,132)(36,131)(37,130)(38,129)(39,128)(40,127)(41,126)(42,125)(43,124)(44,123)(45,103)(46,102)(47,101)(48,100)(49,99)(50,98)(51,97)(52,96)(53,95)(54,94)(55,93)(56,92)(57,91)(58,90)(59,89)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,164)(68,163)(69,162)(70,161)(71,160)(72,159)(73,158)(74,157)(75,156)(76,155)(77,176)(78,175)(79,174)(80,173)(81,172)(82,171)(83,170)(84,169)(85,168)(86,167)(87,166)(88,165), (1,92,146,57)(2,93,147,58)(3,94,148,59)(4,95,149,60)(5,96,150,61)(6,97,151,62)(7,98,152,63)(8,99,153,64)(9,100,154,65)(10,101,133,66)(11,102,134,45)(12,103,135,46)(13,104,136,47)(14,105,137,48)(15,106,138,49)(16,107,139,50)(17,108,140,51)(18,109,141,52)(19,110,142,53)(20,89,143,54)(21,90,144,55)(22,91,145,56)(23,161,128,87)(24,162,129,88)(25,163,130,67)(26,164,131,68)(27,165,132,69)(28,166,111,70)(29,167,112,71)(30,168,113,72)(31,169,114,73)(32,170,115,74)(33,171,116,75)(34,172,117,76)(35,173,118,77)(36,174,119,78)(37,175,120,79)(38,176,121,80)(39,155,122,81)(40,156,123,82)(41,157,124,83)(42,158,125,84)(43,159,126,85)(44,160,127,86), (1,120,12,131)(2,121,13,132)(3,122,14,111)(4,123,15,112)(5,124,16,113)(6,125,17,114)(7,126,18,115)(8,127,19,116)(9,128,20,117)(10,129,21,118)(11,130,22,119)(23,143,34,154)(24,144,35,133)(25,145,36,134)(26,146,37,135)(27,147,38,136)(28,148,39,137)(29,149,40,138)(30,150,41,139)(31,151,42,140)(32,152,43,141)(33,153,44,142)(45,78,56,67)(46,79,57,68)(47,80,58,69)(48,81,59,70)(49,82,60,71)(50,83,61,72)(51,84,62,73)(52,85,63,74)(53,86,64,75)(54,87,65,76)(55,88,66,77)(89,161,100,172)(90,162,101,173)(91,163,102,174)(92,164,103,175)(93,165,104,176)(94,166,105,155)(95,167,106,156)(96,168,107,157)(97,169,108,158)(98,170,109,159)(99,171,110,160) );
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)], [(1,134),(2,133),(3,154),(4,153),(5,152),(6,151),(7,150),(8,149),(9,148),(10,147),(11,146),(12,145),(13,144),(14,143),(15,142),(16,141),(17,140),(18,139),(19,138),(20,137),(21,136),(22,135),(23,122),(24,121),(25,120),(26,119),(27,118),(28,117),(29,116),(30,115),(31,114),(32,113),(33,112),(34,111),(35,132),(36,131),(37,130),(38,129),(39,128),(40,127),(41,126),(42,125),(43,124),(44,123),(45,103),(46,102),(47,101),(48,100),(49,99),(50,98),(51,97),(52,96),(53,95),(54,94),(55,93),(56,92),(57,91),(58,90),(59,89),(60,110),(61,109),(62,108),(63,107),(64,106),(65,105),(66,104),(67,164),(68,163),(69,162),(70,161),(71,160),(72,159),(73,158),(74,157),(75,156),(76,155),(77,176),(78,175),(79,174),(80,173),(81,172),(82,171),(83,170),(84,169),(85,168),(86,167),(87,166),(88,165)], [(1,92,146,57),(2,93,147,58),(3,94,148,59),(4,95,149,60),(5,96,150,61),(6,97,151,62),(7,98,152,63),(8,99,153,64),(9,100,154,65),(10,101,133,66),(11,102,134,45),(12,103,135,46),(13,104,136,47),(14,105,137,48),(15,106,138,49),(16,107,139,50),(17,108,140,51),(18,109,141,52),(19,110,142,53),(20,89,143,54),(21,90,144,55),(22,91,145,56),(23,161,128,87),(24,162,129,88),(25,163,130,67),(26,164,131,68),(27,165,132,69),(28,166,111,70),(29,167,112,71),(30,168,113,72),(31,169,114,73),(32,170,115,74),(33,171,116,75),(34,172,117,76),(35,173