metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.10D22, Q8.11D22, C44.27C23, C22.13C24, D22.8C23, C11⋊22- 1+4, D44.14C22, Dic11.8C23, Dic22.14C22, C4○D4⋊4D11, (Q8×D11)⋊5C2, (C2×C4).25D22, C11⋊D4.C22, D4⋊2D11⋊5C2, D44⋊5C2⋊9C2, (C2×C22).5C23, (C2×Dic22)⋊14C2, (C2×C44).49C22, (C4×D11).6C22, C2.14(C23×D11), C4.34(C22×D11), (D4×C11).10C22, (Q8×C11).11C22, C22.4(C22×D11), (C2×Dic11).22C22, (C11×C4○D4)⋊5C2, SmallGroup(352,185)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.10D22
G = < a,b,c,d | a4=b2=1, c22=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c21 >
Subgroups: 706 in 146 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C2×C4, C2×C4, D4, D4, Q8, Q8, C11, C2×Q8, C4○D4, C4○D4, D11, C22, C22, 2- 1+4, Dic11, C44, C44, D22, C2×C22, Dic22, C4×D11, D44, C2×Dic11, C11⋊D4, C2×C44, D4×C11, Q8×C11, C2×Dic22, D44⋊5C2, D4⋊2D11, Q8×D11, C11×C4○D4, D4.10D22
Quotients: C1, C2, C22, C23, C24, D11, 2- 1+4, D22, C22×D11, C23×D11, D4.10D22
►On 176 points
Generators in S176
(1 61 23 83)(2 84 24 62)(3 63 25 85)(4 86 26 64)(5 65 27 87)(6 88 28 66)(7 67 29 45)(8 46 30 68)(9 69 31 47)(10 48 32 70)(11 71 33 49)(12 50 34 72)(13 73 35 51)(14 52 36 74)(15 75 37 53)(16 54 38 76)(17 77 39 55)(18 56 40 78)(19 79 41 57)(20 58 42 80)(21 81 43 59)(22 60 44 82)(89 143 111 165)(90 166 112 144)(91 145 113 167)(92 168 114 146)(93 147 115 169)(94 170 116 148)(95 149 117 171)(96 172 118 150)(97 151 119 173)(98 174 120 152)(99 153 121 175)(100 176 122 154)(101 155 123 133)(102 134 124 156)(103 157 125 135)(104 136 126 158)(105 159 127 137)(106 138 128 160)(107 161 129 139)(108 140 130 162)(109 163 131 141)(110 142 132 164)
(1 124)(2 103)(3 126)(4 105)(5 128)(6 107)(7 130)(8 109)(9 132)(10 111)(11 90)(12 113)(13 92)(14 115)(15 94)(16 117)(17 96)(18 119)(19 98)(20 121)(21 100)(22 123)(23 102)(24 125)(25 104)(26 127)(27 106)(28 129)(29 108)(30 131)(31 110)(32 89)(33 112)(34 91)(35 114)(36 93)(37 116)(38 95)(39 118)(40 97)(41 120)(42 99)(43 122)(44 101)(45 162)(46 141)(47 164)(48 143)(49 166)(50 145)(51 168)(52 147)(53 170)(54 149)(55 172)(56 151)(57 174)(58 153)(59 176)(60 155)(61 134)(62 157)(63 136)(64 159)(65 138)(66 161)(67 140)(68 163)(69 142)(70 165)(71 144)(72 167)(73 146)(74 169)(75 148)(76 171)(77 150)(78 173)(79 152)(80 175)(81 154)(82 133)(83 156)(84 135)(85 158)(86 137)(87 160)(88 139)
(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)
(1 166 23 144)(2 143 24 165)(3 164 25 142)(4 141 26 163)(5 162 27 140)(6 139 28 161)(7 160 29 138)(8 137 30 159)(9 158 31 136)(10 135 32 157)(11 156 33 134)(12 133 34 155)(13 154 35 176)(14 175 36 153)(15 152 37 174)(16 173 38 151)(17 150 39 172)(18 171 40 149)(19 148 41 170)(20 169 42 147)(21 146 43 168)(22 167 44 145)(45 106 67 128)(46 127 68 105)(47 104 69 126)(48 125 70 103)(49 102 71 124)(50 123 72 101)(51 100 73 122)(52 121 74 99)(53 98 75 120)(54 119 76 97)(55 96 77 118)(56 117 78 95)(57 94 79 116)(58 115 80 93)(59 92 81 114)(60 113 82 91)(61 90 83 112)(62 111 84 89)(63 132 85 110)(64 109 86 131)(65 130 87 108)(66 107 88 129)
