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## G = C28⋊D4order 224 = 25·7

### 3rd semidirect product of C28 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C28⋊D4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×D28 — C28⋊D4
 Lower central C7 — C2×C14 — C28⋊D4
 Upper central C1 — C22 — C2×D4

Generators and relations for C28⋊D4
G = < a,b,c | a28=b4=c2=1, bab-1=a13, cac=a-1, cbc=b-1 >

Subgroups: 510 in 108 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C14, C42, C2×D4, C2×D4, Dic7, C28, D14, C2×C14, C2×C14, C41D4, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, C4×Dic7, C2×D28, C2×C7⋊D4, D4×C14, C28⋊D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C41D4, C7⋊D4, C22×D7, D4×D7, C2×C7⋊D4, C28⋊D4

Smallest permutation representation of C28⋊D4
On 112 points
Generators in S112
```(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 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 62 95 50)(2 75 96 35)(3 60 97 48)(4 73 98 33)(5 58 99 46)(6 71 100 31)(7 84 101 44)(8 69 102 29)(9 82 103 42)(10 67 104 55)(11 80 105 40)(12 65 106 53)(13 78 107 38)(14 63 108 51)(15 76 109 36)(16 61 110 49)(17 74 111 34)(18 59 112 47)(19 72 85 32)(20 57 86 45)(21 70 87 30)(22 83 88 43)(23 68 89 56)(24 81 90 41)(25 66 91 54)(26 79 92 39)(27 64 93 52)(28 77 94 37)
(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(29 83)(30 82)(31 81)(32 80)(33 79)(34 78)(35 77)(36 76)(37 75)(38 74)(39 73)(40 72)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(56 84)(85 105)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(92 98)(93 97)(94 96)(106 112)(107 111)(108 110)```

`G:=sub<Sym(112)| (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,62,95,50)(2,75,96,35)(3,60,97,48)(4,73,98,33)(5,58,99,46)(6,71,100,31)(7,84,101,44)(8,69,102,29)(9,82,103,42)(10,67,104,55)(11,80,105,40)(12,65,106,53)(13,78,107,38)(14,63,108,51)(15,76,109,36)(16,61,110,49)(17,74,111,34)(18,59,112,47)(19,72,85,32)(20,57,86,45)(21,70,87,30)(22,83,88,43)(23,68,89,56)(24,81,90,41)(25,66,91,54)(26,79,92,39)(27,64,93,52)(28,77,94,37), (2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,83)(30,82)(31,81)(32,80)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(56,84)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(106,112)(107,111)(108,110)>;`

`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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,62,95,50)(2,75,96,35)(3,60,97,48)(4,73,98,33)(5,58,99,46)(6,71,100,31)(7,84,101,44)(8,69,102,29)(9,82,103,42)(10,67,104,55)(11,80,105,40)(12,65,106,53)(13,78,107,38)(14,63,108,51)(15,76,109,36)(16,61,110,49)(17,74,111,34)(18,59,112,47)(19,72,85,32)(20,57,86,45)(21,70,87,30)(22,83,88,43)(23,68,89,56)(24,81,90,41)(25,66,91,54)(26,79,92,39)(27,64,93,52)(28,77,94,37), (2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,83)(30,82)(31,81)(32,80)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(56,84)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(106,112)(107,111)(108,110) );`

`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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,62,95,50),(2,75,96,35),(3,60,97,48),(4,73,98,33),(5,58,99,46),(6,71,100,31),(7,84,101,44),(8,69,102,29),(9,82,103,42),(10,67,104,55),(11,80,105,40),(12,65,106,53),(13,78,107,38),(14,63,108,51),(15,76,109,36),(16,61,110,49),(17,74,111,34),(18,59,112,47),(19,72,85,32),(20,57,86,45),(21,70,87,30),(22,83,88,43),(23,68,89,56),(24,81,90,41),(25,66,91,54),(26,79,92,39),(27,64,93,52),(28,77,94,37)], [(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16),(29,83),(30,82),(31,81),(32,80),(33,79),(34,78),(35,77),(36,76),(37,75),(38,74),(39,73),(40,72),(41,71),(42,70),(43,69),(44,68),(45,67),(46,66),(47,65),(48,64),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(56,84),(85,105),(86,104),(87,103),(88,102),(89,101),(90,100),(91,99),(92,98),(93,97),(94,96),(106,112),(107,111),(108,110)]])`

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 7A 7B 7C 14A ··· 14I 14J ··· 14U 28A ··· 28F order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 4 4 28 28 2 2 14 14 14 14 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D7 D14 D14 C7⋊D4 D4×D7 kernel C28⋊D4 C4×Dic7 C2×D28 C2×C7⋊D4 D4×C14 Dic7 C28 C2×D4 C2×C4 C23 C4 C2 # reps 1 1 1 4 1 4 2 3 3 6 12 6

Matrix representation of C28⋊D4 in GL4(𝔽29) generated by

 1 26 0 0 3 21 0 0 0 0 28 17 0 0 5 1
,
 5 16 0 0 2 24 0 0 0 0 28 0 0 0 0 28
,
 1 0 0 0 3 28 0 0 0 0 1 0 0 0 24 28
`G:=sub<GL(4,GF(29))| [1,3,0,0,26,21,0,0,0,0,28,5,0,0,17,1],[5,2,0,0,16,24,0,0,0,0,28,0,0,0,0,28],[1,3,0,0,0,28,0,0,0,0,1,24,0,0,0,28] >;`

C28⋊D4 in GAP, Magma, Sage, TeX

`C_{28}\rtimes D_4`
`% in TeX`

`G:=Group("C28:D4");`
`// GroupNames label`

`G:=SmallGroup(224,135);`
`// by ID`

`G=gap.SmallGroup(224,135);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,103,218,188,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^28=b^4=c^2=1,b*a*b^-1=a^13,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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