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## G = D4.12D20order 320 = 26·5

### 2nd non-split extension by D4 of D20 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D4.12D20
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — D4⋊8D10 — D4.12D20
 Lower central C5 — C10 — C20 — D4.12D20
 Upper central C1 — C2 — C4○D4 — C8○D4

Generators and relations for D4.12D20
G = < a,b,c,d | a4=b2=d2=1, c20=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c19 >

Subgroups: 1286 in 268 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C8○D4, C2×D8, C4○D8, C8⋊C22, 2+ 1+4, C40, C40, Dic10, C4×D5, D20, D20, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, D4○D8, C40⋊C2, D40, Dic20, C2×C40, C5×M4(2), C2×D20, C4○D20, D4×D5, Q82D5, C5×C4○D4, C2×D40, D407C2, C8⋊D10, C5×C8○D4, D48D10, D4.12D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, D20, C22×D5, D4○D8, C2×D20, C23×D5, C22×D20, D4.12D20

Smallest permutation representation of D4.12D20
On 80 points
Generators in S80
```(1 41 21 61)(2 42 22 62)(3 43 23 63)(4 44 24 64)(5 45 25 65)(6 46 26 66)(7 47 27 67)(8 48 28 68)(9 49 29 69)(10 50 30 70)(11 51 31 71)(12 52 32 72)(13 53 33 73)(14 54 34 74)(15 55 35 75)(16 56 36 76)(17 57 37 77)(18 58 38 78)(19 59 39 79)(20 60 40 80)
(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 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 80)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)```

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E ··· 2J 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 8E 10A 10B 10C ··· 10H 20A 20B 20C 20D 20E ··· 20J 40A ··· 40H 40I ··· 40T order 1 2 2 2 2 2 ··· 2 4 4 4 4 4 4 5 5 8 8 8 8 8 10 10 10 ··· 10 20 20 20 20 20 ··· 20 40 ··· 40 40 ··· 40 size 1 1 2 2 2 20 ··· 20 2 2 2 2 20 20 2 2 2 2 4 4 4 2 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 D20 D20 D4○D8 D4.12D20 kernel D4.12D20 C2×D40 D40⋊7C2 C8⋊D10 C5×C8○D4 D4⋊8D10 C5×D4 C5×Q8 C8○D4 C2×C8 M4(2) C4○D4 D4 Q8 C5 C1 # reps 1 3 3 6 1 2 3 1 2 6 6 2 12 4 2 8

Matrix representation of D4.12D20 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 20 0 0 0 38 0 27 40 0 0 4 0 40 0 0 0 23 1 36 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 4 0 40 0 0 0 20 0 0 40
,
 0 40 0 0 0 0 1 35 0 0 0 0 0 0 40 13 0 0 0 0 8 18 0 0 0 0 37 26 35 26 0 0 23 7 12 23
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 40 13 0 0 0 0 0 1 0 0 0 0 37 26 35 26 0 0 21 7 16 6

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,38,4,23,0,0,0,0,0,1,0,0,20,27,40,36,0,0,0,40,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,4,20,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,40,35,0,0,0,0,0,0,40,8,37,23,0,0,13,18,26,7,0,0,0,0,35,12,0,0,0,0,26,23],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,37,21,0,0,13,1,26,7,0,0,0,0,35,16,0,0,0,0,26,6] >;`

D4.12D20 in GAP, Magma, Sage, TeX

`D_4._{12}D_{20}`
`% in TeX`

`G:=Group("D4.12D20");`
`// GroupNames label`

`G:=SmallGroup(320,1424);`
`// by ID`

`G=gap.SmallGroup(320,1424);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,192,1684,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^20=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c^19>;`
`// generators/relations`

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