G:=sub<Sym(176)| (1,61,23,83)(2,84,24,62)(3,63,25,85)(4,86,26,64)(5,65,27,87)(6,88,28,66)(7,67,29,45)(8,46,30,68)(9,69,31,47)(10,48,32,70)(11,71,33,49)(12,50,34,72)(13,73,35,51)(14,52,36,74)(15,75,37,53)(16,54,38,76)(17,77,39,55)(18,56,40,78)(19,79,41,57)(20,58,42,80)(21,81,43,59)(22,60,44,82)(89,143,111,165)(90,166,112,144)(91,145,113,167)(92,168,114,146)(93,147,115,169)(94,170,116,148)(95,149,117,171)(96,172,118,150)(97,151,119,173)(98,174,120,152)(99,153,121,175)(100,176,122,154)(101,155,123,133)(102,134,124,156)(103,157,125,135)(104,136,126,158)(105,159,127,137)(106,138,128,160)(107,161,129,139)(108,140,130,162)(109,163,131,141)(110,142,132,164), (1,124)(2,103)(3,126)(4,105)(5,128)(6,107)(7,130)(8,109)(9,132)(10,111)(11,90)(12,113)(13,92)(14,115)(15,94)(16,117)(17,96)(18,119)(19,98)(20,121)(21,100)(22,123)(23,102)(24,125)(25,104)(26,127)(27,106)(28,129)(29,108)(30,131)(31,110)(32,89)(33,112)(34,91)(35,114)(36,93)(37,116)(38,95)(39,118)(40,97)(41,120)(42,99)(43,122)(44,101)(45,162)(46,141)(47,164)(48,143)(49,166)(50,145)(51,168)(52,147)(53,170)(54,149)(55,172)(56,151)(57,174)(58,153)(59,176)(60,155)(61,134)(62,157)(63,136)(64,159)(65,138)(66,161)(67,140)(68,163)(69,142)(70,165)(71,144)(72,167)(73,146)(74,169)(75,148)(76,171)(77,150)(78,173)(79,152)(80,175)(81,154)(82,133)(83,156)(84,135)(85,158)(86,137)(87,160)(88,139), (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), (1,166,23,144)(2,143,24,165)(3,164,25,142)(4,141,26,163)(5,162,27,140)(6,139,28,161)(7,160,29,138)(8,137,30,159)(9,158,31,136)(10,135,32,157)(11,156,33,134)(12,133,34,155)(13,154,35,176)(14,175,36,153)(15,152,37,174)(16,173,38,151)(17,150,39,172)(18,171,40,149)(19,148,41,170)(20,169,42,147)(21,146,43,168)(22,167,44,145)(45,106,67,128)(46,127,68,105)(47,104,69,126)(48,125,70,103)(49,102,71,124)(50,123,72,101)(51,100,73,122)(52,121,74,99)(53,98,75,120)(54,119,76,97)(55,96,77,118)(56,117,78,95)(57,94,79,116)(58,115,80,93)(59,92,81,114)(60,113,82,91)(61,90,83,112)(62,111,84,89)(63,132,85,110)(64,109,86,131)(65,130,87,108)(66,107,88,129)>;
