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## G = C8.D22order 352 = 25·11

### 1st non-split extension by C8 of D22 acting via D22/C11=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — C8.D22
 Chief series C1 — C11 — C22 — C44 — D44 — D44⋊5C2 — C8.D22
 Lower central C11 — C22 — C44 — C8.D22
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C8.D22
G = < a,b,c | a8=1, b22=c2=a4, bab-1=a5, cac-1=a-1, cbc-1=b21 >

Subgroups: 394 in 60 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C11, M4(2), SD16, Q16, C2×Q8, C4○D4, D11, C22, C22, C8.C22, Dic11, C44, D22, C2×C22, C88, Dic22, Dic22, Dic22, C4×D11, D44, C2×Dic11, C11⋊D4, C2×C44, C8⋊D11, Dic44, C11×M4(2), C2×Dic22, D445C2, C8.D22
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C8.C22, D22, D44, C22×D11, C2×D44, C8.D22

Smallest permutation representation of C8.D22
On 176 points
Generators in S176
```(1 129 86 165 23 107 64 143)(2 108 87 144 24 130 65 166)(3 131 88 167 25 109 66 145)(4 110 45 146 26 132 67 168)(5 89 46 169 27 111 68 147)(6 112 47 148 28 90 69 170)(7 91 48 171 29 113 70 149)(8 114 49 150 30 92 71 172)(9 93 50 173 31 115 72 151)(10 116 51 152 32 94 73 174)(11 95 52 175 33 117 74 153)(12 118 53 154 34 96 75 176)(13 97 54 133 35 119 76 155)(14 120 55 156 36 98 77 134)(15 99 56 135 37 121 78 157)(16 122 57 158 38 100 79 136)(17 101 58 137 39 123 80 159)(18 124 59 160 40 102 81 138)(19 103 60 139 41 125 82 161)(20 126 61 162 42 104 83 140)(21 105 62 141 43 127 84 163)(22 128 63 164 44 106 85 142)
(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 60 67 82)(46 81 68 59)(47 58 69 80)(48 79 70 57)(49 56 71 78)(50 77 72 55)(51 54 73 76)(52 75 74 53)(61 88 83 66)(62 65 84 87)(63 86 85 64)(89 138 111 160)(90 159 112 137)(91 136 113 158)(92 157 114 135)(93 134 115 156)(94 155 116 133)(95 176 117 154)(96 153 118 175)(97 174 119 152)(98 151 120 173)(99 172 121 150)(100 149 122 171)(101 170 123 148)(102 147 124 169)(103 168 125 146)(104 145 126 167)(105 166 127 144)(106 143 128 165)(107 164 129 142)(108 141 130 163)(109 162 131 140)(110 139 132 161)```

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

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

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

61 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 8A 8B 11A ··· 11E 22A ··· 22E 22F ··· 22J 44A ··· 44J 44K ··· 44O 88A ··· 88T 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 2 44 2 2 44 44 44 4 4 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

61 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D4 D4 D11 D22 D22 D44 D44 C8.C22 C8.D22 kernel C8.D22 C8⋊D11 Dic44 C11×M4(2) C2×Dic22 D44⋊5C2 C44 C2×C22 M4(2) C8 C2×C4 C4 C22 C11 C1 # reps 1 2 2 1 1 1 1 1 5 10 5 10 10 1 10

Matrix representation of C8.D22 in GL4(𝔽89) generated by

 0 0 87 0 0 0 0 87 46 15 0 0 74 43 0 0
,
 36 36 75 13 53 11 76 19 70 41 53 53 48 44 36 78
,
 53 53 14 76 78 36 2 75 48 19 82 7 45 41 44 7
`G:=sub<GL(4,GF(89))| [0,0,46,74,0,0,15,43,87,0,0,0,0,87,0,0],[36,53,70,48,36,11,41,44,75,76,53,36,13,19,53,78],[53,78,48,45,53,36,19,41,14,2,82,44,76,75,7,7] >;`

C8.D22 in GAP, Magma, Sage, TeX

`C_8.D_{22}`
`% in TeX`

`G:=Group("C8.D22");`
`// GroupNames label`

`G:=SmallGroup(352,104);`
`// by ID`

`G=gap.SmallGroup(352,104);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,103,218,188,50,579,69,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=1,b^22=c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^21>;`
`// generators/relations`

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