C5xS3xD4 - GroupNames

G = C₅xS₃xD₄ order 240 = 2⁴·3·5

Direct product of C₅, S₃ and D₄

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

Aliases: C₅xS₃xD₄, C₂₀:₆D₆, D₁₂:₃C₁₀, C₆₀:₇C₂², C₃₀.₅₂C₂³, C₁₂:(C₂xC₁₀), C₃:₂(D₄xC₁₀), C₄:₁(S₃xC₁₀), (C₂xC₁₀):₇D₆, C₁₅:₁₅(C₂xD₄), (S₃xC₂₀):₆C₂, (C₄xS₃):₁C₁₀, D₆:₂(C₂xC₁₀), (D₄xC₁₅):₈C₂, (C₃xD₄):₂C₁₀, (C₅xD₁₂):₉C₂, C₃:D₄:₁C₁₀, (C₂xC₃₀):₇C₂², C₂²:₂(S₃xC₁₀), (C₂²xS₃):₂C₁₀, Dic₃:₁(C₂xC₁₀), C₆.₅(C₂²xC₁₀), (S₃xC₁₀):₁₀C₂², C₁₀.₄₂(C₂²xS₃), (C₅xDic₃):₈C₂², (C₂xC₆):(C₂xC₁₀), (S₃xC₂xC₁₀):₆C₂, C₂.₆(S₃xC₂xC₁₀), (C₅xC₃:D₄):₅C₂, SmallGroup(240,169)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₅xS₃xD₄

Chief series

C₁ — C₃ — C₆ — C₃₀ — S₃xC₁₀ — S₃xC₂xC₁₀ — C₅xS₃xD₄

Lower central

C₃ — C₆ — C₅xS₃xD₄

Upper central

C₁ — C₁₀ — C₅xD₄

Generators and relations for C₅xS₃xD₄
G = < a,b,c,d,e | a⁵=b³=c²=d⁴=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 240 in 108 conjugacy classes, 50 normal (26 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₅, S₃, S₃, C₆, C₆, C₂xC₄, D₄, D₄, C₂³, C₁₀, C₁₀, Dic₃, C₁₂, D₆, D₆, D₆, C₂xC₆, C₁₅, C₂xD₄, C₂₀, C₂₀, C₂xC₁₀, C₂xC₁₀, C₄xS₃, D₁₂, C₃:D₄, C₃xD₄, C₂²xS₃, C₅xS₃, C₅xS₃, C₃₀, C₃₀, C₂xC₂₀, C₅xD₄, C₅xD₄, C₂²xC₁₀, S₃xD₄, C₅xDic₃, C₆₀, S₃xC₁₀, S₃xC₁₀, S₃xC₁₀, C₂xC₃₀, D₄xC₁₀, S₃xC₂₀, C₅xD₁₂, C₅xC₃:D₄, D₄xC₁₅, S₃xC₂xC₁₀, C₅xS₃xD₄
Quotients: C₁, C₂, C₂², C₅, S₃, D₄, C₂³, C₁₀, D₆, C₂xD₄, C₂xC₁₀, C₂²xS₃, C₅xS₃, C₅xD₄, C₂²xC₁₀, S₃xD₄, S₃xC₁₀, D₄xC₁₀, S₃xC₂xC₁₀, C₅xS₃xD₄

Smallest permutation representation of C₅xS₃xD₄
►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 39 33)(2 40 34)(3 36 35)(4 37 31)(5 38 32)(6 20 13)(7 16 14)(8 17 15)(9 18 11)(10 19 12)(21 58 51)(22 59 52)(23 60 53)(24 56 54)(25 57 55)(26 41 47)(27 42 48)(28 43 49)(29 44 50)(30 45 46)

(6 13)(7 14)(8 15)(9 11)(10 12)(21 51)(22 52)(23 53)(24 54)(25 55)(31 37)(32 38)(33 39)(34 40)(35 36)(41 47)(42 48)(43 49)(44 50)(45 46)

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

(6 21)(7 22)(8 23)(9 24)(10 25)(11 54)(12 55)(13 51)(14 52)(15 53)(16 59)(17 60)(18 56)(19 57)(20 58)

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

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

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

C₅xS₃xD₄ is a maximal subgroup of D₂₀:₁₀D₆ D₁₂.₉D₁₀ D₂₀:₁₃D₆ D₁₂:₁₄D₁₀

75 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3	4A	4B	5A	5B	5C	5D	6A	6B	6C	10A	10B	10C	10D	10E	···	10L	10M	···	10T	10U	···	10AB	12	15A	15B	15C	15D	20A	20B	20C	20D	20E	20F	20G	20H	30A	30B	30C	30D	30E	···	30L	60A	60B	60C	60D
order	1	2	2	2	2	2	2	2	3	4	4	5	5	5	5	6	6	6	10	10	10	10	10	···	10	10	···	10	10	···	10	12	15	15	15	15	20	20	20	20	20	20	20	20	30	30	30	30	30	···	30	60	60	60	60
size	1	1	2	2	3	3	6	6	2	2	6	1	1	1	1	2	4	4	1	1	1	1	2	···	2	3	···	3	6	···	6	4	2	2	2	2	2	2	2	2	6	6	6	6	2	2	2	2	4	···	4	4	4	4	4

75 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	S₃	D₄	D₆	D₆	C₅xS₃	C₅xD₄	S₃xC₁₀	S₃xC₁₀	S₃xD₄	C₅xS₃xD₄
kernel	C₅xS₃xD₄	S₃xC₂₀	C₅xD₁₂	C₅xC₃:D₄	D₄xC₁₅	S₃xC₂xC₁₀	S₃xD₄	C₄xS₃	D₁₂	C₃:D₄	C₃xD₄	C₂²xS₃	C₅xD₄	C₅xS₃	C₂₀	C₂xC₁₀	D₄	S₃	C₄	C₂²	C₅	C₁
# reps	1	1	1	2	1	2	4	4	4	8	4	8	1	2	1	2	4	8	4	8	1	4

Matrix representation of C₅xS₃xD₄ ►in GL₄(F₆₁) generated by

20	0	0	0
0	20	0	0
0	0	34	0
0	0	0	34

0	60	0	0
1	60	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

60	0	0	0
0	60	0	0
0	0	1	2
0	0	60	60

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

G:=sub<GL(4,GF(61))| [20,0,0,0,0,20,0,0,0,0,34,0,0,0,0,34],[0,1,0,0,60,60,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,1,60,0,0,2,60],[1,0,0,0,0,1,0,0,0,0,1,60,0,0,0,60] >;

C₅xS₃xD₄ in GAP, Magma, Sage, TeX

C_5\times S_3\times D_4

% in TeX

G:=Group("C5xS3xD4");

// GroupNames label

G:=SmallGroup(240,169);

// by ID

G=gap.SmallGroup(240,169);

# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-3,404,5765]);

// Polycyclic

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

// generators/relations

G = C5xS3xD4 order 240 = 24·3·5

Direct product of C5, S3 and D4

G = C₅xS₃xD₄ order 240 = 2⁴·3·5

Direct product of C₅, S₃ and D₄