metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.9D22, C44.50D4, Q8.9D22, C44.19C23, Dic22.12C22, D4.D11⋊6C2, C4○D4.2D11, C11⋊Q16⋊6C2, (C2×C22).10D4, (C2×C4).23D22, C22.61(C2×D4), C11⋊C8.4C22, C11⋊5(C8.C22), C44.C4⋊10C2, (C2×Dic22)⋊11C2, C4.25(C11⋊D4), (C2×C44).44C22, (D4×C11).9C22, C4.19(C22×D11), (Q8×C11).9C22, C22.6(C11⋊D4), (C11×C4○D4).3C2, C2.25(C2×C11⋊D4), SmallGroup(352,146)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.9D22
G = < a,b,c,d | a4=b2=c22=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >
Subgroups: 274 in 60 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C11, M4(2), SD16, Q16, C2×Q8, C4○D4, C22, C22, C8.C22, Dic11, C44, C44, C2×C22, C2×C22, C11⋊C8, Dic22, Dic22, C2×Dic11, C2×C44, C2×C44, D4×C11, D4×C11, Q8×C11, C44.C4, D4.D11, C11⋊Q16, C2×Dic22, C11×C4○D4, D4.9D22
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C8.C22, D22, C11⋊D4, C22×D11, C2×C11⋊D4, D4.9D22
►On 176 points
Generators in S176
(1 32 137 118)(2 33 138 119)(3 34 139 120)(4 35 140 121)(5 36 141 122)(6 37 142 123)(7 38 143 124)(8 39 144 125)(9 40 145 126)(10 41 146 127)(11 42 147 128)(12 43 148 129)(13 44 149 130)(14 23 150 131)(15 24 151 132)(16 25 152 111)(17 26 153 112)(18 27 154 113)(19 28 133 114)(20 29 134 115)(21 30 135 116)(22 31 136 117)(45 169 106 78)(46 170 107 79)(47 171 108 80)(48 172 109 81)(49 173 110 82)(50 174 89 83)(51 175 90 84)(52 176 91 85)(53 155 92 86)(54 156 93 87)(55 157 94 88)(56 158 95 67)(57 159 96 68)(58 160 97 69)(59 161 98 70)(60 162 99 71)(61 163 100 72)(62 164 101 73)(63 165 102 74)(64 166 103 75)(65 167 104 76)(66 168 105 77)
(1 118)(2 33)(3 120)(4 35)(5 122)(6 37)(7 124)(8 39)(9 126)(10 41)(11 128)(12 43)(13 130)(14 23)(15 132)(16 25)(17 112)(18 27)(19 114)(20 29)(21 116)(22 31)(24 151)(26 153)(28 133)(30 135)(32 137)(34 139)(36 141)(38 143)(40 145)(42 147)(44 149)(45 106)(47 108)(49 110)(51 90)(53 92)(55 94)(57 96)(59 98)(61 100)(63 102)(65 104)(67 158)(69 160)(71 162)(73 164)(75 166)(77 168)(79 170)(81 172)(83 174)(85 176)(87 156)(111 152)(113 154)(115 134)(117 136)(119 138)(121 140)(123 142)(125 144)(127 146)(129 148)(131 150)
(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 82 137 173)(2 81 138 172)(3 80 139 171)(4 79 140 170)(5 78 141 169)(6 77 142 168)(7 76 143 167)(8 75 144 166)(9 74 145 165)(10 73 146 164)(11 72 147 163)(12 71 148 162)(13 70 149 161)(14 69 150 160)(15 68 151 159)(16 67 152 158)(17 88 153 157)(18 87 154 156)(19 86 133 155)(20 85 134 176)(21 84 135 175)(22 83 136 174)(23 97 131 58)(24 96 132 57)(25 95 111 56)(26 94 112 55)(27 93 113 54)(28 92 114 53)(29 91 115 52)(30 90 116 51)(31 89 117 50)(32 110 118 49)(33 109 119 48)(34 108 120 47)(35 107 121 46)(36 106 122 45)(37 105 123 66)(38 104 124 65)(39 103 125 64)(40 102 126 63)(41 101 127 62)(42 100 128 61)(43 99 129 60)(44 98 130 59)
G:=sub<Sym(176)| (1,32,137,118)(2,33,138,119)(3,34,139,120)(4,35,140,121)(5,36,141,122)(6,37,142,123)(7,38,143,124)(8,39,144,125)(9,40,145,126)(10,41,146,127)(11,42,147,128)(12,43,148,129)(13,44,149,130)(14,23,150,131)(15,24,151,132)(16,25,152,111)(17,26,153,112)(18,27,154,113)(19,28,133,114)(20,29,134,115)(21,30,135,116)(22,31,136,117)(45,169,106,78)(46,170,107,79)(47,171,108,80)(48,172,109,81)(49,173,110,82)(50,174,89,83)(51,175,90,84)(52,176,91,85)(53,155,92,86)(54,156,93,87)(55,157,94,88)(56,158,95,67)(57,159,96,68)(58,160,97,69)