D10:(C4:C4) - GroupNames

G = D₁₀:(C₄:C₄) order 320 = 2⁶·5

The semidirect product of D₁₀ and C₄:C₄ acting via C₄:C₄/C₂²=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₀:(C₄:C₄), C₂.₉(D₄xF₅), C₁₀.₆(C₄xD₄), C₂²:C₄:₄F₅, (C₂xF₅).₃D₄, C₂²:₂(C₄:F₅), C₂³.D₅:₉C₄, D₁₀:C₄:₁C₄, C₅:(C₂³.₈Q₈), D₁₀.₂₇(C₂xD₄), D₅.₁C₂²wrC₂, D₁₀.₁₀(C₂xQ₈), (C₂³xF₅).₂C₂, C₂³.₂₈(C₂xF₅), D₁₀.₃Q₈:₆C₂, (C₂²xD₅).₆₂D₄, D₅.₁(C₂²:Q₈), D₁₀.₄₃(C₄oD₄), (C₂²xD₅).₁₀Q₈, (C₂²xF₅).₄C₂², C₂².₇₄(C₂²xF₅), (C₂³xD₅).₈₄C₂², D₅.₁(C₂².D₄), (C₂²xD₅).₂₆₉C₂³, (C₂xC₁₀):(C₄:C₄), (C₂xC₄:F₅):₆C₂, (C₂xC₄):₂(C₂xF₅), (C₂xC₂₀):₈(C₂xC₄), C₁₀.₈(C₂xC₄:C₄), C₂.₁₁(C₂xC₄:F₅), (C₅xC₂²:C₄):₄C₄, (D₅xC₂²:C₄).₅C₂, (C₂xC₂²:F₅).₄C₂, (C₂xDic₅):₁₅(C₂xC₄), (C₂xC₄xD₅).₂₇₅C₂², (C₂²xC₁₀).₂₀(C₂xC₄), (C₂xC₁₀).₃₇(C₂²xC₄), (C₂²xD₅).₄₄(C₂xC₄), SmallGroup(320,1037)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₀ — D₁₀:(C₄:C₄)

Chief series

C₁ — C₅ — D₅ — D₁₀ — C₂²xD₅ — C₂²xF₅ — C₂³xF₅ — D₁₀:(C₄:C₄)

Lower central

C₅ — C₂xC₁₀ — D₁₀:(C₄:C₄)

Upper central

C₁ — C₂² — C₂²:C₄

Generators and relations for D₁₀:(C₄:C₄)
G = < a,b,c,d | a¹⁰=b²=c⁴=d⁴=1, bab=cac^-1=a^-1, dad^-1=a³, cbc^-1=a³b, dbd^-1=a²b, dcd^-1=c^-1 >

Subgroups: 1018 in 234 conjugacy classes, 64 normal (24 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂xC₄, C₂xC₄, C₂³, C₂³, D₅, D₅, D₅, C₁₀, C₁₀, C₁₀, C₂²:C₄, C₂²:C₄, C₄:C₄, C₂²xC₄, C₂⁴, Dic₅, C₂₀, F₅, D₁₀, D₁₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₂xC₁₀, C₂.C₄², C₂xC₂²:C₄, C₂xC₄:C₄, C₂³xC₄, C₄xD₅, C₂xDic₅, C₂xC₂₀, C₂xF₅, C₂xF₅, C₂²xD₅, C₂²xD₅, C₂²xD₅, C₂²xC₁₀, C₂³.₈Q₈, D₁₀:C₄, C₂³.D₅, C₅xC₂²:C₄, C₄:F₅, C₂²:F₅, C₂xC₄xD₅, C₂²xF₅, C₂²xF₅, C₂²xF₅, C₂³xD₅, D₁₀.₃Q₈, D₅xC₂²:C₄, C₂xC₄:F₅, C₂xC₂²:F₅, C₂³xF₅, D₁₀:(C₄:C₄)
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, Q₈, C₂³, C₄:C₄, C₂²xC₄, C₂xD₄, C₂xQ₈, C₄oD₄, F₅, C₂xC₄:C₄, C₄xD₄, C₂²wrC₂, C₂²:Q₈, C₂².D₄, C₂xF₅, C₂³.₈Q₈, C₄:F₅, C₂²xF₅, C₂xC₄:F₅, D₄xF₅, D₁₀:(C₄:C₄)

Smallest permutation representation of D₁₀:(C₄:C₄)
►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 12)(2 11)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(30 40)

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4	4	4	8
type	+	+	+	+	+	+				+	+	-		+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	D₄	Q₈	C₄oD₄	F₅	C₂xF₅	C₂xF₅	C₄:F₅	D₄xF₅
kernel	D₁₀:(C₄:C₄)	D₁₀.₃Q₈	D₅xC₂²:C₄	C₂xC₄:F₅	C₂xC₂²:F₅	C₂³xF₅	D₁₀:C₄	C₂³.D₅	C₅xC₂²:C₄	C₂xF₅	C₂²xD₅	C₂²xD₅	D₁₀	C₂²:C₄	C₂xC₄	C₂³	C₂²	C₂
# reps	1	2	1	2	1	1	4	2	2	4	2	2	4	1	2	1	4	2

Matrix representation of D₁₀:(C₄:C₄) ►in GL₆(F₄₁)

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

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

0	40	0	0	0	0
40	0	0	0	0	0
0	0	27	0	14	34
0	0	27	14	7	0
0	0	0	7	14	27
0	0	34	14	0	27

9	0	0	0	0	0
0	9	0	0	0	0
0	0	0	0	1	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	1	0	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40,0,0,1,0,0,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,1,1,1,1,0,0,0,0,0,40],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,27,27,0,34,0,0,0,14,7,14,0,0,14,7,14,0,0,0,34,0,27,27],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

D₁₀:(C₄:C₄) in GAP, Magma, Sage, TeX

D_{10}\rtimes (C_4\rtimes C_4)

% in TeX

G:=Group("D10:(C4:C4)");

// GroupNames label

G:=SmallGroup(320,1037);

// by ID

G=gap.SmallGroup(320,1037);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,100,6278,1595]);

// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^4=d^4=1,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=a^3,c*b*c^-1=a^3*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;

// generators/relations

G = D10:(C4:C4) order 320 = 26·5

The semidirect product of D10 and C4:C4 acting via C4:C4/C22=C22

G = D₁₀:(C₄:C₄) order 320 = 2⁶·5

The semidirect product of D₁₀ and C₄:C₄ acting via C₄:C₄/C₂²=C₂²