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G = D40.4C4order 320 = 26·5

2nd non-split extension by D40 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D40.4C4
 Chief series C1 — C5 — C10 — C20 — C40 — C2×C40 — C2×D40 — D40.4C4
 Lower central C5 — C10 — C20 — C40 — D40.4C4
 Upper central C1 — C2 — C2×C4 — C2×C8 — M5(2)

Generators and relations for D40.4C4
G = < a,b,c | a40=b2=1, c4=a10, bab=a-1, cac-1=a21, cbc-1=a15b >

Subgroups: 430 in 62 conjugacy classes, 25 normal (23 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C8, C2×C4, D4, C23, D5, C10, C10, C16, C2×C8, M4(2), D8, C2×D4, C20, D10, C2×C10, C8.C4, M5(2), C2×D8, C52C8, C40, D20, C2×C20, C22×D5, M5(2)⋊C2, C80, D40, D40, C4.Dic5, C2×C40, C2×D20, C40.6C4, C5×M5(2), C2×D40, D40.4C4
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, D10, D4⋊C4, C4×D5, D20, C5⋊D4, M5(2)⋊C2, C40⋊C2, D40, D10⋊C4, D205C4, D40.4C4

Smallest permutation representation of D40.4C4
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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(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)(72 80)(73 79)(74 78)(75 77)
(1 64 26 69 11 74 36 79 21 44 6 49 31 54 16 59)(2 45 27 50 12 55 37 60 22 65 7 70 32 75 17 80)(3 66 28 71 13 76 38 41 23 46 8 51 33 56 18 61)(4 47 29 52 14 57 39 62 24 67 9 72 34 77 19 42)(5 68 30 73 15 78 40 43 25 48 10 53 35 58 20 63)```

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 5A 5B 8A 8B 8C 8D 8E 10A 10B 10C 10D 16A 16B 16C 16D 20A 20B 20C 20D 20E 20F 40A ··· 40H 40I 40J 40K 40L 80A ··· 80P order 1 2 2 2 2 4 4 5 5 8 8 8 8 8 10 10 10 10 16 16 16 16 20 20 20 20 20 20 40 ··· 40 40 40 40 40 80 ··· 80 size 1 1 2 40 40 2 2 2 2 2 2 4 40 40 2 2 4 4 4 4 4 4 2 2 2 2 4 4 2 ··· 2 4 4 4 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C4 D4 D4 D5 D8 SD16 D10 C4×D5 C5⋊D4 D20 D40 C40⋊C2 M5(2)⋊C2 D40.4C4 kernel D40.4C4 C40.6C4 C5×M5(2) C2×D40 D40 C40 C2×C20 M5(2) C20 C2×C10 C2×C8 C8 C8 C2×C4 C4 C22 C5 C1 # reps 1 1 1 1 4 1 1 2 2 2 2 4 4 4 8 8 2 8

Matrix representation of D40.4C4 in GL4(𝔽241) generated by

 221 47 0 0 194 14 0 0 221 47 20 194 194 14 47 227
,
 20 212 0 0 47 221 0 0 131 106 240 0 170 110 189 1
,
 1 0 239 0 0 1 0 239 142 106 240 0 135 111 0 240
`G:=sub<GL(4,GF(241))| [221,194,221,194,47,14,47,14,0,0,20,47,0,0,194,227],[20,47,131,170,212,221,106,110,0,0,240,189,0,0,0,1],[1,0,142,135,0,1,106,111,239,0,240,0,0,239,0,240] >;`

D40.4C4 in GAP, Magma, Sage, TeX

`D_{40}._4C_4`
`% in TeX`

`G:=Group("D40.4C4");`
`// GroupNames label`

`G:=SmallGroup(320,74);`
`// by ID`

`G=gap.SmallGroup(320,74);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,85,92,422,387,268,570,1684,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=b^2=1,c^4=a^10,b*a*b=a^-1,c*a*c^-1=a^21,c*b*c^-1=a^15*b>;`
`// generators/relations`

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