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## G = D4⋊6D22order 352 = 25·11

### 2nd semidirect product of D4 and D22 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — D4⋊6D22
 Chief series C1 — C11 — C22 — D22 — C22×D11 — D4×D11 — D4⋊6D22
 Lower central C11 — C22 — D4⋊6D22
 Upper central C1 — C2 — C2×D4

Generators and relations for D46D22
G = < a,b,c,d | a4=b2=c22=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 970 in 166 conjugacy classes, 85 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C11, C2×D4, C2×D4, C4○D4, D11, C22, C22, 2+ 1+4, Dic11, C44, D22, D22, C2×C22, C2×C22, C2×C22, Dic22, C4×D11, D44, C2×Dic11, C11⋊D4, C2×C44, D4×C11, C22×D11, C22×C22, D445C2, D4×D11, D42D11, C2×C11⋊D4, D4×C22, D46D22
Quotients: C1, C2, C22, C23, C24, D11, 2+ 1+4, D22, C22×D11, C23×D11, D46D22

Smallest permutation representation of D46D22
On 88 points
Generators in S88
```(1 77 20 88)(2 67 21 78)(3 79 22 68)(4 69 12 80)(5 81 13 70)(6 71 14 82)(7 83 15 72)(8 73 16 84)(9 85 17 74)(10 75 18 86)(11 87 19 76)(23 60 34 49)(24 50 35 61)(25 62 36 51)(26 52 37 63)(27 64 38 53)(28 54 39 65)(29 66 40 55)(30 56 41 45)(31 46 42 57)(32 58 43 47)(33 48 44 59)
(1 57)(2 47)(3 59)(4 49)(5 61)(6 51)(7 63)(8 53)(9 65)(10 55)(11 45)(12 60)(13 50)(14 62)(15 52)(16 64)(17 54)(18 66)(19 56)(20 46)(21 58)(22 48)(23 80)(24 70)(25 82)(26 72)(27 84)(28 74)(29 86)(30 76)(31 88)(32 78)(33 68)(34 69)(35 81)(36 71)(37 83)(38 73)(39 85)(40 75)(41 87)(42 77)(43 67)(44 79)
(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 16)(13 15)(17 22)(18 21)(19 20)(23 38)(24 37)(25 36)(26 35)(27 34)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(45 46)(47 66)(48 65)(49 64)(50 63)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)(67 75)(68 74)(69 73)(70 72)(76 88)(77 87)(78 86)(79 85)(80 84)(81 83)```

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

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

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

67 conjugacy classes

 class 1 2A 2B ··· 2F 2G 2H 2I 2J 4A 4B 4C 4D 4E 4F 11A ··· 11E 22A ··· 22O 22P ··· 22AI 44A ··· 44J order 1 2 2 ··· 2 2 2 2 2 4 4 4 4 4 4 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 size 1 1 2 ··· 2 22 22 22 22 2 2 22 22 22 22 2 ··· 2 2 ··· 2 4 ··· 4 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⋊6D22 kernel D4⋊6D22 D44⋊5C2 D4×D11 D4⋊2D11 C2×C11⋊D4 D4×C22 C2×D4 C2×C4 D4 C23 C11 C1 # reps 1 2 4 4 4 1 5 5 20 10 1 10

Matrix representation of D46D22 in GL4(𝔽89) generated by

 47 28 85 1 61 42 50 54 70 2 43 23 11 8 28 46
,
 88 0 0 0 0 88 0 0 59 3 1 0 63 68 0 1
,
 21 34 1 52 55 34 2 14 0 0 12 47 0 0 34 22
,
 34 21 84 10 34 55 87 75 0 0 17 82 0 0 3 72
`G:=sub<GL(4,GF(89))| [47,61,70,11,28,42,2,8,85,50,43,28,1,54,23,46],[88,0,59,63,0,88,3,68,0,0,1,0,0,0,0,1],[21,55,0,0,34,34,0,0,1,2,12,34,52,14,47,22],[34,34,0,0,21,55,0,0,84,87,17,3,10,75,82,72] >;`

D46D22 in GAP, Magma, Sage, TeX

`D_4\rtimes_6D_{22}`
`% in TeX`

`G:=Group("D4:6D22");`
`// GroupNames label`

`G:=SmallGroup(352,179);`
`// by ID`

`G=gap.SmallGroup(352,179);`
`# by ID`

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

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

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