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## G = C44.46D4order 352 = 25·11

### 3rd non-split extension by C44 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C22 — C44.46D4
 Chief series C1 — C11 — C22 — C44 — C2×C44 — C2×D44 — C44.46D4
 Lower central C11 — C22 — C2×C22 — C44.46D4
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C44.46D4
G = < a,b,c | a44=c2=1, b4=a22, bab-1=cac=a-1, cbc=a11b3 >

Smallest permutation representation of C44.46D4
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 81 12 70 23 59 34 48)(2 80 13 69 24 58 35 47)(3 79 14 68 25 57 36 46)(4 78 15 67 26 56 37 45)(5 77 16 66 27 55 38 88)(6 76 17 65 28 54 39 87)(7 75 18 64 29 53 40 86)(8 74 19 63 30 52 41 85)(9 73 20 62 31 51 42 84)(10 72 21 61 32 50 43 83)(11 71 22 60 33 49 44 82)
(1 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(45 84)(46 83)(47 82)(48 81)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 66)(64 65)(85 88)(86 87)```

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

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

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

61 conjugacy classes

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

61 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C4 D4 D11 D22 D44 C11⋊D4 C4×D11 C4.D4 C44.46D4 kernel C44.46D4 C44.C4 C11×M4(2) C2×D44 C22×D11 C44 M4(2) C2×C4 C4 C4 C22 C11 C1 # reps 1 1 1 1 4 2 5 5 10 10 10 1 10

Matrix representation of C44.46D4 in GL4(𝔽89) generated by

 16 73 0 0 75 53 0 0 0 0 24 73 0 0 16 45
,
 54 10 25 7 11 35 34 16 6 40 20 9 32 14 5 69
,
 60 45 0 0 11 29 0 0 69 15 86 59 12 72 24 3
`G:=sub<GL(4,GF(89))| [16,75,0,0,73,53,0,0,0,0,24,16,0,0,73,45],[54,11,6,32,10,35,40,14,25,34,20,5,7,16,9,69],[60,11,69,12,45,29,15,72,0,0,86,24,0,0,59,3] >;`

C44.46D4 in GAP, Magma, Sage, TeX

`C_{44}._{46}D_4`
`% in TeX`

`G:=Group("C44.46D4");`
`// GroupNames label`

`G:=SmallGroup(352,29);`
`// by ID`

`G=gap.SmallGroup(352,29);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,121,31,362,86,297,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^44=c^2=1,b^4=a^22,b*a*b^-1=c*a*c=a^-1,c*b*c=a^11*b^3>;`
`// generators/relations`

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