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## G = C40.D4order 320 = 26·5

### 48th non-split extension by C40 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C40.D4
 Chief series C1 — C5 — C10 — C20 — C40 — C2×C40 — C20.4C8 — C40.D4
 Lower central C5 — C10 — C2×C10 — C40.D4
 Upper central C1 — C4 — C2×C8 — C2×M4(2)

Generators and relations for C40.D4
G = < a,b,c | a40=1, b4=a10, c2=a25, bab-1=a29, cac-1=a9, cbc-1=a15b3 >

Subgroups: 118 in 58 conjugacy classes, 31 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, C10, C10, C16, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, M5(2), C2×M4(2), C40, C40, C2×C20, C2×C20, C22×C10, C23.C8, C52C16, C2×C40, C5×M4(2), C22×C20, C20.4C8, C10×M4(2), C40.D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D5, C22⋊C4, C2×C8, M4(2), Dic5, D10, C22⋊C8, C52C8, C2×Dic5, C5⋊D4, C23.C8, C2×C52C8, C4.Dic5, C23.D5, C20.55D4, C40.D4

Smallest permutation representation of C40.D4
On 80 points
Generators in S80
```(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)
(1 70 26 75 11 80 36 45 21 50 6 55 31 60 16 65)(2 59 27 64 12 69 37 74 22 79 7 44 32 49 17 54)(3 48 28 53 13 58 38 63 23 68 8 73 33 78 18 43)(4 77 29 42 14 47 39 52 24 57 9 62 34 67 19 72)(5 66 30 71 15 76 40 41 25 46 10 51 35 56 20 61)
(1 50 26 75 11 60 36 45 21 70 6 55 31 80 16 65)(2 59 27 44 12 69 37 54 22 79 7 64 32 49 17 74)(3 68 28 53 13 78 38 63 23 48 8 73 33 58 18 43)(4 77 29 62 14 47 39 72 24 57 9 42 34 67 19 52)(5 46 30 71 15 56 40 41 25 66 10 51 35 76 20 61)```

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 8E 8F 10A ··· 10F 10G 10H 10I 10J 16A ··· 16H 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P order 1 2 2 2 4 4 4 4 5 5 8 8 8 8 8 8 10 ··· 10 10 10 10 10 16 ··· 16 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 2 4 1 1 2 4 2 2 2 2 2 2 4 4 2 ··· 2 4 4 4 4 20 ··· 20 2 ··· 2 4 4 4 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + - + - image C1 C2 C2 C4 C4 C8 C8 D4 D5 M4(2) Dic5 D10 Dic5 C5⋊D4 C5⋊2C8 C5⋊2C8 C4.Dic5 C23.C8 C40.D4 kernel C40.D4 C20.4C8 C10×M4(2) C2×C40 C22×C20 C2×C20 C22×C10 C40 C2×M4(2) C20 C2×C8 C2×C8 C22×C4 C8 C2×C4 C23 C4 C5 C1 # reps 1 2 1 2 2 4 4 2 2 2 2 2 2 8 4 4 8 2 8

Matrix representation of C40.D4 in GL4(𝔽241) generated by

 0 205 143 64 106 0 151 75 0 0 25 210 0 0 191 216
,
 87 0 1 124 134 0 0 109 177 1 0 114 128 0 0 154
,
 154 0 1 118 107 0 0 132 64 1 0 27 113 0 0 87
`G:=sub<GL(4,GF(241))| [0,106,0,0,205,0,0,0,143,151,25,191,64,75,210,216],[87,134,177,128,0,0,1,0,1,0,0,0,124,109,114,154],[154,107,64,113,0,0,1,0,1,0,0,0,118,132,27,87] >;`

C40.D4 in GAP, Magma, Sage, TeX

`C_{40}.D_4`
`% in TeX`

`G:=Group("C40.D4");`
`// GroupNames label`

`G:=SmallGroup(320,111);`
`// by ID`

`G=gap.SmallGroup(320,111);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,387,100,1123,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=1,b^4=a^10,c^2=a^25,b*a*b^-1=a^29,c*a*c^-1=a^9,c*b*c^-1=a^15*b^3>;`
`// generators/relations`

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