C5xD4:2D5 - GroupNames

G = C₅×D₄⋊₂D₅ order 400 = 2⁴·5²

Direct product of C₅ and D₄⋊₂D₅

direct product, metabelian, supersoluble, monomial

Aliases: C₅×D₄⋊₂D₅, C₂₀.₃₇D₁₀, Dic₁₀⋊₃C₁₀, C₁₀².₁₃C₂², (C₅×D₄)⋊₅D₅, D₄⋊₂(C₅×D₅), (C₅×D₄)⋊₃C₁₀, (C₄×D₅)⋊₂C₁₀, (D₅×C₂₀)⋊₆C₂, C₅⋊D₄⋊₂C₁₀, C₄.₅(D₅×C₁₀), C₂₀.₅(C₂×C₁₀), (C₂×C₁₀).₆D₁₀, (D₄×C₅²)⋊₄C₂, (C₂×Dic₅)⋊₃C₁₀, (C₁₀×Dic₅)⋊₉C₂, (C₅×Dic₁₀)⋊₈C₂, D₁₀.₂(C₂×C₁₀), C₅²⋊₁₀(C₄○D₄), C₂².₁(D₅×C₁₀), (C₅×C₁₀).₂₄C₂³, C₁₀.₆(C₂²×C₁₀), (C₅×C₂₀).₂₁C₂², Dic₅.₃(C₂×C₁₀), C₁₀.₄₅(C₂²×D₅), (D₅×C₁₀).₁₅C₂², (C₅×Dic₅).₂₅C₂², C₅⋊₂(C₅×C₄○D₄), C₂.₇(D₅×C₂×C₁₀), (C₂×C₁₀).(C₂×C₁₀), (C₅×C₅⋊D₄)⋊₆C₂, SmallGroup(400,186)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₅×D₄⋊₂D₅

Chief series

C₁ — C₅ — C₁₀ — C₅×C₁₀ — D₅×C₁₀ — D₅×C₂₀ — C₅×D₄⋊₂D₅

Lower central

C₅ — C₁₀ — C₅×D₄⋊₂D₅

Upper central

C₁ — C₁₀ — C₅×D₄

Generators and relations for C₅×D₄⋊₂D₅
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, ece=b²c, ede=d^-1 >

Subgroups: 236 in 96 conjugacy classes, 46 normal (26 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₅, C₂×C₄, D₄, D₄, Q₈, D₅, C₁₀, C₁₀, C₄○D₄, Dic₅, Dic₅, C₂₀, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₅², Dic₁₀, C₄×D₅, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₅×D₄, C₅×D₄, C₅×Q₈, C₅×D₅, C₅×C₁₀, C₅×C₁₀, D₄⋊₂D₅, C₅×C₄○D₄, C₅×Dic₅, C₅×Dic₅, C₅×C₂₀, D₅×C₁₀, C₁₀², C₅×Dic₁₀, D₅×C₂₀, C₁₀×Dic₅, C₅×C₅⋊D₄, D₄×C₅², C₅×D₄⋊₂D₅
Quotients: C₁, C₂, C₂², C₅, C₂³, D₅, C₁₀, C₄○D₄, D₁₀, C₂×C₁₀, C₂²×D₅, C₂²×C₁₀, C₅×D₅, D₄⋊₂D₅, C₅×C₄○D₄, D₅×C₁₀, D₅×C₂×C₁₀, C₅×D₄⋊₂D₅

Smallest permutation representation of C₅×D₄⋊₂D₅
►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 18 8 13)(2 19 9 14)(3 20 10 15)(4 16 6 11)(5 17 7 12)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)

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

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

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

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

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

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

100 conjugacy classes

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	5A	5B	5C	5D	5E	···	5N	10A	10B	10C	10D	10E	···	10V	10W	···	10AP	10AQ	10AR	10AS	10AT	20A	20B	20C	20D	20E	···	20N	20O	···	20V	20W	···	20AD
order	1	2	2	2	2	4	4	4	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	10	2	5	5	10	10	1	1	1	1	2	···	2	1	1	1	1	2	···	2	4	···	4	10	10	10	10	2	2	2	2	4	···	4	5	···	5	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	D₅	C₄○D₄	D₁₀	D₁₀	C₅×D₅	C₅×C₄○D₄	D₅×C₁₀	D₅×C₁₀	D₄⋊₂D₅	C₅×D₄⋊₂D₅
kernel	C₅×D₄⋊₂D₅	C₅×Dic₁₀	D₅×C₂₀	C₁₀×Dic₅	C₅×C₅⋊D₄	D₄×C₅²	D₄⋊₂D₅	Dic₁₀	C₄×D₅	C₂×Dic₅	C₅⋊D₄	C₅×D₄	C₅×D₄	C₅²	C₂₀	C₂×C₁₀	D₄	C₅	C₄	C₂²	C₅	C₁
# reps	1	1	1	2	2	1	4	4	4	8	8	4	2	2	2	4	8	8	8	16	2	8

Matrix representation of C₅×D₄⋊₂D₅ ►in GL₄(𝔽₄₁) generated by

16	0	0	0
0	16	0	0
0	0	37	0
0	0	0	37

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

0	1	0	0
1	0	0	0
0	0	1	0
0	0	0	1

1	0	0	0
0	1	0	0
0	0	18	0
0	0	19	16

0	32	0	0
9	0	0	0
0	0	16	35
0	0	22	25

G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,37,0,0,0,0,37],[0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,18,19,0,0,0,16],[0,9,0,0,32,0,0,0,0,0,16,22,0,0,35,25] >;

C₅×D₄⋊₂D₅ in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes_2D_5

% in TeX

G:=Group("C5xD4:2D5");

// GroupNames label

G:=SmallGroup(400,186);

// by ID

G=gap.SmallGroup(400,186);

# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,247,794,404,11525]);

// 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,e*c*e=b^2*c,e*d*e=d^-1>;

// generators/relations

G = C5×D4⋊2D5 order 400 = 24·52

Direct product of C5 and D4⋊2D5

G = C₅×D₄⋊₂D₅ order 400 = 2⁴·5²

Direct product of C₅ and D₄⋊₂D₅