Copied to
clipboard

## G = D10.D10order 400 = 24·52

### 5th non-split extension by D10 of D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — D10.D10
 Chief series C1 — C5 — C52 — C5×D5 — D5×C10 — C2×D5.D5 — D10.D10
 Lower central C52 — C5×C10 — D10.D10
 Upper central C1 — C2 — C22

Generators and relations for D10.D10
G = < a,b,c,d | a10=b2=c10=1, d2=a4b, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=a5c-1 >

Subgroups: 424 in 73 conjugacy classes, 25 normal (19 characteristic)
C1, C2, C2, C4, C22, C22, C5, C5, C2×C4, C23, D5, D5, C10, C10, C22⋊C4, Dic5, F5, D10, D10, C2×C10, C2×C10, C52, C2×Dic5, C2×F5, C22×D5, C22×C10, C5×D5, C5×D5, C5×C10, C5×C10, C23.D5, C22⋊F5, D5.D5, D5×C10, D5×C10, C102, C2×D5.D5, D5×C2×C10, D10.D10
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, Dic5, F5, D10, C2×Dic5, C5⋊D4, C2×F5, C23.D5, C22⋊F5, D5.D5, C2×D5.D5, D10.D10

Smallest permutation representation of D10.D10
On 40 points
Generators in S40
```(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)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 20)(8 19)(9 18)(10 17)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 40)(30 39)
(1 9 7 5 3)(2 10 8 6 4)(11 13 15 17 19)(12 14 16 18 20)(21 30 29 28 27 26 25 24 23 22)(31 32 33 34 35 36 37 38 39 40)
(1 38 12 27)(2 31 11 24)(3 34 20 21)(4 37 19 28)(5 40 18 25)(6 33 17 22)(7 36 16 29)(8 39 15 26)(9 32 14 23)(10 35 13 30)```

`G:=sub<Sym(40)| (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,20)(8,19)(9,18)(10,17)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,40)(30,39), (1,9,7,5,3)(2,10,8,6,4)(11,13,15,17,19)(12,14,16,18,20)(21,30,29,28,27,26,25,24,23,22)(31,32,33,34,35,36,37,38,39,40), (1,38,12,27)(2,31,11,24)(3,34,20,21)(4,37,19,28)(5,40,18,25)(6,33,17,22)(7,36,16,29)(8,39,15,26)(9,32,14,23)(10,35,13,30)>;`

`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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,20)(8,19)(9,18)(10,17)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,40)(30,39), (1,9,7,5,3)(2,10,8,6,4)(11,13,15,17,19)(12,14,16,18,20)(21,30,29,28,27,26,25,24,23,22)(31,32,33,34,35,36,37,38,39,40), (1,38,12,27)(2,31,11,24)(3,34,20,21)(4,37,19,28)(5,40,18,25)(6,33,17,22)(7,36,16,29)(8,39,15,26)(9,32,14,23)(10,35,13,30) );`

`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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,20),(8,19),(9,18),(10,17),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,40),(30,39)], [(1,9,7,5,3),(2,10,8,6,4),(11,13,15,17,19),(12,14,16,18,20),(21,30,29,28,27,26,25,24,23,22),(31,32,33,34,35,36,37,38,39,40)], [(1,38,12,27),(2,31,11,24),(3,34,20,21),(4,37,19,28),(5,40,18,25),(6,33,17,22),(7,36,16,29),(8,39,15,26),(9,32,14,23),(10,35,13,30)]])`

46 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 5C ··· 5G 10A ··· 10F 10G ··· 10U 10V ··· 10AC order 1 2 2 2 2 2 4 4 4 4 5 5 5 ··· 5 10 ··· 10 10 ··· 10 10 ··· 10 size 1 1 2 5 5 10 50 50 50 50 2 2 4 ··· 4 2 ··· 2 4 ··· 4 10 ··· 10

46 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + - + - + + + image C1 C2 C2 C4 C4 D4 D5 Dic5 D10 Dic5 C5⋊D4 F5 C2×F5 C22⋊F5 D5.D5 C2×D5.D5 D10.D10 kernel D10.D10 C2×D5.D5 D5×C2×C10 D5×C10 C102 C5×D5 C22×D5 D10 D10 C2×C10 D5 C2×C10 C10 C5 C22 C2 C1 # reps 1 2 1 2 2 2 2 2 2 2 8 1 1 2 4 4 8

Matrix representation of D10.D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 10 0 0 0 0 0 4 37 0 0 0 0 31 0 16 0 0 0 14 0 0 18
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 37 14 0 0 0 0 37 4 0 0 0 0 18 37 0 18 0 0 35 37 16 0
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 24 0 18 0 0 0 24 0 0 18
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 23 0 19 0 0 0 0 0 23 18 0 0 24 16 18 0 0 0 24 0 18 0

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,10,4,31,14,0,0,0,37,0,0,0,0,0,0,16,0,0,0,0,0,0,18],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,37,37,18,35,0,0,14,4,37,37,0,0,0,0,0,16,0,0,0,0,18,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,16,0,24,24,0,0,0,16,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,23,0,24,24,0,0,0,0,16,0,0,0,19,23,18,18,0,0,0,18,0,0] >;`

D10.D10 in GAP, Magma, Sage, TeX

`D_{10}.D_{10}`
`% in TeX`

`G:=Group("D10.D10");`
`// GroupNames label`

`G:=SmallGroup(400,148);`
`// by ID`

`G=gap.SmallGroup(400,148);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,1924,8645,2897]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=c^10=1,d^2=a^4*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=a^5*c^-1>;`
`// generators/relations`

׿
×
𝔽