D10:S4 - GroupNames

G = D₁₀:S₄ order 480 = 2⁵·3·5

1^st semidirect product of D₁₀ and S₄ acting via S₄/A₄=C₂

Aliases: D₁₀:₁S₄, (C₅xA₄):₂D₄, (C₂xS₄):₂D₅, (C₁₀xS₄):₂C₂, C₂.₁₄(D₅xS₄), A₄:₂(C₅:D₄), C₅:₂(A₄:D₄), C₁₀.₁₄(C₂xS₄), A₄:Dic₅:₄C₂, (C₂xA₄).₆D₁₀, C₂²:(C₁₅:D₄), (C₂³xD₅):₁S₃, C₂³.₆(S₃xD₅), (C₂²xC₁₀).₆D₆, (C₁₀xA₄).₆C₂², (C₂xD₅xA₄):₁C₂, (C₂xC₁₀):₁(C₃:D₄), SmallGroup(480,980)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₁₀xA₄ — D₁₀:S₄

Chief series

C₁ — C₂² — C₂xC₁₀ — C₅xA₄ — C₁₀xA₄ — C₂xD₅xA₄ — D₁₀:S₄

Lower central

C₅xA₄ — C₁₀xA₄ — D₁₀:S₄

Upper central

C₁ — C₂

Generators and relations for D₁₀:S₄
G = < a,b,c,d,e,f | a¹⁰=b²=c²=d²=e³=f²=1, bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a⁵b, ece^-1=fcf=cd=dc, ede^-1=c, df=fd, fef=e^-1 >

Subgroups: 1012 in 124 conjugacy classes, 19 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₅, S₃, C₆, C₂xC₄, D₄, C₂³, C₂³, D₅, C₁₀, C₁₀, Dic₃, A₄, D₆, C₂xC₆, C₁₅, C₂²:C₄, C₂xD₄, C₂⁴, Dic₅, C₂₀, D₁₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₃:D₄, S₄, C₂xA₄, C₂xA₄, C₅xS₃, C₃xD₅, C₃₀, C₂²wrC₂, C₂xDic₅, C₅:D₄, C₂xC₂₀, C₅xD₄, C₂²xD₅, C₂²xC₁₀, C₂²xC₁₀, A₄:C₄, C₂xS₄, C₂²xA₄, Dic₁₅, C₅xA₄, C₆xD₅, S₃xC₁₀, D₁₀:C₄, C₂³.D₅, C₂xC₅:D₄, D₄xC₁₀, C₂³xD₅, A₄:D₄, C₁₅:D₄, C₅xS₄, D₅xA₄, C₁₀xA₄, C₂³:D₁₀, A₄:Dic₅, C₁₀xS₄, C₂xD₅xA₄, D₁₀:S₄
Quotients: C₁, C₂, C₂², S₃, D₄, D₅, D₆, D₁₀, C₃:D₄, S₄, C₅:D₄, C₂xS₄, S₃xD₅, A₄:D₄, C₁₅:D₄, D₅xS₄, D₁₀:S₄

Smallest permutation representation of D₁₀:S₄
►On 60 points

Generators in S₆₀

(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)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)

(1 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 40)(8 39)(9 38)(10 37)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 50)(20 49)(21 58)(22 57)(23 56)(24 55)(25 54)(26 53)(27 52)(28 51)(29 60)(30 59)

(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)

(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)

(1 13 23)(2 14 24)(3 15 25)(4 16 26)(5 17 27)(6 18 28)(7 19 29)(8 20 30)(9 11 21)(10 12 22)(31 41 51)(32 42 52)(33 43 53)(34 44 54)(35 45 55)(36 46 56)(37 47 57)(38 48 58)(39 49 59)(40 50 60)

(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)

G:=sub<Sym(60)| (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)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,40)(8,39)(9,38)(10,37)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,50)(20,49)(21,58)(22,57)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,60)(30,59), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50), (11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60), (1,13,23)(2,14,24)(3,15,25)(4,16,26)(5,17,27)(6,18,28)(7,19,29)(8,20,30)(9,11,21)(10,12,22)(31,41,51)(32,42,52)(33,43,53)(34,44,54)(35,45,55)(36,46,56)(37,47,57)(38,48,58)(39,49,59)(40,50,60), (11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)>;

