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## G = D4.8D22order 352 = 25·11

### 3rd non-split extension by D4 of D22 acting via D22/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — D4.8D22
 Chief series C1 — C11 — C22 — C44 — D44 — D44⋊5C2 — D4.8D22
 Lower central C11 — C22 — C44 — D4.8D22
 Upper central C1 — C4 — C2×C4 — C4○D4

Generators and relations for D4.8D22
G = < a,b,c,d | a4=b2=1, c22=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c21 >

Subgroups: 322 in 62 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C11, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, D11, C22, C22, C4○D8, Dic11, C44, C44, D22, C2×C22, C2×C22, C11⋊C8, Dic22, C4×D11, D44, C11⋊D4, C2×C44, C2×C44, D4×C11, D4×C11, Q8×C11, C2×C11⋊C8, D4⋊D11, D4.D11, Q8⋊D11, C11⋊Q16, D445C2, C11×C4○D4, D4.8D22
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C4○D8, D22, C11⋊D4, C22×D11, C2×C11⋊D4, D4.8D22

Smallest permutation representation of D4.8D22
On 176 points
Generators in S176
```(1 116 23 94)(2 117 24 95)(3 118 25 96)(4 119 26 97)(5 120 27 98)(6 121 28 99)(7 122 29 100)(8 123 30 101)(9 124 31 102)(10 125 32 103)(11 126 33 104)(12 127 34 105)(13 128 35 106)(14 129 36 107)(15 130 37 108)(16 131 38 109)(17 132 39 110)(18 89 40 111)(19 90 41 112)(20 91 42 113)(21 92 43 114)(22 93 44 115)(45 176 67 154)(46 133 68 155)(47 134 69 156)(48 135 70 157)(49 136 71 158)(50 137 72 159)(51 138 73 160)(52 139 74 161)(53 140 75 162)(54 141 76 163)(55 142 77 164)(56 143 78 165)(57 144 79 166)(58 145 80 167)(59 146 81 168)(60 147 82 169)(61 148 83 170)(62 149 84 171)(63 150 85 172)(64 151 86 173)(65 152 87 174)(66 153 88 175)
(1 152)(2 153)(3 154)(4 155)(5 156)(6 157)(7 158)(8 159)(9 160)(10 161)(11 162)(12 163)(13 164)(14 165)(15 166)(16 167)(17 168)(18 169)(19 170)(20 171)(21 172)(22 173)(23 174)(24 175)(25 176)(26 133)(27 134)(28 135)(29 136)(30 137)(31 138)(32 139)(33 140)(34 141)(35 142)(36 143)(37 144)(38 145)(39 146)(40 147)(41 148)(42 149)(43 150)(44 151)(45 96)(46 97)(47 98)(48 99)(49 100)(50 101)(51 102)(52 103)(53 104)(54 105)(55 106)(56 107)(57 108)(58 109)(59 110)(60 111)(61 112)(62 113)(63 114)(64 115)(65 116)(66 117)(67 118)(68 119)(69 120)(70 121)(71 122)(72 123)(73 124)(74 125)(75 126)(76 127)(77 128)(78 129)(79 130)(80 131)(81 132)(82 89)(83 90)(84 91)(85 92)(86 93)(87 94)(88 95)
(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)(45 149 67 171)(46 170 68 148)(47 147 69 169)(48 168 70 146)(49 145 71 167)(50 166 72 144)(51 143 73 165)(52 164 74 142)(53 141 75 163)(54 162 76 140)(55 139 77 161)(56 160 78 138)(57 137 79 159)(58 158 80 136)(59 135 81 157)(60 156 82 134)(61 133 83 155)(62 154 84 176)(63 175 85 153)(64 152 86 174)(65 173 87 151)(66 150 88 172)(89 120 111 98)(90 97 112 119)(91 118 113 96)(92 95 114 117)(93 116 115 94)(99 110 121 132)(100 131 122 109)(101 108 123 130)(102 129 124 107)(103 106 125 128)(104 127 126 105)```

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 8A 8B 8C 8D 11A ··· 11E 22A ··· 22E 22F ··· 22T 44A ··· 44J 44K ··· 44Y order 1 2 2 2 2 4 4 4 4 4 8 8 8 8 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 44 ··· 44 size 1 1 2 4 44 1 1 2 4 44 22 22 22 22 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D11 C4○D8 D22 D22 D22 C11⋊D4 C11⋊D4 D4.8D22 kernel D4.8D22 C2×C11⋊C8 D4⋊D11 D4.D11 Q8⋊D11 C11⋊Q16 D44⋊5C2 C11×C4○D4 C44 C2×C22 C4○D4 C11 C2×C4 D4 Q8 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 1 1 5 4 5 5 5 10 10 10

Matrix representation of D4.8D22 in GL4(𝔽89) generated by

 1 0 0 0 0 1 0 0 0 0 1 87 0 0 1 88
,
 1 0 0 0 0 1 0 0 0 0 0 64 0 0 32 0
,
 56 84 0 0 28 15 0 0 0 0 34 0 0 0 0 34
,
 30 88 0 0 9 59 0 0 0 0 34 0 0 0 34 55
`G:=sub<GL(4,GF(89))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,87,88],[1,0,0,0,0,1,0,0,0,0,0,32,0,0,64,0],[56,28,0,0,84,15,0,0,0,0,34,0,0,0,0,34],[30,9,0,0,88,59,0,0,0,0,34,34,0,0,0,55] >;`

D4.8D22 in GAP, Magma, Sage, TeX

`D_4._8D_{22}`
`% in TeX`

`G:=Group("D4.8D22");`
`// GroupNames label`

`G:=SmallGroup(352,145);`
`// by ID`

`G=gap.SmallGroup(352,145);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,103,218,579,159,69,11525]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=1,c^22=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^21>;`
`// generators/relations`

׿
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