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## G = D4⋊D22order 352 = 25·11

### 2nd semidirect product of D4 and 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 — D4×D11 — D4⋊D22
 Lower central C11 — C22 — C44 — D4⋊D22
 Upper central C1 — C2 — C4 — D8

Generators and relations for D4⋊D22
G = < a,b,c,d | a8=b2=c11=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >

Subgroups: 506 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, D4, Q8, C23, C11, M4(2), D8, D8, SD16, C2×D4, C4○D4, D11, C22, C22, C8⋊C22, Dic11, Dic11, C44, D22, D22, C2×C22, C11⋊C8, C88, Dic22, C4×D11, D44, C2×Dic11, C11⋊D4, D4×C11, C22×D11, C88⋊C2, C8⋊D11, D4⋊D11, D4.D11, C11×D8, D4×D11, D42D11, 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 88 points
Generators in S88
```(1 54 32 87 21 65 43 76)(2 55 33 88 22 66 44 77)(3 45 23 78 12 56 34 67)(4 46 24 79 13 57 35 68)(5 47 25 80 14 58 36 69)(6 48 26 81 15 59 37 70)(7 49 27 82 16 60 38 71)(8 50 28 83 17 61 39 72)(9 51 29 84 18 62 40 73)(10 52 30 85 19 63 41 74)(11 53 31 86 20 64 42 75)
(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 66)(53 65)(54 64)(55 63)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 88)(75 87)(76 86)(77 85)```

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

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

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

46 conjugacy classes

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

46 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D11 D22 D22 C8⋊C22 D4×D11 D4⋊D22 kernel D4⋊D22 C88⋊C2 C8⋊D11 D4⋊D11 D4.D11 C11×D8 D4×D11 D4⋊2D11 Dic11 D22 D8 C8 D4 C11 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 5 5 10 1 5 10

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

 0 0 15 35 0 0 67 59 14 46 18 12 55 82 77 71
,
 1 0 0 0 0 1 0 0 67 57 88 0 1 70 0 88
,
 11 1 0 0 23 75 0 0 0 0 0 1 0 0 88 86
,
 69 81 0 0 61 20 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(89))| [0,0,14,55,0,0,46,82,15,67,18,77,35,59,12,71],[1,0,67,1,0,1,57,70,0,0,88,0,0,0,0,88],[11,23,0,0,1,75,0,0,0,0,0,88,0,0,1,86],[69,61,0,0,81,20,0,0,0,0,0,1,0,0,1,0] >;`

D4⋊D22 in GAP, Magma, Sage, TeX

`D_4\rtimes D_{22}`
`% in TeX`

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

`G:=SmallGroup(352,106);`
`// by ID`

`G=gap.SmallGroup(352,106);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,362,116,297,159,69,11525]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^11=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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