metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.2D22, D22.8D4, D4.4D22, Q8.1D22, Dic44⋊6C2, SD16⋊2D11, C44.6C23, C88.9C22, Dic11.10D4, Dic22.2C22, (Q8×D11)⋊2C2, C88⋊C2⋊2C2, D4.D11⋊4C2, D4⋊2D11.C2, C11⋊Q16⋊1C2, C22.32(C2×D4), C2.20(D4×D11), C11⋊C8.1C22, (C11×SD16)⋊2C2, C11⋊2(C8.C22), C4.6(C22×D11), (D4×C11).4C22, (C4×D11).3C22, (Q8×C11).1C22, SmallGroup(352,110)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D22
G = < a,b,c,d | a4=b2=1, c22=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a-1b, dcd-1=c21 >
Subgroups: 354 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, D4, Q8, Q8, C11, M4(2), SD16, SD16, Q16, C2×Q8, C4○D4, D11, C22, C22, C8.C22, Dic11, Dic11, C44, C44, D22, C2×C22, C11⋊C8, C88, Dic22, Dic22, C4×D11, C4×D11, C2×Dic11, C11⋊D4, D4×C11, Q8×C11, C88⋊C2, Dic44, D4.D11, C11⋊Q16, C11×SD16, D4⋊2D11, Q8×D11, D4.D22
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C8.C22, D22, C22×D11, D4×D11, D4.D22
►On 176 points
Generators in S176
(1 80 23 58)(2 59 24 81)(3 82 25 60)(4 61 26 83)(5 84 27 62)(6 63 28 85)(7 86 29 64)(8 65 30 87)(9 88 31 66)(10 67 32 45)(11 46 33 68)(12 69 34 47)(13 48 35 70)(14 71 36 49)(15 50 37 72)(16 73 38 51)(17 52 39 74)(18 75 40 53)(19 54 41 76)(20 77 42 55)(21 56 43 78)(22 79 44 57)(89 136 111 158)(90 159 112 137)(91 138 113 160)(92 161 114 139)(93 140 115 162)(94 163 116 141)(95 142 117 164)(96 165 118 143)(97 144 119 166)(98 167 120 145)(99 146 121 168)(100 169 122 147)(101 148 123 170)(102 171 124 149)(103 150 125 172)(104 173 126 151)(105 152 127 174)(106 175 128 153)(107 154 129 176)(108 133 130 155)(109 156 131 134)(110 135 132 157)
(1 145)(2 121)(3 147)(4 123)(5 149)(6 125)(7 151)(8 127)(9 153)(10 129)(11 155)(12 131)(13 157)(14 89)(15 159)(16 91)(17 161)(18 93)(19 163)(20 95)(21 165)(22 97)(23 167)(24 99)(25 169)(26 101)(27 171)(28 103)(29 173)(30 105)(31 175)(32 107)(33 133)(34 109)(35 135)(36 111)(37 137)(38 113)(39 139)(40 115)(41 141)(42 117)(43 143)(44 119)(45 176)(46 130)(47 134)(48 132)(49 136)(50 90)(51 138)(52 92)(53 140)(54 94)(55 142)(56 96)(57 144)(58 98)(59 146)(60 100)(61 148)(62 102)(63 150)(64 104)(65 152)(66 106)(67 154)(68 108)(69 156)(70 110)(71 158)(72 112)(73 160)(74 114)(75 162)(76 116)(77 164)(78 118)(79 166)(80 120)(81 168)(82 122)(83 170)(84 124)(85 172)(86 126)(87 174)(88 128)
(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 22 23 44)(2 43 24 21)(3 20 25 42)(4 41 26 19)(5 18 27 40)(6 39 28 17)(7 16 29 38)(8 37 30 15)(9 14 31 36)(10 35 32 13)(11 12 33 34)(45 70 67 48)(46 47 68 69)(49 66 71 88)(50 87 72 65)(51 64 73 86)(52 85 74 63)(53 62 75 84)(54 83 76 61)(55 60 77 82)(56 81 78 59)(57 58 79 80)(89 128 111 106)(90 105 112 127)(91 126 113 104)(92 103 114 125)(93 124 115 102)(94 101 116 123)(95 122 117 100)(96 99 118 121)(97 120 119 98)(107 110 129 132)(108 131 130 109)(133 156 155 134)(135 154 157 176)(136 175 158 153)(137 152 159 174)(138 173 160 151)(139 150 161 172)(140 171 162 149)(141 148 163 170)(142 169 164 147)(143 146 165 168)(144 167 166 145)