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)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,40)(8,39)(9,38)(10,37)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,50)(20,49)(21,58)(22,57)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,60)(30,59), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50), (11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60), (1,13,23)(2,14,24)(3,15,25)(4,16,26)(5,17,27)(6,18,28)(7,19,29)(8,20,30)(9,11,21)(10,12,22)(31,41,51)(32,42,52)(33,43,53)(34,44,54)(35,45,55)(36,46,56)(37,47,57)(38,48,58)(39,49,59)(40,50,60), (11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55) );

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),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,40),(8,39),(9,38),(10,37),(11,48),(12,47),(13,46),(14,45),(15,44),(16,43),(17,42),(18,41),(19,50),(20,49),(21,58),(22,57),(23,56),(24,55),(25,54),(26,53),(27,52),(28,51),(29,60),(30,59)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(31,36),(32,37),(33,38),(34,39),(35,40),(41,46),(42,47),(43,48),(44,49),(45,50)], [(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(41,46),(42,47),(43,48),(44,49),(45,50),(51,56),(52,57),(53,58),(54,59),(55,60)], [(1,13,23),(2,14,24),(3,15,25),(4,16,26),(5,17,27),(6,18,28),(7,19,29),(8,20,30),(9,11,21),(10,12,22),(31,41,51),(32,42,52),(33,43,53),(34,44,54),(35,45,55),(36,46,56),(37,47,57),(38,48,58),(39,49,59),(40,50,60)], [(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,56),(42,57),(43,58),(44,59),(45,60),(46,51),(47,52),(48,53),(49,54),(50,55)]])

34 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	3	4A	4B	4C	5A	5B	6A	6B	6C	10A	10B	10C	10D	10E	10F	10G	10H	10I	10J	15A	15B	20A	20B	20C	20D	30A	30B
order	1	2	2	2	2	2	2	3	4	4	4	5	5	6	6	6	10	10	10	10	10	10	10	10	10	10	15	15	20	20	20	20	30	30
size	1	1	3	3	10	12	30	8	12	60	60	2	2	8	40	40	2	2	6	6	6	6	12	12	12	12	16	16	12	12	12	12	16	16

34 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	2	3	3	4	4	6	6	6
type	+	+	+	+	+	+	+	+	+			+	+	+	-	+	+
image	C₁	C₂	C₂	C₂	S₃	D₄	D₅	D₆	D₁₀	C₃:D₄	C₅:D₄	S₄	C₂xS₄	S₃xD₅	C₁₅:D₄	A₄:D₄	D₅xS₄	D₁₀:S₄
kernel	D₁₀:S₄	A₄:Dic₅	C₁₀xS₄	C₂xD₅xA₄	C₂³xD₅	C₅xA₄	C₂xS₄	C₂²xC₁₀	C₂xA₄	C₂xC₁₀	A₄	D₁₀	C₁₀	C₂³	C₂²	C₅	C₂	C₁
# reps	1	1	1	1	1	1	2	1	2	2	4	2	2	2	2	1	4	4

Matrix representation of D₁₀:S₄ ►in GL₅(F₆₁)

3	0	0	0	0
23	41	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

7	54	0	0	0
33	54	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	60	0	0
0	0	60	0	1
0	0	60	1	0

1	0	0	0	0
0	1	0	0	0
0	0	0	1	60
0	0	1	0	60
0	0	0	0	60

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

60	0	0	0	0
59	1	0	0	0
0	0	0	60	0
0	0	60	0	0
0	0	0	0	60

G:=sub<GL(5,GF(61))| [3,23,0,0,0,0,41,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[7,33,0,0,0,54,54,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,60,60,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,60,60,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[60,59,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,60,0,0,0,0,0,0,60] >;

D₁₀:S₄ in GAP, Magma, Sage, TeX

D_{10}\rtimes S_4

% in TeX

G:=Group("D10:S4");

// GroupNames label

G:=SmallGroup(480,980);

// by ID

G=gap.SmallGroup(480,980);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,85,234,3364,5052,1286,2953,2232]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^10=b^2=c^2=d^2=e^3=f^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a^5*b,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;

// generators/relations

G = D10:S4 order 480 = 25·3·5

1st semidirect product of D10 and S4 acting via S4/A4=C2

G = D₁₀:S₄ order 480 = 2⁵·3·5

1^st semidirect product of D₁₀ and S₄ acting via S₄/A₄=C₂