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## G = Q8⋊D22order 352 = 25·11

### 2nd semidirect product of Q8 and D22 acting via D22/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — Q8⋊D22
 Chief series C1 — C11 — C22 — C44 — D44 — C2×D44 — Q8⋊D22
 Lower central C11 — C22 — C44 — Q8⋊D22
 Upper central C1 — C2 — C2×C4 — C4○D4

Generators and relations for Q8⋊D22
G = < a,b,c,d | a4=b2=c22=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, dbd=a-1b, dcd=c-1 >

Subgroups: 466 in 68 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C11, M4(2), D8, SD16, C2×D4, C4○D4, D11, C22, C22, C8⋊C22, C44, C44, D22, C2×C22, C2×C22, C11⋊C8, D44, D44, C2×C44, C2×C44, D4×C11, D4×C11, Q8×C11, C22×D11, C44.C4, D4⋊D11, Q8⋊D11, C2×D44, C11×C4○D4, Q8⋊D22
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C8⋊C22, D22, C11⋊D4, C22×D11, C2×C11⋊D4, Q8⋊D22

Smallest permutation representation of Q8⋊D22
On 88 points
Generators in S88
```(1 30 37 19)(2 31 38 20)(3 32 39 21)(4 33 40 22)(5 23 41 12)(6 24 42 13)(7 25 43 14)(8 26 44 15)(9 27 34 16)(10 28 35 17)(11 29 36 18)(45 72 56 83)(46 73 57 84)(47 74 58 85)(48 75 59 86)(49 76 60 87)(50 77 61 88)(51 78 62 67)(52 79 63 68)(53 80 64 69)(54 81 65 70)(55 82 66 71)
(1 78)(2 68)(3 80)(4 70)(5 82)(6 72)(7 84)(8 74)(9 86)(10 76)(11 88)(12 66)(13 56)(14 46)(15 58)(16 48)(17 60)(18 50)(19 62)(20 52)(21 64)(22 54)(23 55)(24 45)(25 57)(26 47)(27 59)(28 49)(29 61)(30 51)(31 63)(32 53)(33 65)(34 75)(35 87)(36 77)(37 67)(38 79)(39 69)(40 81)(41 71)(42 83)(43 73)(44 85)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 22)(8 21)(9 20)(10 19)(11 18)(23 42)(24 41)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 44)(33 43)(45 55)(46 54)(47 53)(48 52)(49 51)(56 66)(57 65)(58 64)(59 63)(60 62)(67 76)(68 75)(69 74)(70 73)(71 72)(77 88)(78 87)(79 86)(80 85)(81 84)(82 83)```

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

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

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

61 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 8A 8B 11A ··· 11E 22A ··· 22E 22F ··· 22T 44A ··· 44J 44K ··· 44Y order 1 2 2 2 2 2 4 4 4 8 8 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 44 ··· 44 size 1 1 2 4 44 44 2 2 4 44 44 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

61 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D11 D22 D22 D22 C11⋊D4 C11⋊D4 C8⋊C22 Q8⋊D22 kernel Q8⋊D22 C44.C4 D4⋊D11 Q8⋊D11 C2×D44 C11×C4○D4 C44 C2×C22 C4○D4 C2×C4 D4 Q8 C4 C22 C11 C1 # reps 1 1 2 2 1 1 1 1 5 5 5 5 10 10 1 10

Matrix representation of Q8⋊D22 in GL4(𝔽89) generated by

 13 14 0 0 26 76 0 0 0 0 76 75 0 0 63 13
,
 0 0 76 75 0 0 63 13 13 14 0 0 26 76 0 0
,
 0 38 0 0 7 82 0 0 0 0 0 51 0 0 82 7
,
 7 38 0 0 69 82 0 0 0 0 78 40 0 0 86 11
`G:=sub<GL(4,GF(89))| [13,26,0,0,14,76,0,0,0,0,76,63,0,0,75,13],[0,0,13,26,0,0,14,76,76,63,0,0,75,13,0,0],[0,7,0,0,38,82,0,0,0,0,0,82,0,0,51,7],[7,69,0,0,38,82,0,0,0,0,78,86,0,0,40,11] >;`

Q8⋊D22 in GAP, Magma, Sage, TeX

`Q_8\rtimes D_{22}`
`% in TeX`

`G:=Group("Q8:D22");`
`// GroupNames label`

`G:=SmallGroup(352,144);`
`// by ID`

`G=gap.SmallGroup(352,144);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,218,188,579,159,69,11525]);`
`// 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,d*b*d=a^-1*b,d*c*d=c^-1>;`
`// generators/relations`

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