Copied to
clipboard

G = D4.10D22order 352 = 25·11

The non-split extension by D4 of D22 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — D4.10D22
 Chief series C1 — C11 — C22 — D22 — C4×D11 — Q8×D11 — D4.10D22
 Lower central C11 — C22 — D4.10D22
 Upper central C1 — C2 — C4○D4

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, D445C2, D42D11, Q8×D11, C11×C4○D4, D4.10D22
Quotients: C1, C2, C22, C23, C24, D11, 2- 1+4, D22, C22×D11, C23×D11, D4.10D22

Smallest permutation representation of 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`

׿
×
𝔽