G:=sub<Sym(176)| (1,80,23,58)(2,59,24,81)(3,82,25,60)(4,61,26,83)(5,84,27,62)(6,63,28,85)(7,86,29,64)(8,65,30,87)(9,88,31,66)(10,67,32,45)(11,46,33,68)(12,69,34,47)(13,48,35,70)(14,71,36,49)(15,50,37,72)(16,73,38,51)(17,52,39,74)(18,75,40,53)(19,54,41,76)(20,77,42,55)(21,56,43,78)(22,79,44,57)(89,136,111,158)(90,159,112,137)(91,138,113,160)(92,161,114,139)(93,140,115,162)(94,163,116,141)(95,142,117,164)(96,165,118,143)(97,144,119,166)(98,167,120,145)(99,146,121,168)(100,169,122,147)(101,148,123,170)(102,171,124,149)(103,150,125,172)(104,173,126,151)(105,152,127,174)(106,175,128,153)(107,154,129,176)(108,133,130,155)(109,156,131,134)(110,135,132,157), (1,145)(2,121)(3,147)(4,123)(5,149)(6,125)(7,151)(8,127)(9,153)(10,129)(11,155)(12,131)(13,157)(14,89)(15,159)(16,91)(17,161)(18,93)(19,163)(20,95)(21,165)(22,97)(23,167)(24,99)(25,169)(26,101)(27,171)(28,103)(29,173)(30,105)(31,175)(32,107)(33,133)(34,109)(35,135)(36,111)(37,137)(38,113)(39,139)(40,115)(41,141)(42,117)(43,143)(44,119)(45,176)(46,130)(47,134)(48,132)(49,136)(50,90)(51,138)(52,92)(53,140)(54,94)(55,142)(56,96)(57,144)(58,98)(59,146)(60,100)(61,148)(62,102)(63,150)(64,104)(65,152)(66,106)(67,154)(68,108)(69,156)(70,110)(71,158)(72,112)(73,160)(74,114)(75,162)(76,116)(77,164)(78,118)(79,166)(80,120)(81,168)(82,122)(83,170)(84,124)(85,172)(86,126)(87,174)(88,128), (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), (1,22,23,44)(2,43,24,21)(3,20,25,42)(4,41,26,19)(5,18,27,40)(6,39,28,17)(7,16,29,38)(8,37,30,15)(9,14,31,36)(10,35,32,13)(11,12,33,34)(45,70,67,48)(46,47,68,69)(49,66,71,88)(50,87,72,65)(51,64,73,86)(52,85,74,63)(53,62,75,84)(54,83,76,61)(55,60,77,82)(56,81,78,59)(57,58,79,80)(89,128,111,106)(90,105,112,127)(91,126,113,104)(92,103,114,125)(93,124,115,102)(94,101,116,123)(95,122,117,100)(96,99,118,121)(97,120,119,98)(107,110,129,132)(108,131,130,109)(133,156,155,134)(135,154,157,176)(136,175,158,153)(137,152,159,174)(138,173,160,151)(139,150,161,172)(140,171,162,149)(141,148,163,170)(142,169,164,147)(143,146,165,168)(144,167,166,145)>;