,118,77),(36,174,119,78),(37,175,120,79),(38,176,121,80),(39,155,122,81),(40,156,123,82),(41,157,124,83),(42,158,125,84),(43,159,126,85),(44,160,127,86)], [(1,120,12,131),(2,121,13,132),(3,122,14,111),(4,123,15,112),(5,124,16,113),(6,125,17,114),(7,126,18,115),(8,127,19,116),(9,128,20,117),(10,129,21,118),(11,130,22,119),(23,143,34,154),(24,144,35,133),(25,145,36,134),(26,146,37,135),(27,147,38,136),(28,148,39,137),(29,149,40,138),(30,150,41,139),(31,151,42,140),(32,152,43,141),(33,153,44,142),(45,78,56,67),(46,79,57,68),(47,80,58,69),(48,81,59,70),(49,82,60,71),(50,83,61,72),(51,84,62,73),(52,85,63,74),(53,86,64,75),(54,87,65,76),(55,88,66,77),(89,161,100,172),(90,162,101,173),(91,163,102,174),(92,164,103,175),(93,165,104,176),(94,166,105,155),(95,167,106,156),(96,168,107,157),(97,169,108,158),(98,170,109,159),(99,171,110,160)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 11A | ··· | 11E | 22A | ··· | 22O | 22P | ··· | 22Y | 44A | ··· | 44T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 4 | 22 | 22 | 2 | 2 | 4 | 22 | 22 | 44 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
64 irreducible representations
| dim | 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 | D4 | C4○D4 | D11 | D22 | D22 | D44⋊5C2 | D4×D11 | D4⋊2D11 |
| kernel | D22.D4 | Dic11⋊C4 | C44⋊C4 | D22⋊C4 | C23.D11 | C11×C22⋊C4 | C2×C4×D11 | C2×C11⋊D4 | D22 | C22 | C22⋊C4 | C2×C4 | C23 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 5 | 10 | 5 | 20 | 5 | 5 |
Matrix representation of D22.D4 ►in GL6(𝔽89)
| 34 | 34 | 0 | 0 | 0 | 0 |
| 55 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 88 | 0 | 0 | 0 |
| 0 | 0 | 0 | 88 | 0 | 0 |
| 0 | 0 | 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 0 | 0 | 88 |
| 34 | 34 | 0 | 0 | 0 | 0 |
| 21 | 55 | 0 | 0 | 0 | 0 |
| 0 | 0 | 88 | 0 | 0 | 0 |
| 0 | 0 | 10 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 53 |
| 0 | 0 | 0 | 0 | 0 | 88 |
| 88 | 0 | 0 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 | 0 | 0 |
| 0 | 0 | 55 | 11 | 0 | 0 |
| 0 | 0 | 0 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 22 |
| 0 | 0 | 0 | 0 | 81 | 55 |
| 88 | 0 | 0 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 0 | 0 | 0 |
| 0 | 0 | 0 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 55 | 67 |
| 0 | 0 | 0 | 0 | 0 | 34 |
G:=sub<GL(6,GF(89))| [34,55,0,0,0,0,34,21,0,0,0,0,0,0,88,0,0,0,0,0,0,88,0,0,0,0,0,0,88,0,0,0,0,0,0,88],[34,21,0,0,0,0,34,55,0,0,0,0,0,0,88,10,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,53,88],[88,0,0,0,0,0,0,88,0,0,0,0,0,0,55,0,0,0,0,0,11,34,0,0,0,0,0,0,34,81,0,0,0,0,22,55],[88,0,0,0,0,0,0,88,0,0,0,0,0,0,34,0,0,0,0,0,0,34,0,0,0,0,0,0,55,0,0,0,0,0,67,34] >;
D22.D4 in GAP, Magma, Sage, TeX
D_{22}.D_4 % in TeX
G:=Group("D22.D4"); // GroupNames label
G:=SmallGroup(352,78);
// by ID
G=gap.SmallGroup(352,78);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,55,218,188,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^22=b^2=c^4=1,d^2=a^11,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^11*b,b*d=d*b,d*c*d^-1=a^11*c^-1>;
// generators/relations