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## G = D88⋊C2order 352 = 25·11

### 6th semidirect product of D88 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — D88⋊C2
 Chief series C1 — C11 — C22 — C44 — C4×D11 — D4×D11 — D88⋊C2
 Lower central C11 — C22 — C44 — D88⋊C2
 Upper central C1 — C2 — C4 — SD16

Generators and relations for D88⋊C2
G = < a,b,c | a88=b2=c2=1, bab=a-1, cac=a67, bc=cb >

Subgroups: 546 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, D4, Q8, C23, C11, M4(2), D8, SD16, SD16, C2×D4, C4○D4, D11, C22, C22, C8⋊C22, Dic11, C44, C44, D22, D22, C2×C22, C11⋊C8, C88, C4×D11, C4×D11, D44, D44, C11⋊D4, D4×C11, Q8×C11, C22×D11, C88⋊C2, D88, D4⋊D11, Q8⋊D11, C11×SD16, D4×D11, D44⋊C2, D88⋊C2
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C8⋊C22, D22, C22×D11, D4×D11, D88⋊C2

Smallest permutation representation of D88⋊C2
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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 60)(19 59)(20 58)(21 57)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(78 88)(79 87)(80 86)(81 85)(82 84)
(2 68)(3 47)(4 26)(6 72)(7 51)(8 30)(10 76)(11 55)(12 34)(14 80)(15 59)(16 38)(18 84)(19 63)(20 42)(22 88)(23 67)(24 46)(27 71)(28 50)(31 75)(32 54)(35 79)(36 58)(39 83)(40 62)(43 87)(44 66)(48 70)(52 74)(56 78)(60 82)(64 86)```

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

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

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

46 conjugacy classes

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

46 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D11 D22 D22 D22 C8⋊C22 D4×D11 D88⋊C2 kernel D88⋊C2 C88⋊C2 D88 D4⋊D11 Q8⋊D11 C11×SD16 D4×D11 D44⋊C2 Dic11 D22 SD16 C8 D4 Q8 C11 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 5 5 5 5 1 5 10

Matrix representation of D88⋊C2 in GL4(𝔽89) generated by

 0 0 79 76 0 0 77 88 81 13 74 42 74 72 47 70
,
 5 34 0 0 15 84 0 0 32 44 2 13 21 36 34 87
,
 1 0 0 0 0 1 0 0 19 67 88 0 22 0 0 88
`G:=sub<GL(4,GF(89))| [0,0,81,74,0,0,13,72,79,77,74,47,76,88,42,70],[5,15,32,21,34,84,44,36,0,0,2,34,0,0,13,87],[1,0,19,22,0,1,67,0,0,0,88,0,0,0,0,88] >;`

D88⋊C2 in GAP, Magma, Sage, TeX

`D_{88}\rtimes C_2`
`% in TeX`

`G:=Group("D88:C2");`
`// GroupNames label`

`G:=SmallGroup(352,109);`
`// by ID`

`G=gap.SmallGroup(352,109);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,362,116,86,297,159,69,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^88=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^67,b*c=c*b>;`
`// generators/relations`

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