G:=Group( (1,61,23,83)(2,84,24,62)(3,63,25,85)(4,86,26,64)(5,65,27,87)(6,88,28,66)(7,67,29,45)(8,46,30,68)(9,69,31,47)(10,48,32,70)(11,71,33,49)(12,50,34,72)(13,73,35,51)(14,52,36,74)(15,75,37,53)(16,54,38,76)(17,77,39,55)(18,56,40,78)(19,79,41,57)(20,58,42,80)(21,81,43,59)(22,60,44,82)(89,143,111,165)(90,166,112,144)(91,145,113,167)(92,168,114,146)(93,147,115,169)(94,170,116,148)(95,149,117,171)(96,172,118,150)(97,151,119,173)(98,174,120,152)(99,153,121,175)(100,176,122,154)(101,155,123,133)(102,134,124,156)(103,157,125,135)(104,136,126,158)(105,159,127,137)(106,138,128,160)(107,161,129,139)(108,140,130,162)(109,163,131,141)(110,142,132,164), (1,124)(2,103)(3,126)(4,105)(5,128)(6,107)(7,130)(8,109)(9,132)(10,111)(11,90)(12,113)(13,92)(14,115)(15,94)(16,117)(17,96)(18,119)(19,98)(20,121)(21,100)(22,123)(23,102)(24,125)(25,104)(26,127)(27,106)(28,129)(29,108)(30,131)(31,110)(32,89)(33,112)(34,91)(35,114)(36,93)(37,116)(38,95)(39,118)(40,97)(41,120)(42,99)(43,122)(44,101)(45,162)(46,141)(47,164)(48,143)(49,166)(50,145)(51,168)(52,147)(53,170)(54,149)(55,172)(56,151)(57,174)(58,153)(59,176)(60,155)(61,134)(62,157)(63,136)(64,159)(65,138)(66,161)(67,140)(68,163)(69,142)(70,165)(71,144)(72,167)(73,146)(74,169)(75,148)(76,171)(77,150)(78,173)(79,152)(80,175)(81,154)(82,133)(83,156)(84,135)(85,158)(86,137)(87,160)(88,139), (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), (1,166,23,144)(2,143,24,165)(3,164,25,142)(4,141,26,163)(5,162,27,140)(6,139,28,161)(7,160,29,138)(8,137,30,159)(9,158,31,136)(10,135,32,157)(11,156,33,134)(12,133,34,155)(13,154,35,176)(14,175,36,153)(15,152,37,174)(16,173,38,151)(17,150,39,172)(18,171,40,149)(19,148,41,170)(20,169,42,147)(21,146,43,168)(22,167,44,145)(45,106,67,128)(46,127,68,105)(47,104,69,126)(48,125,70,103)(49,102,71,124)(50,123,72,101)(51,100,73,122)(52,121,74,99)(53,98,75,120)(54,119,76,97)(55,96,77,118)(56,117,78,95)(57,94,79,116)(58,115,80,93)(59,92,81,114)(60,113,82,91)(61,90,83,112)(62,111,84,89)(63,132,85,110)(64,109,86,131)(65,130,87,108)(66,107,88,129) );
G=PermutationGroup([[(1,61,23,83),(2,84,24,62),(3,63,25,85),(4,86,26,64),(5,65,27,87),(6,88,28,66),(7,67,29,45),(8,46,30,68),(9,69,31,47),(10,48,32,70),(11,71,33,49),(12,50,34,72),(13,73,35,51),(14,52,36,74),(15,75,37,53),(16,54,38,76),(17,77,39,55),(18,56,40,78),(19,79,41,57),(20,58,42,80),(21,81,43,59),(22,60,44,82),(89,143,111,165),(90,166,112,144),(91,145,113,167),(92,168,114,146),(93,147,115,169),(94,170,116,148),(95,149,117,171),(96,172,118,150),(97,151,119,173),(98,174,120,152),(99,153,121,175),(100,176,122,154),(101,155,123,133),(102,134,124,156),(103,157,125,135),(104,136,126,158),(105,159,127,137),(106,138,128,160),(107,161,129,139),(108,140,130,162),(109,163,131,141),(110,142,132,164)], [(1,124),(2,103),(3,126),(4,105),(5,128),(6,107),(7,130),(8,109),(9,132),(10,111),(11,90),(12,113),(13,92),(14,115),(15,94),(16,117),(17,96),(18,119),(19,98),(20,121),(21,100),(22,123),(23,102),(24,125),(25,104),(26,127),(27,106),(28,129),(29,108),(30,131),(31,110),(32,89),(33,112),(34,91),(35,