C10xD20 - GroupNames

G = C₁₀×D₂₀ order 400 = 2⁴·5²

Direct product of C₁₀ and D₂₀

direct product, metabelian, supersoluble, monomial

Aliases: C₁₀×D₂₀, C₂₀⋊₈D₁₀, C₁₀².₂₈C₂², (C₅×C₁₀)⋊₄D₄, (C₂×C₂₀)⋊₆D₅, C₁₀⋊₁(C₅×D₄), C₄⋊₂(D₅×C₁₀), C₅⋊₁(D₄×C₁₀), C₅²⋊₈(C₂×D₄), (C₁₀×C₂₀)⋊₇C₂, (C₂×C₂₀)⋊₃C₁₀, C₂₀⋊₂(C₂×C₁₀), D₁₀⋊₁(C₂×C₁₀), (C₅×C₂₀)⋊₇C₂², (C₂×C₁₀).₄₇D₁₀, (D₅×C₁₀)⋊₇C₂², (C₂²×D₅)⋊₁C₁₀, (C₅×C₁₀).₂₁C₂³, C₁₀.₃(C₂²×C₁₀), C₁₀.₄₂(C₂²×D₅), C₂².₁₀(D₅×C₁₀), (D₅×C₂×C₁₀)⋊₄C₂, (C₂×C₄)⋊₂(C₅×D₅), C₂.₄(D₅×C₂×C₁₀), (C₂×C₁₀).₁₂(C₂×C₁₀), SmallGroup(400,183)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₁₀×D₂₀

Chief series

C₁ — C₅ — C₁₀ — C₅×C₁₀ — D₅×C₁₀ — D₅×C₂×C₁₀ — C₁₀×D₂₀

Lower central

C₅ — C₁₀ — C₁₀×D₂₀

Upper central

C₁ — C₂×C₁₀ — C₂×C₂₀

Generators and relations for C₁₀×D₂₀
G = < a,b,c | a¹⁰=b²⁰=c²=1, ab=ba, ac=ca, cbc=b^-1 >

Subgroups: 388 in 124 conjugacy classes, 54 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₅, C₅, C₂×C₄, D₄, C₂³, D₅, C₁₀, C₁₀, C₁₀, C₂×D₄, C₂₀, C₂₀, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₅², D₂₀, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₂²×D₅, C₂²×C₁₀, C₅×D₅, C₅×C₁₀, C₅×C₁₀, C₂×D₂₀, D₄×C₁₀, C₅×C₂₀, D₅×C₁₀, D₅×C₁₀, C₁₀², C₅×D₂₀, C₁₀×C₂₀, D₅×C₂×C₁₀, C₁₀×D₂₀
Quotients: C₁, C₂, C₂², C₅, D₄, C₂³, D₅, C₁₀, C₂×D₄, D₁₀, C₂×C₁₀, D₂₀, C₅×D₄, C₂²×D₅, C₂²×C₁₀, C₅×D₅, C₂×D₂₀, D₄×C₁₀, D₅×C₁₀, C₅×D₂₀, D₅×C₂×C₁₀, C₁₀×D₂₀

Smallest permutation representation of C₁₀×D₂₀
►On 80 points

Generators in S₈₀

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

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

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

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

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

130 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	5A	5B	5C	5D	5E	···	5N	10A	···	10L	10M	···	10AP	10AQ	···	10BF	20A	···	20AV
order	1	2	2	2	2	2	2	2	4	4	5	5	5	5	5	···	5	10	···	10	10	···	10	10	···	10	20	···	20
size	1	1	1	1	10	10	10	10	2	2	1	1	1	1	2	···	2	1	···	1	2	···	2	10	···	10	2	···	2

130 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2
type	+	+	+	+					+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	D₄	D₅	D₁₀	D₁₀	D₂₀	C₅×D₄	C₅×D₅	D₅×C₁₀	D₅×C₁₀	C₅×D₂₀
kernel	C₁₀×D₂₀	C₅×D₂₀	C₁₀×C₂₀	D₅×C₂×C₁₀	C₂×D₂₀	D₂₀	C₂×C₂₀	C₂²×D₅	C₅×C₁₀	C₂×C₂₀	C₂₀	C₂×C₁₀	C₁₀	C₁₀	C₂×C₄	C₄	C₂²	C₂
# reps	1	4	1	2	4	16	4	8	2	2	4	2	8	8	8	16	8	32

Matrix representation of C₁₀×D₂₀ ►in GL₄(𝔽₄₁) generated by

23	0	0	0
0	23	0	0
0	0	16	0
0	0	0	16

25	0	0	0
21	23	0	0
0	0	2	0
0	0	0	21

6	17	0	0
10	35	0	0
0	0	0	21
0	0	2	0

G:=sub<GL(4,GF(41))| [23,0,0,0,0,23,0,0,0,0,16,0,0,0,0,16],[25,21,0,0,0,23,0,0,0,0,2,0,0,0,0,21],[6,10,0,0,17,35,0,0,0,0,0,2,0,0,21,0] >;

C₁₀×D₂₀ in GAP, Magma, Sage, TeX

C_{10}\times D_{20}

% in TeX

G:=Group("C10xD20");

// GroupNames label

G:=SmallGroup(400,183);

// by ID

G=gap.SmallGroup(400,183);

# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,794,194,11525]);

// Polycyclic

G:=Group<a,b,c|a^10=b^20=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;

// generators/relations

G = C10×D20 order 400 = 24·52

Direct product of C10 and D20

G = C₁₀×D₂₀ order 400 = 2⁴·5²

Direct product of C₁₀ and D₂₀