G:=Group( (1,80,23,58)(2,59,24,81)(3,82,25,60)(4,61,26,83)(5,84,27,62)(6,63,28,85)(7,86,29,64)(8,65,30,87)(9,88,31,66)(10,67,32,45)(11,46,33,68)(12,69,34,47)(13,48,35,70)(14,71,36,49)(15,50,37,72)(16,73,38,51)(17,52,39,74)(18,75,40,53)(19,54,41,76)(20,77,42,55)(21,56,43,78)(22,79,44,57)(89,136,111,158)(90,159,112,137)(91,138,113,160)(92,161,114,139)(93,140,115,162)(94,163,116,141)(95,142,117,164)(96,165,118,143)(97,144,119,166)(98,167,120,145)(99,146,121,168)(100,169,122,147)(101,148,123,170)(102,171,124,149)(103,150,125,172)(104,173,126,151)(105,152,127,174)(106,175,128,153)(107,154,129,176)(108,133,130,155)(109,156,131,134)(110,135,132,157), (1,145)(2,121)(3,147)(4,123)(5,149)(6,125)(7,151)(8,127)(9,153)(10,129)(11,155)(12,131)(13,157)(14,89)(15,159)(16,91)(17,161)(18,93)(19,163)(20,95)(21,165)(22,97)(23,167)(24,99)(25,169)(26,101)(27,171)(28,103)(29,173)(30,105)(31,175)(32,107)(33,133)(34,109)(35,135)(36,111)(37,137)(38,113)(39,139)(40,115)(41,141)(42,117)(43,143)(44,119)(45,176)(46,130)(47,134)(48,132)(49,136)(50,90)(51,138)(52,92)(53,140)(54,94)(55,142)(56,96)(57,144)(58,98)(59,146)(60,100)(61,148)(62,102)(63,150)(64,104)(65,152)(66,106)(67,154)(68,108)(69,156)(70,110)(71,158)(72,112)(73,160)(74,114)(75,162)(76,116)(77,164)(78,118)(79,166)(80,120)(81,168)(82,122)(83,170)(84,124)(85,172)(86,126)(87,174)(88,128), (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), (1,22,23,44)(2,43,24,21)(3,20,25,42)(4,41,26,19)(5,18,27,40)(6,39,28,17)(7,16,29,38)(8,37,30,15)(9,14,31,36)(10,35,32,13)(11,12,33,34)(45,70,67,48)(46,47,68,69)(49,66,71,88)(50,87,72,65)(51,64,73,86)(52,85,74,63)(53,62,75,84)(54,83,76,61)(55,60,77,82)(56,81,78,59)(57,58,79,80)(89,128,111,106)(90,105,112,127)(91,126,113,104)(92,103,114,125)(93,124,115,102)(94,101,116,123)(95,122,117,100)(96,99,118,121)(97,120,119,98)(107,110,129,132)(108,131,130,109)(133,156,155,134)(135,154,157,176)(136,175,158,153)(137,152,159,174)(138,173,160,151)(139,150,161,172)(140,171,162,149)(141,148,163,170)(142,169,164,147)(143,146,165,168)(144,167,166,145) );
G=PermutationGroup([[(1,80,23,58),(2,59,24,81),(3,82,25,60),(4,61,26,83),(5,84,27,62),(6,63,28,85),(7,86,29,64),(8,65,30,87),(9,88,31,66),(10,67,32,45),(11,46,33,68),(12,69,34,47),(13,48,35,70),(14,71,36,49),(15,50,37,72),(16,73,38,51),(17,52,39,74),(18,75,40,53),(19,54,41,76),(20,77,42,55),(21,56,43,78),(22,79,44,57),(89,136,111,158),(90,159,112,137),(91,138,113,160),(92,161,114,139),(93,140,115,162),(94,163,116,141),(95,142,117,164),(96,165,118,143),(97,144,119,166),(98,167,120,145),(99,146,121,168),(100,169,122,147),(101,148,123,170),(102,171,124,149),(103,150,125,172),(104,173,126,151),(105,152,127,174),(106,175,128,153),(107,154,129,176),(108,133,130,155),(109,156,131,134),(110,135,132,157)], [(1,145),(2,121),(3,147),(4,123),(5,149),(6,125),(7,151),(8,127),(9,153),(10,129),(11,155),(12,131),(13,157),(14,89),(15,159),(16,91),(17,161),(18,93),(19,163),(20,95),(21,165),(22,97),(23,167),(24,99),(25,169),(26,101),(27,171),(28,103),(29,173),(30,105),(31,175),(32,107),(33,133),(34,109),(35,135),(36,111),(37,137),(38,113),(39,139