114),(36,93),(37,116),(38,95),(39,118),(40,97),(41,120),(42,99),(43,122),(44,101),(45,162),(46,141),(47,164),(48,143),(49,166),(50,145),(51,168),(52,147),(53,170),(54,149),(55,172),(56,151),(57,174),(58,153),(59,176),(60,155),(61,134),(62,157),(63,136),(64,159),(65,138),(66,161),(67,140),(68,163),(69,142),(70,165),(71,144),(72,167),(73,146),(74,169),(75,148),(76,171),(77,150),(78,173),(79,152),(80,175),(81,154),(82,133),(83,156),(84,135),(85,158),(86,137),(87,160),(88,139)], [(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)], [(1,166,23,144),(2,143,24,165),(3,164,25,142),(4,141,26,163),(5,162,27,140),(6,139,28,161),(7,160,29,138),(8,137,30,159),(9,158,31,136),(10,135,32,157),(11,156,33,134),(12,133,34,155),(13,154,35,176),(14,175,36,153),(15,152,37,174),(16,173,38,151),(17,150,39,172),(18,171,40,149),(19,148,41,170),(20,169,42,147),(21,146,43,168),(22,167,44,145),(45,106,67,128),(46,127,68,105),(47,104,69,126),(48,125,70,103),(49,102,71,124),(50,123,72,101),(51,100,73,122),(52,121,74,99),(53,98,75,120),(54,119,76,97),(55,96,77,118),(56,117,78,95),(57,94,79,116),(58,115,80,93),(59,92,81,114),(60,113,82,91),(61,90,83,112),(62,111,84,89),(63,132,85,110),(64,109,86,131),(65,130,87,108),(66,107,88,129)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 11A | ··· | 11E | 22A | ··· | 22E | 22F | ··· | 22T | 44A | ··· | 44J | 44K | ··· | 44Y |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 |
| size | 1 | 1 | 2 | 2 | 2 | 22 | 22 | 2 | 2 | 2 | 2 | 22 | ··· | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D11 | D22 | D22 | D22 | 2- 1+4 | D4.10D22 |
| kernel | D4.10D22 | C2×Dic22 | D44⋊5C2 | D4⋊2D11 | Q8×D11 | C11×C4○D4 | C4○D4 | C2×C4 | D4 | Q8 | C11 | C1 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 5 | 15 | 15 | 5 | 1 | 10 |
Matrix representation of D4.10D22 ►in GL4(𝔽89) generated by
| 88 | 0 | 40 | 0 |
| 0 | 88 | 0 | 40 |
| 40 | 0 | 1 | 0 |
| 0 | 40 | 0 | 1 |
| 71 | 12 | 4 | 27 |
| 77 | 18 | 62 | 85 |
| 4 | 27 | 18 | 77 |
| 62 | 85 | 12 | 71 |
| 0 | 0 | 21 | 34 |
| 0 | 0 | 55 | 34 |
| 68 | 55 | 0 | 0 |
| 34 | 55 | 0 | 0 |
| 0 | 0 | 39 | 65 |
| 0 | 0 | 4 | 50 |
| 50 | 24 | 0 | 0 |
| 85 | 39 | 0 | 0 |
G:=sub<GL(4,GF(89))| [88,0,40,0,0,88,0,40,40,0,1,0,0,40,0,1],[71,77,4,62,12,18,27,85,4,62,18,12,27,85,77,71],[0,0,68,34,0,0,55,55,21,55,0,0,34,34,0,0],[0,0,50,85,0,0,24,39,39,4,0,0,65,50,0,0] >;
D4.10D22 in GAP, Magma, Sage, TeX
D_4._{10}D_{22} % in TeX
G:=Group("D4.10D22"); // GroupNames label
G:=SmallGroup(352,185);
// by ID
G=gap.SmallGroup(352,185);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,103,188,86,579,69,11525]);
// 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^2*b,b*d=d*b,d*c*d^-1=c^21>;
// generators/relations