C2xD5xF5 - GroupNames

G = C₂×D₅×F₅ order 400 = 2⁴·5²

Direct product of C₂, D₅ and F₅

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C₂×D₅×F₅, D₁₀.₈D₁₀, D₅²⋊₂C₄, D₅⋊(C₄×D₅), C₁₀⋊(C₄×D₅), C₁₀⋊₄(C₂×F₅), (C₅×F₅)⋊C₂², (D₅×C₁₀)⋊₃C₄, (C₁₀×F₅)⋊₃C₂, D₅².₃C₂², D₅.D₅⋊C₂², (C₅×D₅).C₂³, C₅⋊₄(C₂²×F₅), C₅²⋊₁(C₂²×C₄), D₅.₁(C₂²×D₅), (D₅×C₁₀).₈C₂², C₅⋊(C₂×C₄×D₅), (C₂×C₅⋊D₅)⋊₂C₄, C₅⋊D₅⋊₁(C₂×C₄), (C₂×D₅²).₄C₂, (C₅×C₁₀)⋊₁(C₂×C₄), (C₅×D₅)⋊₁(C₂×C₄), (C₂×D₅.D₅)⋊₃C₂, SmallGroup(400,209)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅² — C₂×D₅×F₅

Chief series

C₁ — C₅ — C₅² — C₅×D₅ — C₅×F₅ — D₅×F₅ — C₂×D₅×F₅

Lower central

C₅² — C₂×D₅×F₅

Upper central

C₁ — C₂

Generators and relations for C₂×D₅×F₅
G = < a,b,c,d,e | a²=b⁵=c²=d⁵=e⁴=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d³ >

Subgroups: 760 in 113 conjugacy classes, 42 normal (22 characteristic)
C₁, C₂, C₂, C₄, C₂², C₅, C₅, C₂×C₄, C₂³, D₅, D₅, D₅, C₁₀, C₁₀, C₂²×C₄, Dic₅, C₂₀, F₅, F₅, D₁₀, D₁₀, C₂×C₁₀, C₅², C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂×F₅, C₂×F₅, C₂²×D₅, C₅×D₅, C₅×D₅, C₅⋊D₅, C₅×C₁₀, C₂×C₄×D₅, C₂²×F₅, C₅×F₅, D₅.D₅, D₅², D₅×C₁₀, C₂×C₅⋊D₅, D₅×F₅, C₁₀×F₅, C₂×D₅.D₅, C₂×D₅², C₂×D₅×F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, D₅, C₂²×C₄, F₅, D₁₀, C₄×D₅, C₂×F₅, C₂²×D₅, C₂×C₄×D₅, C₂²×F₅, D₅×F₅, C₂×D₅×F₅

Smallest permutation representation of C₂×D₅×F₅
►On 40 points

Generators in S₄₀

(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)

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

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

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

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

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

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

40 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	5A	5B	5C	5D	5E	10A	10B	10C	10D	10E	10F	10G	10H	10I	10J	10K	20A	···	20H
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	5	5	5	5	5	10	10	10	10	10	10	10	10	10	10	10	20	···	20
size	1	1	5	5	5	5	25	25	5	5	5	5	25	25	25	25	2	2	4	8	8	2	2	4	8	8	10	10	10	10	20	20	10	···	10

40 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	4	4	4	8	8
type	+	+	+	+	+				+	+	+			+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₅	D₁₀	D₁₀	C₄×D₅	C₄×D₅	F₅	C₂×F₅	C₂×F₅	D₅×F₅	C₂×D₅×F₅
kernel	C₂×D₅×F₅	D₅×F₅	C₁₀×F₅	C₂×D₅.D₅	C₂×D₅²	D₅²	D₅×C₁₀	C₂×C₅⋊D₅	C₂×F₅	F₅	D₁₀	D₅	C₁₀	D₁₀	D₅	C₁₀	C₂	C₁
# reps	1	4	1	1	1	4	2	2	2	4	2	4	4	1	2	1	2	2

Matrix representation of C₂×D₅×F₅ ►in GL₆(𝔽₄₁)

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

34	40	0	0	0	0
1	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

7	1	0	0	0	0
34	34	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

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

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[34,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,34,0,0,0,0,1,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,1,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0] >;

C₂×D₅×F₅ in GAP, Magma, Sage, TeX

C_2\times D_5\times F_5

% in TeX

G:=Group("C2xD5xF5");

// GroupNames label

G:=SmallGroup(400,209);

// by ID

G=gap.SmallGroup(400,209);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,970,5765,1463]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^5=e^4=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^-1=d^3>;

// generators/relations

G = C2×D5×F5 order 400 = 24·52

Direct product of C2, D5 and F5

G = C₂×D₅×F₅ order 400 = 2⁴·5²

Direct product of C₂, D₅ and F₅