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## G = D4.D11order 176 = 24·11

### The non-split extension by D4 of D11 acting via D11/C11=C2

Aliases: D4.D11, C22.8D4, C4.2D22, C112SD16, Dic222C2, C44.2C22, C11⋊C82C2, (D4×C11).1C2, C2.5(C11⋊D4), SmallGroup(176,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — D4.D11
 Chief series C1 — C11 — C22 — C44 — Dic22 — D4.D11
 Lower central C11 — C22 — C44 — D4.D11
 Upper central C1 — C2 — C4 — D4

Generators and relations for D4.D11
G = < a,b,c,d | a4=b2=c11=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Smallest permutation representation of D4.D11
On 88 points
Generators in S88
```(1 43 21 32)(2 44 22 33)(3 34 12 23)(4 35 13 24)(5 36 14 25)(6 37 15 26)(7 38 16 27)(8 39 17 28)(9 40 18 29)(10 41 19 30)(11 42 20 31)(45 67 56 78)(46 68 57 79)(47 69 58 80)(48 70 59 81)(49 71 60 82)(50 72 61 83)(51 73 62 84)(52 74 63 85)(53 75 64 86)(54 76 65 87)(55 77 66 88)
(1 32)(2 33)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)
(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 66 21 55)(2 65 22 54)(3 64 12 53)(4 63 13 52)(5 62 14 51)(6 61 15 50)(7 60 16 49)(8 59 17 48)(9 58 18 47)(10 57 19 46)(11 56 20 45)(23 86 34 75)(24 85 35 74)(25 84 36 73)(26 83 37 72)(27 82 38 71)(28 81 39 70)(29 80 40 69)(30 79 41 68)(31 78 42 67)(32 88 43 77)(33 87 44 76)```

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

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

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

D4.D11 is a maximal subgroup of   D4⋊D22  D83D11  SD16×D11  D4.D22  D446C22  D4.8D22  D4.9D22
D4.D11 is a maximal quotient of   C4.Dic22  C22.Q16  D4⋊Dic11

32 conjugacy classes

 class 1 2A 2B 4A 4B 8A 8B 11A ··· 11E 22A ··· 22E 22F ··· 22O 44A ··· 44E order 1 2 2 4 4 8 8 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 size 1 1 4 2 44 22 22 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 4 type + + + + + + + - image C1 C2 C2 C2 D4 SD16 D11 D22 C11⋊D4 D4.D11 kernel D4.D11 C11⋊C8 Dic22 D4×C11 C22 C11 D4 C4 C2 C1 # reps 1 1 1 1 1 2 5 5 10 5

Matrix representation of D4.D11 in GL4(𝔽89) generated by

 1 0 0 0 0 1 0 0 0 0 88 85 0 0 45 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 44 88
,
 0 1 0 0 88 86 0 0 0 0 1 0 0 0 0 1
,
 68 34 0 0 8 21 0 0 0 0 0 80 0 0 10 0
`G:=sub<GL(4,GF(89))| [1,0,0,0,0,1,0,0,0,0,88,45,0,0,85,1],[1,0,0,0,0,1,0,0,0,0,1,44,0,0,0,88],[0,88,0,0,1,86,0,0,0,0,1,0,0,0,0,1],[68,8,0,0,34,21,0,0,0,0,0,10,0,0,80,0] >;`

D4.D11 in GAP, Magma, Sage, TeX

`D_4.D_{11}`
`% in TeX`

`G:=Group("D4.D11");`
`// GroupNames label`

`G:=SmallGroup(176,15);`
`// by ID`

`G=gap.SmallGroup(176,15);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-11,40,61,182,97,42,4004]);`
`// Polycyclic`

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

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