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## G = D4.D22order 352 = 25·11

### 4th non-split extension by D4 of D22 acting via D22/D11=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — D4.D22
 Chief series C1 — C11 — C22 — C44 — C4×D11 — Q8×D11 — D4.D22
 Lower central C11 — C22 — C44 — D4.D22
 Upper central C1 — C2 — C4 — SD16

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, D42D11, Q8×D11, D4.D22
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C8.C22, D22, C22×D11, D4×D11, D4.D22

Smallest permutation representation of 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`

׿
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