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## G = C8⋊D22order 352 = 25·11

### 1st semidirect product of C8 and 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 — C2×D44 — 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=b22=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >

Subgroups: 586 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C11, M4(2), D8, SD16, C2×D4, C4○D4, D11, C22, C22, C8⋊C22, Dic11, C44, D22, C2×C22, C88, Dic22, C4×D11, D44, D44, D44, C11⋊D4, C2×C44, C22×D11, C8⋊D11, D88, C11×M4(2), C2×D44, 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 88 points
Generators in S88
```(1 77 21 64 28 88 44 53)(2 67 22 54 29 78 34 65)(3 79 12 66 30 68 35 55)(4 69 13 56 31 80 36 45)(5 81 14 46 32 70 37 57)(6 71 15 58 33 82 38 47)(7 83 16 48 23 72 39 59)(8 73 17 60 24 84 40 49)(9 85 18 50 25 74 41 61)(10 75 19 62 26 86 42 51)(11 87 20 52 27 76 43 63)
(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)
(1 14)(2 13)(3 12)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(23 42)(24 41)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 44)(33 43)(45 54)(46 53)(47 52)(48 51)(49 50)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(67 69)(70 88)(71 87)(72 86)(73 85)(74 84)(75 83)(76 82)(77 81)(78 80)```

`G:=sub<Sym(88)| (1,77,21,64,28,88,44,53)(2,67,22,54,29,78,34,65)(3,79,12,66,30,68,35,55)(4,69,13,56,31,80,36,45)(5,81,14,46,32,70,37,57)(6,71,15,58,33,82,38,47)(7,83,16,48,23,72,39,59)(8,73,17,60,24,84,40,49)(9,85,18,50,25,74,41,61)(10,75,19,62,26,86,42,51)(11,87,20,52,27,76,43,63), (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), (1,14)(2,13)(3,12)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,44)(33,43)(45,54)(46,53)(47,52)(48,51)(49,50)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(67,69)(70,88)(71,87)(72,86)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80)>;`

`G:=Group( (1,77,21,64,28,88,44,53)(2,67,22,54,29,78,34,65)(3,79,12,66,30,68,35,55)(4,69,13,56,31,80,36,45)(5,81,14,46,32,70,37,57)(6,71,15,58,33,82,38,47)(7,83,16,48,23,72,39,59)(8,73,17,60,24,84,40,49)(9,85,18,50,25,74,41,61)(10,75,19,62,26,86,42,51)(11,87,20,52,27,76,43,63), (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), (1,14)(2,13)(3,12)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,44)(33,43)(45,54)(46,53)(47,52)(48,51)(49,50)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(67,69)(70,88)(71,87)(72,86)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80) );`

`G=PermutationGroup([[(1,77,21,64,28,88,44,53),(2,67,22,54,29,78,34,65),(3,79,12,66,30,68,35,55),(4,69,13,56,31,80,36,45),(5,81,14,46,32,70,37,57),(6,71,15,58,33,82,38,47),(7,83,16,48,23,72,39,59),(8,73,17,60,24,84,40,49),(9,85,18,50,25,74,41,61),(10,75,19,62,26,86,42,51),(11,87,20,52,27,76,43,63)], [(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)], [(1,14),(2,13),(3,12),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(23,42),(24,41),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,44),(33,43),(45,54),(46,53),(47,52),(48,51),(49,50),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61),(67,69),(70,88),(71,87),(72,86),(73,85),(74,84),(75,83),(76,82),(77,81),(78,80)]])`

61 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 8A 8B 11A ··· 11E 22A ··· 22E 22F ··· 22J 44A ··· 44J 44K ··· 44O 88A ··· 88T order 1 2 2 2 2 2 4 4 4 8 8 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 44 ··· 44 88 ··· 88 size 1 1 2 44 44 44 2 2 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 D88 C11×M4(2) C2×D44 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 GL6(𝔽89)

 17 30 0 0 0 0 20 72 0 0 0 0 0 0 73 0 79 0 0 0 26 0 48 1 0 0 66 1 16 0 0 0 15 0 26 0
,
 54 53 0 0 0 0 65 77 0 0 0 0 0 0 88 0 0 0 0 0 0 88 0 0 0 0 21 0 1 0 0 0 23 0 0 1
,
 73 11 0 0 0 0 82 16 0 0 0 0 0 0 88 0 0 0 0 0 63 1 0 0 0 0 22 0 0 1 0 0 22 0 1 0

`G:=sub<GL(6,GF(89))| [17,20,0,0,0,0,30,72,0,0,0,0,0,0,73,26,66,15,0,0,0,0,1,0,0,0,79,48,16,26,0,0,0,1,0,0],[54,65,0,0,0,0,53,77,0,0,0,0,0,0,88,0,21,23,0,0,0,88,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[73,82,0,0,0,0,11,16,0,0,0,0,0,0,88,63,22,22,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C8⋊D22 in GAP, Magma, Sage, TeX

`C_8\rtimes D_{22}`
`% in TeX`

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

`G:=SmallGroup(352,103);`
`// by ID`

`G=gap.SmallGroup(352,103);`
`# by ID`

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

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

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