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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

Aliases: D4×C20, C424C10, C4⋊C47C10, C41(C2×C20), C209(C2×C4), (C4×C20)⋊11C2, C2.3(D4×C10), C22⋊C46C10, (C22×C20)⋊4C2, C221(C2×C20), (C22×C4)⋊2C10, (C2×D4).7C10, C10.66(C2×D4), (D4×C10).14C2, C2.4(C22×C20), C23.7(C2×C10), C10.39(C4○D4), (C2×C10).73C23, C10.45(C22×C4), (C2×C20).121C22, C22.7(C22×C10), (C22×C10).26C22, (C5×C4⋊C4)⋊16C2, (C2×C10)⋊7(C2×C4), (C2×C20)(D4×C10), C2.2(C5×C4○D4), (C5×C22⋊C4)⋊14C2, (C2×C4).15(C2×C10), (C2×C20)(C5×C4⋊C4), (C2×C20)(C5×C22⋊C4), SmallGroup(160,179)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C20
C1C2C22C2×C10C2×C20C5×C22⋊C4 — D4×C20
C1C2 — D4×C20
C1C2×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)])

D4×C20 is a maximal subgroup of
C20.57D8  C20.50D8  C20.38SD16  D4.3Dic10  C42.47D10  C207M4(2)  C42.48D10  C207D8  D4.1D20  C42.51D10  D4.2D20  C42.102D10  D45Dic10  C42.104D10  C42.105D10  C42.106D10  D46Dic10  C4211D10  C42.108D10  C4212D10  C42.228D10  D2023D4  D2024D4  Dic1023D4  Dic1024D4  D45D20  D46D20  C4216D10  C42.229D10  C42.113D10  C42.114D10  C4217D10  C42.115D10  C42.116D10  C42.117D10  C42.118D10  C42.119D10

100 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4L5A5B5C5D10A···10L10M···10AB20A···20P20Q···20AV
order1222222244444···4555510···1010···1020···2020···20
size1111222211112···211111···12···21···12···2

100 irreducible representations

dim111111111111112222
type+++++++
imageC1C2C2C2C2C2C4C5C10C10C10C10C10C20D4C4○D4C5×D4C5×C4○D4
kernelD4×C20C4×C20C5×C22⋊C4C5×C4⋊C4C22×C20D4×C10C5×D4C4×D4C42C22⋊C4C4⋊C4C22×C4C2×D4D4C20C10C4C2
# reps1121218448484322288

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

3900
0400
0040
,
4000
0040
010
,
100
010
0040
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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