S3xD4xD5 - GroupNames

G = S₃xD₄xD₅ order 480 = 2⁵·3·5

Direct product of S₃, D₄ and D₅

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: S₃xD₄xD₅, C₆₀:C₂³, D₂₀:₁₁D₆, D₁₂:₁₁D₁₀, D₆₀:₇C₂², D₃₀:₃C₂³, C₃₀.₂₅C₂⁴, Dic₁₅:₁C₂³, (C₅xD₄):₉D₆, (C₄xD₅):₇D₆, C₅:D₄:₁D₆, (C₂xC₃₀):C₂³, (D₄xD₁₅):₄C₂, (C₃xD₄):₉D₁₀, D₁₅:₂(C₂xD₄), (C₄xS₃):₇D₁₀, (S₃xD₂₀):₄C₂, (D₅xD₁₂):₄C₂, C₃:D₄:₁D₁₀, C₁₅:₄(C₂²xD₄), C₂₀:D₆:₄C₂, C₂₀:₁(C₂²xS₃), (C₆xD₅):₃C₂³, D₆:₃(C₂²xD₅), C₁₂:₁(C₂²xD₅), D₁₀:D₆:₂C₂, (S₃xC₁₀):₃C₂³, (S₃xC₂₀):₁C₂², (C₃xD₂₀):₇C₂², (D₅xC₁₂):₁C₂², (C₅xD₁₂):₇C₂², D₁₀:₃(C₂²xS₃), (D₄xC₁₅):₇C₂², (C₄xD₁₅):₁C₂², (C₂²xD₅):₁₁D₆, C₅:D₁₂:₃C₂², C₃:D₂₀:₃C₂², C₁₅:D₄:₃C₂², C₁₅:₇D₄:₁C₂², C₆.₂₅(C₂³xD₅), (C₂²xS₃):₁₀D₁₀, C₁₀.₂₅(S₃xC₂³), D₃₀.C₂:₉C₂², (S₃xDic₅):₉C₂², Dic₃:₁(C₂²xD₅), (C₃xDic₅):₁C₂³, (D₅xDic₃):₉C₂², Dic₅:₁(C₂²xS₃), (C₅xDic₃):₁C₂³, (C₂²xD₁₅):₁₁C₂², C₃:₃(C₂xD₄xD₅), C₅:₃(C₂xS₃xD₄), C₄:₁(C₂xS₃xD₅), (C₃xD₄xD₅):₄C₂, (C₄xS₃xD₅):₁C₂, (C₅xS₃xD₄):₄C₂, C₂²:₂(C₂xS₃xD₅), (C₃xD₅):₂(C₂xD₄), (C₅xS₃):₂(C₂xD₄), (D₅xC₃:D₄):₂C₂, (S₃xC₅:D₄):₂C₂, (D₅xC₂xC₆):₇C₂², (C₂²xS₃xD₅):₈C₂, (S₃xC₂xC₁₀):₇C₂², (C₂xS₃xD₅):₁₂C₂², (C₂xC₆):₁(C₂²xD₅), C₂.₂₈(C₂²xS₃xD₅), (C₂xC₁₀):₄(C₂²xS₃), (C₅xC₃:D₄):₁C₂², (C₃xC₅:D₄):₁C₂², SmallGroup(480,1097)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₀ — S₃xD₄xD₅

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₆xD₅ — C₂xS₃xD₅ — C₂²xS₃xD₅ — S₃xD₄xD₅

