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## G = D4×C20order 160 = 25·5

### Direct product of C20 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C20
 Chief series C1 — C2 — C22 — C2×C10 — C2×C20 — C5×C22⋊C4 — D4×C20
 Lower central C1 — C2 — D4×C20
 Upper central C1 — C2×C20 — D4×C20

Generators and relations for D4×C20
G = < a,b,c | a20=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 124 in 94 conjugacy classes, 64 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×3], C22, C22 [×4], C22 [×4], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×4], C23 [×2], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C20 [×4], C20 [×3], C2×C10, C2×C10 [×4], C2×C10 [×4], C4×D4, C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C22×C10 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C22×C20 [×2], D4×C10, D4×C20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×2], C23, C10 [×7], C22×C4, C2×D4, C4○D4, C20 [×4], C2×C10 [×7], C4×D4, C2×C20 [×6], C5×D4 [×2], C22×C10, C22×C20, D4×C10, C5×C4○D4, D4×C20

Smallest permutation representation of D4×C20
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 41 25 64)(2 42 26 65)(3 43 27 66)(4 44 28 67)(5 45 29 68)(6 46 30 69)(7 47 31 70)(8 48 32 71)(9 49 33 72)(10 50 34 73)(11 51 35 74)(12 52 36 75)(13 53 37 76)(14 54 38 77)(15 55 39 78)(16 56 40 79)(17 57 21 80)(18 58 22 61)(19 59 23 62)(20 60 24 63)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 61)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4L 5A 5B 5C 5D 10A ··· 10L 10M ··· 10AB 20A ··· 20P 20Q ··· 20AV order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 2 2 1 1 1 1 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C5 C10 C10 C10 C10 C10 C20 D4 C4○D4 C5×D4 C5×C4○D4 kernel D4×C20 C4×C20 C5×C22⋊C4 C5×C4⋊C4 C22×C20 D4×C10 C5×D4 C4×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C20 C10 C4 C2 # reps 1 1 2 1 2 1 8 4 4 8 4 8 4 32 2 2 8 8

Matrix representation of D4×C20 in GL3(𝔽41) generated by

 39 0 0 0 40 0 0 0 40
,
 40 0 0 0 0 40 0 1 0
,
 1 0 0 0 1 0 0 0 40
G:=sub<GL(3,GF(41))| [39,0,0,0,40,0,0,0,40],[40,0,0,0,0,1,0,40,0],[1,0,0,0,1,0,0,0,40] >;

D4×C20 in GAP, Magma, Sage, TeX

D_4\times C_{20}
% in TeX

G:=Group("D4xC20");
// GroupNames label

G:=SmallGroup(160,179);
// by ID

G=gap.SmallGroup(160,179);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,480,505,374]);
// Polycyclic

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

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