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## G = D22.D4order 352 = 25·11

### 1st non-split extension by D22 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

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

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, D445C2, D4×D11, D42D11, D22.D4

Smallest permutation representation of 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`

׿
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