C5xD4xD5 - GroupNames

G = C₅×D₄×D₅ order 400 = 2⁴·5²

Direct product of C₅, D₄ and D₅

direct product, metabelian, supersoluble, monomial

Aliases: C₅×D₄×D₅, C₂₀⋊₅D₁₀, D₂₀⋊₃C₁₀, C₁₀²⋊₃C₂², C₂₀⋊(C₂×C₁₀), C₄⋊₁(D₅×C₁₀), C₅⋊₂(D₄×C₁₀), (C₅×D₄)⋊₂C₁₀, (D₅×C₂₀)⋊₅C₂, (C₄×D₅)⋊₁C₁₀, (C₅×D₂₀)⋊₈C₂, (C₂×C₁₀)⋊₄D₁₀, C₅⋊D₄⋊₁C₁₀, C₅²⋊₁₀(C₂×D₄), D₁₀⋊₂(C₂×C₁₀), (C₅×C₂₀)⋊₃C₂², (D₄×C₅²)⋊₃C₂, C₂²⋊₁(D₅×C₁₀), Dic₅⋊₁(C₂×C₁₀), (C₂²×D₅)⋊₂C₁₀, (D₅×C₁₀)⋊₈C₂², (C₅×C₁₀).₂₃C₂³, C₁₀.₅(C₂²×C₁₀), (C₅×Dic₅)⋊₈C₂², C₁₀.₄₄(C₂²×D₅), (C₂×C₁₀)⋊(C₂×C₁₀), (D₅×C₂×C₁₀)⋊₅C₂, C₂.₆(D₅×C₂×C₁₀), (C₅×C₅⋊D₄)⋊₅C₂, SmallGroup(400,185)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₅×D₄×D₅

Chief series

C₁ — C₅ — C₁₀ — C₅×C₁₀ — D₅×C₁₀ — D₅×C₂×C₁₀ — C₅×D₄×D₅

Lower central

C₅ — C₁₀ — C₅×D₄×D₅

Upper central

C₁ — C₁₀ — C₅×D₄

Generators and relations for C₅×D₄×D₅
G = < a,b,c,d,e | a⁵=b⁴=c²=d⁵=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 380 in 124 conjugacy classes, 50 normal (26 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₅, C₂×C₄, D₄, D₄, C₂³, D₅, D₅, C₁₀, C₁₀, C₂×D₄, Dic₅, C₂₀, C₂₀, D₁₀, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₅², C₄×D₅, D₂₀, C₅⋊D₄, C₂×C₂₀, C₅×D₄, C₅×D₄, C₂²×D₅, C₂²×C₁₀, C₅×D₅, C₅×D₅, C₅×C₁₀, C₅×C₁₀, D₄×D₅, D₄×C₁₀, C₅×Dic₅, C₅×C₂₀, D₅×C₁₀, D₅×C₁₀, D₅×C₁₀, C₁₀², D₅×C₂₀, C₅×D₂₀, C₅×C₅⋊D₄, D₄×C₅², D₅×C₂×C₁₀, C₅×D₄×D₅
Quotients: C₁, C₂, C₂², C₅, D₄, C₂³, D₅, C₁₀, C₂×D₄, D₁₀, C₂×C₁₀, C₅×D₄, C₂²×D₅, C₂²×C₁₀, C₅×D₅, D₄×D₅, D₄×C₁₀, D₅×C₁₀, D₅×C₂×C₁₀, C₅×D₄×D₅

Smallest permutation representation of C₅×D₄×D₅
►On 40 points

Generators in S₄₀

(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)

(1 11 6 16)(2 12 7 17)(3 13 8 18)(4 14 9 19)(5 15 10 20)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)

(11 16)(12 17)(13 18)(14 19)(15 20)(31 36)(32 37)(33 38)(34 39)(35 40)

(1 5 4 3 2)(6 10 9 8 7)(11 15 14 13 12)(16 20 19 18 17)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)

(1 25)(2 21)(3 22)(4 23)(5 24)(6 30)(7 26)(8 27)(9 28)(10 29)(11 35)(12 31)(13 32)(14 33)(15 34)(16 40)(17 36)(18 37)(19 38)(20 39)

