Copied to
clipboard

## G = C7⋊D28order 392 = 23·72

### The semidirect product of C7 and D28 acting via D28/D14=C2

Aliases: C72D28, Dic7⋊D7, C723D4, D142D7, C14.4D14, C2.4D72, (D7×C14)⋊2C2, C71(C7⋊D4), (C7×Dic7)⋊1C2, (C7×C14).4C22, (C2×C7⋊D7)⋊1C2, SmallGroup(392,21)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7×C14 — C7⋊D28
 Chief series C1 — C7 — C72 — C7×C14 — D7×C14 — C7⋊D28
 Lower central C72 — C7×C14 — C7⋊D28
 Upper central C1 — C2

Generators and relations for C7⋊D28
G = < a,b,c | a7=b28=c2=1, bab-1=cac=a-1, cbc=b-1 >

14C2
98C2
2C7
2C7
2C7
7C22
7C4
49C22
2D7
2C14
2C14
2C14
14D7
14D7
14C14
14D7
14D7
14D7
14D7
14D7
14D7
49D4
7C28
7D14
7D14
14D14
14D14
14D14
7D28

Permutation representations of C7⋊D28
On 28 points - transitive group 28T50
Generators in S28
```(1 13 25 9 21 5 17)(2 18 6 22 10 26 14)(3 15 27 11 23 7 19)(4 20 8 24 12 28 16)
(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)
(1 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)```

`G:=sub<Sym(28)| (1,13,25,9,21,5,17)(2,18,6,22,10,26,14)(3,15,27,11,23,7,19)(4,20,8,24,12,28,16), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)>;`

`G:=Group( (1,13,25,9,21,5,17)(2,18,6,22,10,26,14)(3,15,27,11,23,7,19)(4,20,8,24,12,28,16), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19) );`

`G=PermutationGroup([[(1,13,25,9,21,5,17),(2,18,6,22,10,26,14),(3,15,27,11,23,7,19),(4,20,8,24,12,28,16)], [(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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19)]])`

`G:=TransitiveGroup(28,50);`

47 conjugacy classes

 class 1 2A 2B 2C 4 7A ··· 7F 7G ··· 7O 14A ··· 14F 14G ··· 14O 14P ··· 14U 28A ··· 28F order 1 2 2 2 4 7 ··· 7 7 ··· 7 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 14 98 14 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 14 ··· 14 14 ··· 14

47 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 D4 D7 D7 D14 D28 C7⋊D4 D72 C7⋊D28 kernel C7⋊D28 C7×Dic7 D7×C14 C2×C7⋊D7 C72 Dic7 D14 C14 C7 C7 C2 C1 # reps 1 1 1 1 1 3 3 6 6 6 9 9

Matrix representation of C7⋊D28 in GL6(𝔽29)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 21 18 0 0 0 0 19 26 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 21 1 0 0 0 0 20 1 0 0 0 0 0 0 26 11 0 0 0 0 23 3 0 0 0 0 0 0 28 27 0 0 0 0 1 1
,
 0 19 0 0 0 0 26 0 0 0 0 0 0 0 26 11 0 0 0 0 23 3 0 0 0 0 0 0 28 0 0 0 0 0 1 1

`G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,19,0,0,0,0,18,26,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[21,20,0,0,0,0,1,1,0,0,0,0,0,0,26,23,0,0,0,0,11,3,0,0,0,0,0,0,28,1,0,0,0,0,27,1],[0,26,0,0,0,0,19,0,0,0,0,0,0,0,26,23,0,0,0,0,11,3,0,0,0,0,0,0,28,1,0,0,0,0,0,1] >;`

C7⋊D28 in GAP, Magma, Sage, TeX

`C_7\rtimes D_{28}`
`% in TeX`

`G:=Group("C7:D28");`
`// GroupNames label`

`G:=SmallGroup(392,21);`
`// by ID`

`G=gap.SmallGroup(392,21);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-7,-7,61,26,488,8404]);`
`// Polycyclic`

`G:=Group<a,b,c|a^7=b^28=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

Export

׿
×
𝔽