(59,161,98,70)(60,162,99,71)(61,163,100,72)(62,164,101,73)(63,165,102,74)(64,166,103,75)(65,167,104,76)(66,168,105,77), (1,118)(2,33)(3,120)(4,35)(5,122)(6,37)(7,124)(8,39)(9,126)(10,41)(11,128)(12,43)(13,130)(14,23)(15,132)(16,25)(17,112)(18,27)(19,114)(20,29)(21,116)(22,31)(24,151)(26,153)(28,133)(30,135)(32,137)(34,139)(36,141)(38,143)(40,145)(42,147)(44,149)(45,106)(47,108)(49,110)(51,90)(53,92)(55,94)(57,96)(59,98)(61,100)(63,102)(65,104)(67,158)(69,160)(71,162)(73,164)(75,166)(77,168)(79,170)(81,172)(83,174)(85,176)(87,156)(111,152)(113,154)(115,134)(117,136)(119,138)(121,140)(123,142)(125,144)(127,146)(129,148)(131,150), (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), (1,82,137,173)(2,81,138,172)(3,80,139,171)(4,79,140,170)(5,78,141,169)(6,77,142,168)(7,76,143,167)(8,75,144,166)(9,74,145,165)(10,73,146,164)(11,72,147,163)(12,71,148,162)(13,70,149,161)(14,69,150,160)(15,68,151,159)(16,67,152,158)(17,88,153,157)(18,87,154,156)(19,86,133,155)(20,85,134,176)(21,84,135,175)(22,83,136,174)(23,97,131,58)(24,96,132,57)(25,95,111,56)(26,94,112,55)(27,93,113,54)(28,92,114,53)(29,91,115,52)(30,90,116,51)(31,89,117,50)(32,110,118,49)(33,109,119,48)(34,108,120,47)(35,107,121,46)(36,106,122,45)(37,105,123,66)(38,104,124,65)(39,103,125,64)(40,102,126,63)(41,101,127,62)(42,100,128,61)(43,99,129,60)(44,98,130,59)>;
G:=Group( (1,32,137,118)(2,33,138,119)(3,34,139,120)(4,35,140,121)(5,36,141,122)(6,37,142,123)(7,38,143,124)(8,39,144,125)(9,40,145,126)(10,41,146,127)(11,42,147,128)(12,43,148,129)(13,44,149,130)(14,23,150,131)(15,24,151,132)(16,25,152,111)(17,26,153,112)(18,27,154,113)(19,28,133,114)(20,29,134,115)(21,30,135,116)(22,31,136,117)(45,169,106,78)(46,170,107,79)(47,171,108,80)(48,172,109,81)(49,173,110,82)(50,174,89,83)(51,175,90,84)(52,176,91,85)(53,155,92,86)(54,156,93,87)(55,157,94,88)(56,158,95,67)(57,159,96,68)(58,160,97,69)(59,161,98,70)(60,162,99,71)(61,163,100,72)(62,164,101,73)(63,165,102,74)(64,166,103,75)(65,167,104,76)(66,168,105,77), (1,118)(2,33)(3,120)(4,35)(5,122)(6,37)(7,124)(8,39)(9,126)(10,41)(11,128)(12,43)(13,130)(14,23)(15,132)(16,25)(17,112)(18,27)(19,114)(20,29)(21,116)(22,31)(24,151)(26,153)(28,133)(30,135)(32,137)(34,139)(36,141)(38,143)(40,145)(42,147)(44,149)(45,106)(47,108)(49,110)(51,90)(53,92)(55,94)(57,96)(59,98)(61,100)(63,102)(65,104)(67,158)(69,160)(71,162)(73,164)(75,166)(77,168)(79,170)(81,172)(83,174)(85,176)(87,156)(111,152)(113,154)(115,134)(117,136)(119,138)(121,140)(123,142)(125,144)(127,146)(129,148)(131,150), (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), (1,82,137,173)(2,81,138,172)(3,80,139,171)(4,79,140,170)(5,78,141,169)(6,77,142,168)(7,76,143,167)(8,75,144,166)(9,74,145,165)(10,73,146,164)(11,72,147,163)(12,71,148,162)(13,70,149,161)(14,69,150,160)(15,68,151,159)(16,67,152,158)(17,88,153,157)(18,87,154,156)(19,86,133,155)(20,85,134,176)(21,84,135,175)(22,83,136,174)(23,97,131,58)(24,96,132,57)(25,95,111,56)(26,94,112,55)(27,93,113,54)(28,92,114,53)(29,91,115,52)(30,90,116,51)(31,89,117,50)(32,110,118,49)(33,109,119,48)(34,108,120,47)(35,107,121,46)(36,106,122,45)(37,105,123,66)(38,104,124,65)(39,103,125,64)(40,102,126,63)(41,101,127,62)(42,100,128,61)(43,99,129,60)(44,98,130,59) );