),(40,115),(41,141),(42,117),(43,143),(44,119),(45,176),(46,130),(47,134),(48,132),(49,136),(50,90),(51,138),(52,92),(53,140),(54,94),(55,142),(56,96),(57,144),(58,98),(59,146),(60,100),(61,148),(62,102),(63,150),(64,104),(65,152),(66,106),(67,154),(68,108),(69,156),(70,110),(71,158),(72,112),(73,160),(74,114),(75,162),(76,116),(77,164),(78,118),(79,166),(80,120),(81,168),(82,122),(83,170),(84,124),(85,172),(86,126),(87,174),(88,128)], [(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)], [(1,22,23,44),(2,43,24,21),(3,20,25,42),(4,41,26,19),(5,18,27,40),(6,39,28,17),(7,16,29,38),(8,37,30,15),(9,14,31,36),(10,35,32,13),(11,12,33,34),(45,70,67,48),(46,47,68,69),(49,66,71,88),(50,87,72,65),(51,64,73,86),(52,85,74,63),(53,62,75,84),(54,83,76,61),(55,60,77,82),(56,81,78,59),(57,58,79,80),(89,128,111,106),(90,105,112,127),(91,126,113,104),(92,103,114,125),(93,124,115,102),(94,101,116,123),(95,122,117,100),(96,99,118,121),(97,120,119,98),(107,110,129,132),(108,131,130,109),(133,156,155,134),(135,154,157,176),(136,175,158,153),(137,152,159,174),(138,173,160,151),(139,150,161,172),(140,171,162,149),(141,148,163,170),(142,169,164,147),(143,146,165,168),(144,167,166,145)]])
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 11A | ··· | 11E | 22A | ··· | 22E | 22F | ··· | 22J | 44A | ··· | 44E | 44F | ··· | 44J | 88A | ··· | 88J |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 4 | 22 | 2 | 4 | 22 | 44 | 44 | 4 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D11 | D22 | D22 | D22 | C8.C22 | D4×D11 | D4.D22 |
| kernel | D4.D22 | C88⋊C2 | Dic44 | D4.D11 | C11⋊Q16 | C11×SD16 | D4⋊2D11 | Q8×D11 | Dic11 | D22 | SD16 | C8 | D4 | Q8 | C11 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 5 | 10 |
Matrix representation of D4.D22 ►in GL4(𝔽89) generated by
| 1 | 0 | 47 | 16 |
| 0 | 1 | 83 | 62 |
| 30 | 87 | 88 | 0 |
| 23 | 17 | 0 | 88 |
| 51 | 3 | 68 | 9 |
| 8 | 38 | 85 | 82 |
| 78 | 24 | 26 | 86 |
| 19 | 56 | 3 | 63 |
| 81 | 62 | 34 | 1 |
| 17 | 20 | 46 | 85 |
| 21 | 58 | 27 | 27 |
| 41 | 42 | 62 | 50 |
| 32 | 17 | 1 | 34 |
| 83 | 57 | 85 | 46 |
| 21 | 58 | 27 | 27 |
| 56 | 14 | 50 | 62 |
G:=sub<GL(4,GF(89))| [1,0,30,23,0,1,87,17,47,83,88,0,16,62,0,88],[51,8,78,19,3,38,24,56,68,85,26,3,9,82,86,63],[81,17,21,41,62,20,58,42,34,46,27,62,1,85,27,50],[32,83,21,56,17,57,58,14,1,85,27,50,34,46,27,62] >;
D4.D22 in GAP, Magma, Sage, TeX
D_4.D_{22} % in TeX
G:=Group("D4.D22"); // GroupNames label
G:=SmallGroup(352,110);
// by ID
G=gap.SmallGroup(352,110);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,103,362,116,297,159,69,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^22=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^21>;
// generators/relations