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## G = D4⋊8D22order 352 = 25·11

### 4th semidirect product of D4 and D22 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — D4⋊8D22
 Chief series C1 — C11 — C22 — D22 — C22×D11 — D4×D11 — D4⋊8D22
 Lower central C11 — C22 — D4⋊8D22
 Upper central C1 — C2 — C4○D4

Generators and relations for D48D22
G = < a,b,c,d | a4=b2=c22=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, bd=db, dcd=c-1 >

Subgroups: 1090 in 166 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C11, C2×D4, C4○D4, C4○D4, D11, C22, C22, 2+ 1+4, Dic11, C44, C44, D22, D22, C2×C22, Dic22, C4×D11, D44, C11⋊D4, C2×C44, D4×C11, Q8×C11, C22×D11, C2×D44, D445C2, D4×D11, D44⋊C2, C11×C4○D4, D48D22
Quotients: C1, C2, C22, C23, C24, D11, 2+ 1+4, D22, C22×D11, C23×D11, D48D22

Smallest permutation representation of D48D22
On 88 points
Generators in S88
```(1 32 69 46)(2 33 70 47)(3 34 71 48)(4 35 72 49)(5 36 73 50)(6 37 74 51)(7 38 75 52)(8 39 76 53)(9 40 77 54)(10 41 78 55)(11 42 79 56)(12 43 80 57)(13 44 81 58)(14 23 82 59)(15 24 83 60)(16 25 84 61)(17 26 85 62)(18 27 86 63)(19 28 87 64)(20 29 88 65)(21 30 67 66)(22 31 68 45)
(1 46)(2 33)(3 48)(4 35)(5 50)(6 37)(7 52)(8 39)(9 54)(10 41)(11 56)(12 43)(13 58)(14 23)(15 60)(16 25)(17 62)(18 27)(19 64)(20 29)(21 66)(22 31)(24 83)(26 85)(28 87)(30 67)(32 69)(34 71)(36 73)(38 75)(40 77)(42 79)(44 81)(45 68)(47 70)(49 72)(51 74)(53 76)(55 78)(57 80)(59 82)(61 84)(63 86)(65 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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 60)(20 59)(21 58)(22 57)(23 88)(24 87)(25 86)(26 85)(27 84)(28 83)(29 82)(30 81)(31 80)(32 79)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 72)(40 71)(41 70)(42 69)(43 68)(44 67)```

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

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

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

67 conjugacy classes

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

67 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D11 D22 D22 D22 2+ 1+4 D4⋊8D22 kernel D4⋊8D22 C2×D44 D44⋊5C2 D4×D11 D44⋊C2 C11×C4○D4 C4○D4 C2×C4 D4 Q8 C11 C1 # reps 1 3 3 6 2 1 5 15 15 5 1 10

Matrix representation of D48D22 in GL4(𝔽89) generated by

 85 14 78 48 38 45 64 8 14 0 31 75 27 14 45 17
,
 85 14 78 48 38 45 64 8 71 14 31 75 16 58 45 17
,
 49 51 50 83 23 53 74 81 51 0 84 38 51 51 76 81
,
 6 68 82 50 16 33 73 57 60 0 28 29 34 68 55 22
`G:=sub<GL(4,GF(89))| [85,38,14,27,14,45,0,14,78,64,31,45,48,8,75,17],[85,38,71,16,14,45,14,58,78,64,31,45,48,8,75,17],[49,23,51,51,51,53,0,51,50,74,84,76,83,81,38,81],[6,16,60,34,68,33,0,68,82,73,28,55,50,57,29,22] >;`

D48D22 in GAP, Magma, Sage, TeX

`D_4\rtimes_8D_{22}`
`% in TeX`

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

`G:=SmallGroup(352,184);`
`// by ID`

`G=gap.SmallGroup(352,184);`
`# by ID`

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

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

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