G:=sub<Sym(40)| (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), (1,11,6,16)(2,12,7,17)(3,13,8,18)(4,14,9,19)(5,15,10,20)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (11,16)(12,17)(13,18)(14,19)(15,20)(31,36)(32,37)(33,38)(34,39)(35,40), (1,5,4,3,2)(6,10,9,8,7)(11,15,14,13,12)(16,20,19,18,17)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40), (1,25)(2,21)(3,22)(4,23)(5,24)(6,30)(7,26)(8,27)(9,28)(10,29)(11,35)(12,31)(13,32)(14,33)(15,34)(16,40)(17,36)(18,37)(19,38)(20,39)>;

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), (1,11,6,16)(2,12,7,17)(3,13,8,18)(4,14,9,19)(5,15,10,20)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (11,16)(12,17)(13,18)(14,19)(15,20)(31,36)(32,37)(33,38)(34,39)(35,40), (1,5,4,3,2)(6,10,9,8,7)(11,15,14,13,12)(16,20,19,18,17)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40), (1,25)(2,21)(3,22)(4,23)(5,24)(6,30)(7,26)(8,27)(9,28)(10,29)(11,35)(12,31)(13,32)(14,33)(15,34)(16,40)(17,36)(18,37)(19,38)(20,39) );

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)], [(1,11,6,16),(2,12,7,17),(3,13,8,18),(4,14,9,19),(5,15,10,20),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40)], [(11,16),(12,17),(13,18),(14,19),(15,20),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,5,4,3,2),(6,10,9,8,7),(11,15,14,13,12),(16,20,19,18,17),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40)], [(1,25),(2,21),(3,22),(4,23),(5,24),(6,30),(7,26),(8,27),(9,28),(10,29),(11,35),(12,31),(13,32),(14,33),(15,34),(16,40),(17,36),(18,37),(19,38),(20,39)]])

100 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	5A	5B	5C	5D	5E	···	5N	10A	10B	10C	10D	10E	···	10V	10W	···	10AP	10AQ	···	10AX	10AY	···	10BF	20A	20B	20C	20D	20E	···	20N	20O	20P	20Q	20R
order	1	2	2	2	2	2	2	2	4	4	5	5	5	5	5	···	5	10	10	10	10	10	···	10	10	···	10	10	···	10	10	···	10	20	20	20	20	20	···	20	20	20	20	20
size	1	1	2	2	5	5	10	10	2	10	1	1	1	1	2	···	2	1	1	1	1	2	···	2	4	···	4	5	···	5	10	···	10	2	2	2	2	4	···	4	10	10	10	10

100 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+							+	+	+	+					+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	C₁₀	C₁₀	D₄	D₅	D₁₀	D₁₀	C₅×D₄	C₅×D₅	D₅×C₁₀	D₅×C₁₀	D₄×D₅	C₅×D₄×D₅
kernel	C₅×D₄×D₅	D₅×C₂₀	C₅×D₂₀	C₅×C₅⋊D₄	D₄×C₅²	D₅×C₂×C₁₀	D₄×D₅	C₄×D₅	D₂₀	C₅⋊D₄	C₅×D₄	C₂²×D₅	C₅×D₅	C₅×D₄	C₂₀	C₂×C₁₀	D₅	D₄	C₄	C₂²	C₅	C₁
# reps	1	1	1	2	1	2	4	4	4	8	4	8	2	2	2	4	8	8	8	16	2	8

Matrix representation of C₅×D₄×D₅ ►in GL₄(𝔽₄₁) generated by

18	0	0	0
0	18	0	0
0	0	1	0
0	0	0	1

40	0	0	0
0	40	0	0
0	0	0	40
0	0	1	0

1	0	0	0
0	1	0	0
0	0	1	0
0	0	0	40

10	0	0	0
0	37	0	0
0	0	1	0
0	0	0	1

0	37	0	0
10	0	0	0
0	0	40	0
0	0	0	40

G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,0,1,0,0,40,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,40],[10,0,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[0,10,0,0,37,0,0,0,0,0,40,0,0,0,0,40] >;

C₅×D₄×D₅ in GAP, Magma, Sage, TeX

C_5\times D_4\times D_5

% in TeX

G:=Group("C5xD4xD5");

// GroupNames label

G:=SmallGroup(400,185);

// by ID

G=gap.SmallGroup(400,185);

# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,404,11525]);

// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;

// generators/relations

G = C5×D4×D5 order 400 = 24·52

Direct product of C5, D4 and D5

G = C₅×D₄×D₅ order 400 = 2⁴·5²

Direct product of C₅, D₄ and D₅