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## G = D44⋊6C22order 352 = 25·11

### 4th semidirect product of D44 and C22 acting via C22/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — D44⋊6C22
 Chief series C1 — C11 — C22 — C44 — D44 — D44⋊5C2 — D44⋊6C22
 Lower central C11 — C22 — C44 — D44⋊6C22
 Upper central C1 — C2 — C2×C4 — C2×D4

Generators and relations for D446C22
G = < a,b,c,d | a44=b2=c2=d2=1, bab=a-1, ac=ca, dad=a23, cbc=a22b, dbd=a11b, cd=dc >

Subgroups: 346 in 68 conjugacy classes, 29 normal (19 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, Dic11, C44, D22, C2×C22, C2×C22, C11⋊C8, Dic22, C4×D11, D44, C11⋊D4, C2×C44, D4×C11, D4×C11, C22×C22, C44.C4, D4⋊D11, D4.D11, D445C2, D4×C22, D446C22
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C8⋊C22, D22, C11⋊D4, C22×D11, C2×C11⋊D4, D446C22

Smallest permutation representation of D446C22
On 88 points
Generators in S88
```(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 62)(2 61)(3 60)(4 59)(5 58)(6 57)(7 56)(8 55)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 88)(20 87)(21 86)(22 85)(23 84)(24 83)(25 82)(26 81)(27 80)(28 79)(29 78)(30 77)(31 76)(32 75)(33 74)(34 73)(35 72)(36 71)(37 70)(38 69)(39 68)(40 67)(41 66)(42 65)(43 64)(44 63)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)
(2 24)(4 26)(6 28)(8 30)(10 32)(12 34)(14 36)(16 38)(18 40)(20 42)(22 44)(45 56)(46 79)(47 58)(48 81)(49 60)(50 83)(51 62)(52 85)(53 64)(54 87)(55 66)(57 68)(59 70)(61 72)(63 74)(65 76)(67 78)(69 80)(71 82)(73 84)(75 86)(77 88)```

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

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

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

61 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 8A 8B 11A ··· 11E 22A ··· 22O 22P ··· 22AI 44A ··· 44J order 1 2 2 2 2 2 4 4 4 8 8 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 size 1 1 2 4 4 44 2 2 44 44 44 2 ··· 2 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 C11⋊D4 C11⋊D4 C8⋊C22 D44⋊6C22 kernel D44⋊6C22 C44.C4 D4⋊D11 D4.D11 D44⋊5C2 D4×C22 C44 C2×C22 C2×D4 C2×C4 D4 C4 C22 C11 C1 # reps 1 1 2 2 1 1 1 1 5 5 10 10 10 1 10

Matrix representation of D446C22 in GL4(𝔽89) generated by

 45 64 0 0 73 44 0 0 30 56 0 87 71 56 2 0
,
 46 0 0 11 41 0 2 2 26 45 0 43 10 0 0 43
,
 88 0 0 0 0 88 0 0 65 0 1 0 65 0 0 1
,
 1 0 0 0 57 88 0 0 24 0 0 88 24 0 88 0
`G:=sub<GL(4,GF(89))| [45,73,30,71,64,44,56,56,0,0,0,2,0,0,87,0],[46,41,26,10,0,0,45,0,0,2,0,0,11,2,43,43],[88,0,65,65,0,88,0,0,0,0,1,0,0,0,0,1],[1,57,24,24,0,88,0,0,0,0,0,88,0,0,88,0] >;`

D446C22 in GAP, Magma, Sage, TeX

`D_{44}\rtimes_6C_2^2`
`% in TeX`

`G:=Group("D44:6C2^2");`
`// GroupNames label`

`G:=SmallGroup(352,127);`
`// by ID`

`G=gap.SmallGroup(352,127);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,103,218,188,579,159,69,11525]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^44=b^2=c^2=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^23,c*b*c=a^22*b,d*b*d=a^11*b,c*d=d*c>;`
`// generators/relations`

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