G=PermutationGroup([[(1,32,137,118),(2,33,138,119),(3,34,139,120),(4,35,140,121),(5,36,141,122),(6,37,142,123),(7,38,143,124),(8,39,144,125),(9,40,145,126),(10,41,146,127),(11,42,147,128),(12,43,148,129),(13,44,149,130),(14,23,150,131),(15,24,151,132),(16,25,152,111),(17,26,153,112),(18,27,154,113),(19,28,133,114),(20,29,134,115),(21,30,135,116),(22,31,136,117),(45,169,106,78),(46,170,107,79),(47,171,108,80),(48,172,109,81),(49,173,110,82),(50,174,89,83),(51,175,90,84),(52,176,91,85),(53,155,92,86),(54,156,93,87),(55,157,94,88),(56,158,95,67),(57,159,96,68),(58,160,97,69),(59,161,98,70),(60,162,99,71),(61,163,100,72),(62,164,101,73),(63,165,102,74),(64,166,103,75),(65,167,104,76),(66,168,105,77)], [(1,118),(2,33),(3,120),(4,35),(5,122),(6,37),(7,124),(8,39),(9,126),(10,41),(11,128),(12,43),(13,130),(14,23),(15,132),(16,25),(17,112),(18,27),(19,114),(20,29),(21,116),(22,31),(24,151),(26,153),(28,133),(30,135),(32,137),(34,139),(36,141),(38,143),(40,145),(42,147),(44,149),(45,106),(47,108),(49,110),(51,90),(53,92),(55,94),(57,96),(59,98),(61,100),(63,102),(65,104),(67,158),(69,160),(71,162),(73,164),(75,166),(77,168),(79,170),(81,172),(83,174),(85,176),(87,156),(111,152),(113,154),(115,134),(117,136),(119,138),(121,140),(123,142),(125,144),(127,146),(129,148),(131,150)], [(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)], [(1,82,137,173),(2,81,138,172),(3,80,139,171),(4,79,140,170),(5,78,141,169),(6,77,142,168),(7,76,143,167),(8,75,144,166),(9,74,145,165),(10,73,146,164),(11,72,147,163),(12,71,148,162),(13,70,149,161),(14,69,150,160),(15,68,151,159),(16,67,152,158),(17,88,153,157),(18,87,154,156),(19,86,133,155),(20,85,134,176),(21,84,135,175),(22,83,136,174),(23,97,131,58),(24,96,132,57),(25,95,111,56),(26,94,112,55),(27,93,113,54),(28,92,114,53),(29,91,115,52),(30,90,116,51),(31,89,117,50),(32,110,118,49),(33,109,119,48),(34,108,120,47),(35,107,121,46),(36,106,122,45),(37,105,123,66),(38,104,124,65),(39,103,125,64),(40,102,126,63),(41,101,127,62),(42,100,128,61),(43,99,129,60),(44,98,130,59)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 11A | ··· | 11E | 22A | ··· | 22E | 22F | ··· | 22T | 44A | ··· | 44J | 44K | ··· | 44Y |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 44 | 44 | 44 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D11 | D22 | D22 | D22 | C11⋊D4 | C11⋊D4 | C8.C22 | D4.9D22 |
| kernel | D4.9D22 | C44.C4 | D4.D11 | C11⋊Q16 | C2×Dic22 | C11×C4○D4 | C44 | C2×C22 | C4○D4 | C2×C4 | D4 | Q8 | C4 | C22 | C11 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 10 | 10 | 1 | 10 |
Matrix representation of D4.9D22 ►in GL4(𝔽89) generated by
| 1 | 0 | 87 | 0 |
| 0 | 1 | 0 | 87 |
| 1 | 0 | 88 | 0 |
| 0 | 1 | 0 | 88 |
| 1 | 0 | 87 | 0 |
| 0 | 1 | 0 | 87 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 7 | 82 | 75 | 14 |
| 7 | 44 | 75 | 1 |
| 7 | 82 | 82 | 7 |
| 7 | 44 | 82 | 45 |
| 64 | 88 | 25 | 1 |
| 4 | 25 | 85 | 64 |
| 32 | 44 | 25 | 1 |
| 2 | 57 | 85 | 64 |
G:=sub<GL(4,GF(89))| [1,0,1,0,0,1,0,1,87,0,88,0,0,87,0,88],[1,0,0,0,0,1,0,0,87,0,88,0,0,87,0,88],[7,7,7,7,82,44,82,44,75,75,82,82,14,1,7,45],[64,4,32,2,88,25,44,57,25,85,25,85,1,64,1,64] >;
D4.9D22 in GAP, Magma, Sage, TeX
D_4._9D_{22} % in TeX
G:=Group("D4.9D22"); // GroupNames label
G:=SmallGroup(352,146);
// by ID
G=gap.SmallGroup(352,146);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,96,218,188,579,159,69,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^22=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations