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## G = D22.5D4order 352 = 25·11

### 2nd non-split extension by D22 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C22 — D22.5D4
 Chief series C1 — C11 — C22 — C2×C22 — C22×D11 — C2×C4×D11 — D22.5D4
 Lower central C11 — C2×C22 — D22.5D4
 Upper central C1 — C22 — C4⋊C4

Generators and relations for D22.5D4
G = < a,b,c,d | a22=b2=c4=1, d2=a11, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a11b, dcd-1=c-1 >

Subgroups: 546 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C11, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, D11, C22, C22.D4, Dic11, C44, D22, D22, C2×C22, C4×D11, D44, C2×Dic11, C2×C44, C22×D11, Dic11⋊C4, D22⋊C4, C11×C4⋊C4, C2×C4×D11, C2×D44, D22.5D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D11, C22.D4, D22, C22×D11, D445C2, D4×D11, D44⋊C2, D22.5D4

Smallest permutation representation of D22.5D4
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)
(1 107)(2 106)(3 105)(4 104)(5 103)(6 102)(7 101)(8 100)(9 99)(10 98)(11 97)(12 96)(13 95)(14 94)(15 93)(16 92)(17 91)(18 90)(19 89)(20 110)(21 109)(22 108)(23 148)(24 147)(25 146)(26 145)(27 144)(28 143)(29 142)(30 141)(31 140)(32 139)(33 138)(34 137)(35 136)(36 135)(37 134)(38 133)(39 154)(40 153)(41 152)(42 151)(43 150)(44 149)(45 71)(46 70)(47 69)(48 68)(49 67)(50 88)(51 87)(52 86)(53 85)(54 84)(55 83)(56 82)(57 81)(58 80)(59 79)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(111 157)(112 156)(113 155)(114 176)(115 175)(116 174)(117 173)(118 172)(119 171)(120 170)(121 169)(122 168)(123 167)(124 166)(125 165)(126 164)(127 163)(128 162)(129 161)(130 160)(131 159)(132 158)
(1 59 36 112)(2 60 37 113)(3 61 38 114)(4 62 39 115)(5 63 40 116)(6 64 41 117)(7 65 42 118)(8 66 43 119)(9 45 44 120)(10 46 23 121)(11 47 24 122)(12 48 25 123)(13 49 26 124)(14 50 27 125)(15 51 28 126)(16 52 29 127)(17 53 30 128)(18 54 31 129)(19 55 32 130)(20 56 33 131)(21 57 34 132)(22 58 35 111)(67 134 166 106)(68 135 167 107)(69 136 168 108)(70 137 169 109)(71 138 170 110)(72 139 171 89)(73 140 172 90)(74 141 173 91)(75 142 174 92)(76 143 175 93)(77 144 176 94)(78 145 155 95)(79 146 156 96)(80 147 157 97)(81 148 158 98)(82 149 159 99)(83 150 160 100)(84 151 161 101)(85 152 162 102)(86 153 163 103)(87 154 164 104)(88 133 165 105)
(1 97 12 108)(2 98 13 109)(3 99 14 110)(4 100 15 89)(5 101 16 90)(6 102 17 91)(7 103 18 92)(8 104 19 93)(9 105 20 94)(10 106 21 95)(11 107 22 96)(23 134 34 145)(24 135 35 146)(25 136 36 147)(26 137 37 148)(27 138 38 149)(28 139 39 150)(29 140 40 151)(30 141 41 152)(31 142 42 153)(32 143 43 154)(33 144 44 133)(45 165 56 176)(46 166 57 155)(47 167 58 156)(48 168 59 157)(49 169 60 158)(50 170 61 159)(51 171 62 160)(52 172 63 161)(53 173 64 162)(54 174 65 163)(55 175 66 164)(67 132 78 121)(68 111 79 122)(69 112 80 123)(70 113 81 124)(71 114 82 125)(72 115 83 126)(73 116 84 127)(74 117 85 128)(75 118 86 129)(76 119 87 130)(77 120 88 131)```

`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), (1,107)(2,106)(3,105)(4,104)(5,103)(6,102)(7,101)(8,100)(9,99)(10,98)(11,97)(12,96)(13,95)(14,94)(15,93)(16,92)(17,91)(18,90)(19,89)(20,110)(21,109)(22,108)(23,148)(24,147)(25,146)(26,145)(27,144)(28,143)(29,142)(30,141)(31,140)(32,139)(33,138)(34,137)(35,136)(36,135)(37,134)(38,133)(39,154)(40,153)(41,152)(42,151)(43,150)(44,149)(45,71)(46,70)(47,69)(48,68)(49,67)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(111,157)(112,156)(113,155)(114,176)(115,175)(116,174)(117,173)(118,172)(119,171)(120,170)(121,169)(122,168)(123,167)(124,166)(125,165)(126,164)(127,163)(128,162)(129,161)(130,160)(131,159)(132,158), (1,59,36,112)(2,60,37,113)(3,61,38,114)(4,62,39,115)(5,63,40,116)(6,64,41,117)(7,65,42,118)(8,66,43,119)(9,45,44,120)(10,46,23,121)(11,47,24,122)(12,48,25,123)(13,49,26,124)(14,50,27,125)(15,51,28,126)(16,52,29,127)(17,53,30,128)(18,54,31,129)(19,55,32,130)(20,56,33,131)(21,57,34,132)(22,58,35,111)(67,134,166,106)(68,135,167,107)(69,136,168,108)(70,137,169,109)(71,138,170,110)(72,139,171,89)(73,140,172,90)(74,141,173,91)(75,142,174,92)(76,143,175,93)(77,144,176,94)(78,145,155,95)(79,146,156,96)(80,147,157,97)(81,148,158,98)(82,149,159,99)(83,150,160,100)(84,151,161,101)(85,152,162,102)(86,153,163,103)(87,154,164,104)(88,133,165,105), (1,97,12,108)(2,98,13,109)(3,99,14,110)(4,100,15,89)(5,101,16,90)(6,102,17,91)(7,103,18,92)(8,104,19,93)(9,105,20,94)(10,106,21,95)(11,107,22,96)(23,134,34,145)(24,135,35,146)(25,136,36,147)(26,137,37,148)(27,138,38,149)(28,139,39,150)(29,140,40,151)(30,141,41,152)(31,142,42,153)(32,143,43,154)(33,144,44,133)(45,165,56,176)(46,166,57,155)(47,167,58,156)(48,168,59,157)(49,169,60,158)(50,170,61,159)(51,171,62,160)(52,172,63,161)(53,173,64,162)(54,174,65,163)(55,175,66,164)(67,132,78,121)(68,111,79,122)(69,112,80,123)(70,113,81,124)(71,114,82,125)(72,115,83,126)(73,116,84,127)(74,117,85,128)(75,118,86,129)(76,119,87,130)(77,120,88,131)>;`

`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), (1,107)(2,106)(3,105)(4,104)(5,103)(6,102)(7,101)(8,100)(9,99)(10,98)(11,97)(12,96)(13,95)(14,94)(15,93)(16,92)(17,91)(18,90)(19,89)(20,110)(21,109)(22,108)(23,148)(24,147)(25,146)(26,145)(27,144)(28,143)(29,142)(30,141)(31,140)(32,139)(33,138)(34,137)(35,136)(36,135)(37,134)(38,133)(39,154)(40,153)(41,152)(42,151)(43,150)(44,149)(45,71)(46,70)(47,69)(48,68)(49,67)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(111,157)(112,156)(113,155)(114,176)(115,175)(116,174)(117,173)(118,172)(119,171)(120,170)(121,169)(122,168)(123,167)(124,166)(125,165)(126,164)(127,163)(128,162)(129,161)(130,160)(131,159)(132,158), (1,59,36,112)(2,60,37,113)(3,61,38,114)(4,62,39,115)(5,63,40,116)(6,64,41,117)(7,65,42,118)(8,66,43,119)(9,45,44,120)(10,46,23,121)(11,47,24,122)(12,48,25,123)(13,49,26,124)(14,50,27,125)(15,51,28,126)(16,52,29,127)(17,53,30,128)(18,54,31,129)(19,55,32,130)(20,56,33,131)(21,57,34,132)(22,58,35,111)(67,134,166,106)(68,135,167,107)(69,136,168,108)(70,137,169,109)(71,138,170,110)(72,139,171,89)(73,140,172,90)(74,141,173,91)(75,142,174,92)(76,143,175,93)(77,144,176,94)(78,145,155,95)(79,146,156,96)(80,147,157,97)(81,148,158,98)(82,149,159,99)(83,150,160,100)(84,151,161,101)(85,152,162,102)(86,153,163,103)(87,154,164,104)(88,133,165,105), (1,97,12,108)(2,98,13,109)(3,99,14,110)(4,100,15,89)(5,101,16,90)(6,102,17,91)(7,103,18,92)(8,104,19,93)(9,105,20,94)(10,106,21,95)(11,107,22,96)(23,134,34,145)(24,135,35,146)(25,136,36,147)(26,137,37,148)(27,138,38,149)(28,139,39,150)(29,140,40,151)(30,141,41,152)(31,142,42,153)(32,143,43,154)(33,144,44,133)(45,165,56,176)(46,166,57,155)(47,167,58,156)(48,168,59,157)(49,169,60,158)(50,170,61,159)(51,171,62,160)(52,172,63,161)(53,173,64,162)(54,174,65,163)(55,175,66,164)(67,132,78,121)(68,111,79,122)(69,112,80,123)(70,113,81,124)(71,114,82,125)(72,115,83,126)(73,116,84,127)(74,117,85,128)(75,118,86,129)(76,119,87,130)(77,120,88,131) );`

`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)], [(1,107),(2,106),(3,105),(4,104),(5,103),(6,102),(7,101),(8,100),(9,99),(10,98),(11,97),(12,96),(13,95),(14,94),(15,93),(16,92),(17,91),(18,90),(19,89),(20,110),(21,109),(22,108),(23,148),(24,147),(25,146),(26,145),(27,144),(28,143),(29,142),(30,141),(31,140),(32,139),(33,138),(34,137),(35,136),(36,135),(37,134),(38,133),(39,154),(40,153),(41,152),(42,151),(43,150),(44,149),(45,71),(46,70),(47,69),(48,68),(49,67),(50,88),(51,87),(52,86),(53,85),(54,84),(55,83),(56,82),(57,81),(58,80),(59,79),(60,78),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(111,157),(112,156),(113,155),(114,176),(115,175),(116,174),(117,173),(118,172),(119,171),(120,170),(121,169),(122,168),(123,167),(124,166),(125,165),(126,164),(127,163),(128,162),(129,161),(130,160),(131,159),(132,158)], [(1,59,36,112),(2,60,37,113),(3,61,38,114),(4,62,39,115),(5,63,40,116),(6,64,41,117),(7,65,42,118),(8,66,43,119),(9,45,44,120),(10,46,23,121),(11,47,24,122),(12,48,25,123),(13,49,26,124),(14,50,27,125),(15,51,28,126),(16,52,29,127),(17,53,30,128),(18,54,31,129),(19,55,32,130),(20,56,33,131),(21,57,34,132),(22,58,35,111),(67,134,166,106),(68,135,167,107),(69,136,168,108),(70,137,169,109),(71,138,170,110),(72,139,171,89),(73,140,172,90),(74,141,173,91),(75,142,174,92),(76,143,175,93),(77,144,176,94),(78,145,155,95),(79,146,156,96),(80,147,157,97),(81,148,158,98),(82,149,159,99),(83,150,160,100),(84,151,161,101),(85,152,162,102),(86,153,163,103),(87,154,164,104),(88,133,165,105)], [(1,97,12,108),(2,98,13,109),(3,99,14,110),(4,100,15,89),(5,101,16,90),(6,102,17,91),(7,103,18,92),(8,104,19,93),(9,105,20,94),(10,106,21,95),(11,107,22,96),(23,134,34,145),(24,135,35,146),(25,136,36,147),(26,137,37,148),(27,138,38,149),(28,139,39,150),(29,140,40,151),(30,141,41,152),(31,142,42,153),(32,143,43,154),(33,144,44,133),(45,165,56,176),(46,166,57,155),(47,167,58,156),(48,168,59,157),(49,169,60,158),(50,170,61,159),(51,171,62,160),(52,172,63,161),(53,173,64,162),(54,174,65,163),(55,175,66,164),(67,132,78,121),(68,111,79,122),(69,112,80,123),(70,113,81,124),(71,114,82,125),(72,115,83,126),(73,116,84,127),(74,117,85,128),(75,118,86,129),(76,119,87,130),(77,120,88,131)]])`

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 11A ··· 11E 22A ··· 22O 44A ··· 44AD order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 1 1 22 22 44 2 2 4 4 22 22 44 2 ··· 2 2 ··· 2 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 C4○D4 D11 D22 D44⋊5C2 D4×D11 D44⋊C2 kernel D22.5D4 Dic11⋊C4 D22⋊C4 C11×C4⋊C4 C2×C4×D11 C2×D44 D22 C22 C4⋊C4 C2×C4 C2 C2 C2 # reps 1 1 3 1 1 1 2 4 5 15 20 5 5

Matrix representation of D22.5D4 in GL6(𝔽89)

 36 36 0 0 0 0 53 11 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
,
 53 53 0 0 0 0 78 36 0 0 0 0 0 0 34 18 0 0 0 0 10 55 0 0 0 0 0 0 34 34 0 0 0 0 21 55
,
 88 0 0 0 0 0 0 88 0 0 0 0 0 0 34 0 0 0 0 0 10 55 0 0 0 0 0 0 1 0 0 0 0 0 87 88
,
 88 0 0 0 0 0 0 88 0 0 0 0 0 0 55 71 0 0 0 0 0 34 0 0 0 0 0 0 34 0 0 0 0 0 21 55

`G:=sub<GL(6,GF(89))| [36,53,0,0,0,0,36,11,0,0,0,0,0,0,88,0,0,0,0,0,0,88,0,0,0,0,0,0,88,0,0,0,0,0,0,88],[53,78,0,0,0,0,53,36,0,0,0,0,0,0,34,10,0,0,0,0,18,55,0,0,0,0,0,0,34,21,0,0,0,0,34,55],[88,0,0,0,0,0,0,88,0,0,0,0,0,0,34,10,0,0,0,0,0,55,0,0,0,0,0,0,1,87,0,0,0,0,0,88],[88,0,0,0,0,0,0,88,0,0,0,0,0,0,55,0,0,0,0,0,71,34,0,0,0,0,0,0,34,21,0,0,0,0,0,55] >;`

D22.5D4 in GAP, Magma, Sage, TeX

`D_{22}._5D_4`
`% in TeX`

`G:=Group("D22.5D4");`
`// GroupNames label`

`G:=SmallGroup(352,89);`
`// by ID`

`G=gap.SmallGroup(352,89);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,55,218,188,86,11525]);`
`// 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=d*b*d^-1=a^11*b,d*c*d^-1=c^-1>;`
`// generators/relations`

׿
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