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⋊₆D₈, D₂₀⋊₂C₁₀, C₂₀.₃₃D₁₀, D₄⋊(C₅×D₅), C₅⋊₂(C₅×D₈), (C₅×D₄)⋊₄D₅, C₅⋊₂C₈⋊₁C₁₀, (C₅×D₄)⋊₁C₁₀, (C₅×D₂₀)⋊₃C₂, C₁₀.₇(C₅×D₄), C₄.₁(D₅×C₁₀), C₂₀.₁(C₂×C₁₀), (C₅×C₁₀).₂₈D₄, (D₄×C₅²)⋊₁C₂, (C₅×C₂₀).₈C₂², C₁₀.₂₉(C₅⋊D₄), (C₅×C₅⋊₂C₈)⋊₈C₂, C₂.₄(C₅×C₅⋊D₄), SmallGroup(400,87)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — C₅×D₄⋊D₅

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₅×C₂₀ — C₅×D₂₀ — 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=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b^-1, bd=db, cd=dc, ece=bc, ede=d^-1 >

Subgroups: 188 in 60 conjugacy classes, 22 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₅, C₅, C₈, D₄, D₄, D₅, C₁₀, C₁₀, D₈, C₂₀, C₂₀, D₁₀, C₂×C₁₀, C₅², C₅⋊₂C₈, C₄₀, D₂₀, C₅×D₄, C₅×D₄, C₅×D₅, C₅×C₁₀, C₅×C₁₀, D₄⋊D₅, C₅×D₈, C₅×C₂₀, D₅×C₁₀, C₁₀², C₅×C₅⋊₂C₈, C₅×D₂₀, D₄×C₅², C₅×D₄⋊D₅
Quotients: C₁, C₂, C₂², C₅, D₄, D₅, C₁₀, D₈, D₁₀, C₂×C₁₀, C₅⋊D₄, C₅×D₄, C₅×D₅, D₄⋊D₅, C₅×D₈, 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 17 7 12)(2 18 8 13)(3 19 9 14)(4 20 10 15)(5 16 6 11)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)

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

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

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

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

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

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

85 conjugacy classes

class	1	2A	2B	2C	4	5A	5B	5C	5D	5E	···	5N	8A	8B	10A	10B	10C	10D	10E	···	10N	10O	···	10AL	10AM	10AN	10AO	10AP	20A	20B	20C	20D	20E	···	20N	40A	···	40H
order	1	2	2	2	4	5	5	5	5	5	···	5	8	8	10	10	10	10	10	···	10	10	···	10	10	10	10	10	20	20	20	20	20	···	20	40	···	40
size	1	1	4	20	2	1	1	1	1	2	···	2	10	10	1	1	1	1	2	···	2	4	···	4	20	20	20	20	2	2	2	2	4	···	4	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	D₈	D₁₀	C₅⋊D₄	C₅×D₄	C₅×D₅	C₅×D₈	D₅×C₁₀	C₅×C₅⋊D₄	D₄⋊D₅	C₅×D₄⋊D₅
kernel	C₅×D₄⋊D₅	C₅×C₅⋊₂C₈	C₅×D₂₀	D₄×C₅²	D₄⋊D₅	C₅⋊₂C₈	D₂₀	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	1	2
0	0	40	40

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

16	0	0	0
0	18	0	0
0	0	1	0
0	0	0	1

0	18	0	0
16	0	0	0
0	0	24	24
0	0	29	17

G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,40,0,0,2,40],[40,0,0,0,0,1,0,0,0,0,40,1,0,0,0,1],[16,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,18,0,0,0,0,0,24,29,0,0,24,17] >;

C₅×D₄⋊D₅ in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes D_5

% in TeX

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

// GroupNames label

G:=SmallGroup(400,87);

// by ID

G=gap.SmallGroup(400,87);

# by ID

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

// 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=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=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