C5xD4.D5 - GroupNames

G = C₅×D₄.D₅ order 400 = 2⁴·5²

Direct product of C₅ and D₄.D₅

direct product, metabelian, supersoluble, monomial

Aliases: C₅×D₄.D₅, C₅²⋊₇SD₁₆, C₂₀.₃₄D₁₀, Dic₁₀⋊₂C₁₀, D₄.(C₅×D₅), C₅⋊₂C₈⋊₂C₁₀, (C₅×D₄).₆D₅, C₅⋊₂(C₅×SD₁₆), C₄.₂(D₅×C₁₀), C₁₀.₈(C₅×D₄), C₂₀.₂(C₂×C₁₀), (C₅×D₄).₁C₁₀, (C₅×C₁₀).₂₉D₄, (C₅×Dic₁₀)⋊₃C₂, (C₅×C₂₀).₉C₂², (D₄×C₅²).₁C₂, C₁₀.₃₀(C₅⋊D₄), (C₅×C₅⋊₂C₈)⋊₉C₂, C₂.₅(C₅×C₅⋊D₄), SmallGroup(400,88)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — C₅×D₄.D₅

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₅×C₂₀ — C₅×Dic₁₀ — C₅×D₄.D₅

Lower central

C₅ — C₁₀ — C₂₀ — C₅×D₄.D₅

Upper central

C₁ — C₁₀ — C₂₀ — C₅×D₄

Generators and relations for C₅×D₄.D₅
G = < a,b,c,d,e | a⁵=b⁴=c²=d⁵=1, e²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe^-1=b^-1, bd=db, cd=dc, ece^-1=bc, ede^-1=d^-1 >

Subgroups: 140 in 56 conjugacy classes, 22 normal (all characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₅, C₅, C₈, D₄, Q₈, C₁₀, C₁₀, SD₁₆, Dic₅, C₂₀, C₂₀, C₂×C₁₀, C₅², C₅⋊₂C₈, C₄₀, Dic₁₀, C₅×D₄, C₅×D₄, C₅×Q₈, C₅×C₁₀, C₅×C₁₀, D₄.D₅, C₅×SD₁₆, C₅×Dic₅, C₅×C₂₀, C₁₀², C₅×C₅⋊₂C₈, C₅×Dic₁₀, D₄×C₅², C₅×D₄.D₅
Quotients: C₁, C₂, C₂², C₅, D₄, D₅, C₁₀, SD₁₆, D₁₀, C₂×C₁₀, C₅⋊D₄, C₅×D₄, C₅×D₅, D₄.D₅, C₅×SD₁₆, D₅×C₁₀, C₅×C₅⋊D₄, 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 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)

(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 26)(22 27)(23 28)(24 29)(25 30)

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

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

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

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

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

85 conjugacy classes

class	1	2A	2B	4A	4B	5A	5B	5C	5D	5E	···	5N	8A	8B	10A	10B	10C	10D	10E	···	10N	10O	···	10AL	20A	20B	20C	20D	20E	···	20N	20O	20P	20Q	20R	40A	···	40H
order	1	2	2	4	4	5	5	5	5	5	···	5	8	8	10	10	10	10	10	···	10	10	···	10	20	20	20	20	20	···	20	20	20	20	20	40	···	40
size	1	1	4	2	20	1	1	1	1	2	···	2	10	10	1	1	1	1	2	···	2	4	···	4	2	2	2	2	4	···	4	20	20	20	20	10	···	10

85 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+					+	+		+							-
image	C₁	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	D₄	D₅	SD₁₆	D₁₀	C₅⋊D₄	C₅×D₄	C₅×D₅	C₅×SD₁₆	D₅×C₁₀	C₅×C₅⋊D₄	D₄.D₅	C₅×D₄.D₅
kernel	C₅×D₄.D₅	C₅×C₅⋊₂C₈	C₅×Dic₁₀	D₄×C₅²	D₄.D₅	C₅⋊₂C₈	Dic₁₀	C₅×D₄	C₅×C₁₀	C₅×D₄	C₅²	C₂₀	C₁₀	C₁₀	D₄	C₅	C₄	C₂	C₅	C₁
# reps	1	1	1	1	4	4	4	4	1	2	2	2	4	4	8	8	8	16	2	8

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

37	0	0	0
0	37	0	0
0	0	1	0
0	0	0	1

40	0	0	0
0	40	0	0
0	0	40	25
0	0	36	1

40	0	0	0
0	1	0	0
0	0	1	0
0	0	5	40

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

0	10	0	0
37	0	0	0
0	0	0	35
0	0	7	0

G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,36,0,0,25,1],[40,0,0,0,0,1,0,0,0,0,1,5,0,0,0,40],[37,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[0,37,0,0,10,0,0,0,0,0,0,7,0,0,35,0] >;

C₅×D₄.D₅ in GAP, Magma, Sage, TeX

C_5\times D_4.D_5

% in TeX

G:=Group("C5xD4.D5");

// GroupNames label

G:=SmallGroup(400,88);

// by ID

G=gap.SmallGroup(400,88);

# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,240,265,1443,729,69,11525]);

// Polycyclic

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

// generators/relations

G = C5×D4.D5 order 400 = 24·52

Direct product of C5 and D4.D5

G = C₅×D₄.D₅ order 400 = 2⁴·5²

Direct product of C₅ and D₄.D₅