Lower central

C₁₅ — C₃₀ — S₃xD₄xD₅

Upper central

C₁ — C₂ — D₄

Generators and relations for S₃xD₄xD₅
G = < a,b,c,d,e,f | a³=b²=c⁴=d²=e⁵=f²=1, bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c^-1, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 2892 in 472 conjugacy classes, 122 normal (50 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₅, S₃, S₃, C₆, C₆, C₂xC₄, D₄, D₄, C₂³, D₅, D₅, C₁₀, C₁₀, Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₆, D₆, C₂xC₆, C₂xC₆, C₁₅, C₂²xC₄, C₂xD₄, C₂⁴, Dic₅, Dic₅, C₂₀, C₂₀, D₁₀, D₁₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₄xS₃, C₄xS₃, D₁₂, D₁₂, C₂xDic₃, C₃:D₄, C₃:D₄, C₂xC₁₂, C₃xD₄, C₃xD₄, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₅xS₃, C₅xS₃, C₃xD₅, C₃xD₅, D₁₅, D₁₅, C₃₀, C₃₀, C₂²xD₄, C₄xD₅, C₄xD₅, D₂₀, D₂₀, C₂xDic₅, C₅:D₄, C₅:D₄, C₂xC₂₀, C₅xD₄, C₅xD₄, C₂²xD₅, C₂²xD₅, C₂²xC₁₀, S₃xC₂xC₄, C₂xD₁₂, S₃xD₄, S₃xD₄, C₂xC₃:D₄, C₆xD₄, S₃xC₂³, C₅xDic₃, C₃xDic₅, Dic₁₅, C₆₀, S₃xD₅, S₃xD₅, C₆xD₅, C₆xD₅, C₆xD₅, S₃xC₁₀, S₃xC₁₀, S₃xC₁₀, D₃₀, D₃₀, D₃₀, C₂xC₃₀, C₂xC₄xD₅, C₂xD₂₀, D₄xD₅, D₄xD₅, C₂xC₅:D₄, D₄xC₁₀, C₂³xD₅, C₂xS₃xD₄, D₅xDic₃, S₃xDic₅, D₃₀.C₂, C₁₅:D₄, C₃:D₂₀, C₅:D₁₂, D₅xC₁₂, C₃xD₂₀, C₃xC₅:D₄, S₃xC₂₀, C₅xD₁₂, C₅xC₃:D₄, C₄xD₁₅, D₆₀, C₁₅:₇D₄, D₄xC₁₅, C₂xS₃xD₅, C₂xS₃xD₅, C₂xS₃xD₅, D₅xC₂xC₆, S₃xC₂xC₁₀, C₂²xD₁₅, C₂xD₄xD₅, C₄xS₃xD₅, D₅xD₁₂, S₃xD₂₀, C₂₀:D₆, D₅xC₃:D₄, S₃xC₅:D₄, D₁₀:D₆, C₃xD₄xD₅, C₅xS₃xD₄, D₄xD₁₅, C₂²xS₃xD₅, S₃xD₄xD₅
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₅, D₆, C₂xD₄, C₂⁴, D₁₀, C₂²xS₃, C₂²xD₄, C₂²xD₅, S₃xD₄, S₃xC₂³, S₃xD₅, D₄xD₅, C₂³xD₅, C₂xS₃xD₄, C₂xS₃xD₅, C₂xD₄xD₅, C₂²xS₃xD₅, S₃xD₄xD₅

Smallest permutation representation of S₃xD₄xD₅
►On 60 points

Generators in S₆₀

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

(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)(36 41)(37 42)(38 43)(39 44)(40 45)(51 56)(52 57)(53 58)(54 59)(55 60)

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

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

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

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

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

60 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	2L	2M	2N	2O	3	4A	4B	4C	4D	5A	5B	6A	6B	6C	6D	6E	6F	6G	10A	10B	10C	10D	10E	10F	10G	10H	10I	10J	10K	10L	10M	10N	12A	12B	15A	15B	20A	20B	20C	20D	30A	30B	30C	30D	30E	30F	60A	60B
order	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	3	4	4	4	4	5	5	6	6	6	6	6	6	6	10	10	10	10	10	10	10	10	10	10	10	10	10	10	12	12	15	15	20	20	20	20	30	30	30	30	30	30	60	60
size	1	1	2	2	3	3	5	5	6	6	10	10	15	15	30	30	2	2	6	10	30	2	2	2	4	4	10	10	20	20	2	2	4	4	4	4	6	6	6	6	12	12	12	12	4	20	4	4	4	4	12	12	4	4	8	8	8	8	8	8

60 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₅

D₆

D₁₀

S₃xD₄

S₃xD₅

D₄xD₅

C₂xS₃xD₅

S₃xD₄xD₅

kernel

S₃xD₄xD₅

C₄xS₃xD₅

D₅xD₁₂

S₃xD₂₀

C₂₀:D₆

D₅xC₃:D₄

S₃xC₅:D₄

D₁₀:D₆

C₃xD₄xD₅

C₅xS₃xD₄

D₄xD₁₅

C₂²xS₃xD₅

D₄xD₅

S₃xD₅

S₃xD₄

C₄xD₅

D₂₀

C₅:D₄

C₅xD₄

C₂²xD₅

C₄xS₃

D₁₂

C₃:D₄

C₃xD₄

C₂²xS₃

D₅

D₄

S₃

C₄

C₂²

C₁

# reps

Matrix representation of S₃xD₄xD₅ ►in GL₆(F₆₁)

0	60	0	0	0	0
1	60	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

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

60	0	0	0	0	0
0	60	0	0	0	0
0	0	60	0	0	0
0	0	0	60	0	0
0	0	0	0	1	41
0	0	0	0	55	60

1	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	0
0	0	0	0	55	60

1	0	0	0	0	0
0	1	0	0	0	0
0	0	44	1	0	0
0	0	16	60	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	60	60	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,55,0,0,0,0,41,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,55,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,44,16,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S₃xD₄xD₅ in GAP, Magma, Sage, TeX

S_3\times D_4\times D_5

% in TeX

G:=Group("S3xD4xD5");

// GroupNames label

G:=SmallGroup(480,1097);

// by ID

G=gap.SmallGroup(480,1097);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,185,1356,18822]);

// Polycyclic

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

// generators/relations

G = S3xD4xD5 order 480 = 25·3·5

Direct product of S3, D4 and D5

G = S₃xD₄xD₅ order 480 = 2⁵·3·5

Direct product of S₃, D₄ and D₅