C3xS3xF5 - GroupNames

G = C₃×S₃×F₅ order 360 = 2³·3²·5

Direct product of C₃, S₃ and F₅

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C₃×S₃×F₅, D₁₅⋊C₁₂, C₃⋊F₅⋊C₆, C₅⋊(S₃×C₁₂), (C₃×F₅)⋊C₆, (C₅×S₃)⋊C₁₂, C₁₅⋊(C₂×C₁₂), (S₃×D₅).C₆, C₃⋊₁(C₆×F₅), C₁₅⋊₄(C₄×S₃), (S₃×C₁₅)⋊₂C₄, C₃²⋊₅(C₂×F₅), (C₃×D₁₅)⋊₁C₄, D₅.₁(S₃×C₆), (C₃×D₅).₅D₆, (C₃²×F₅)⋊₁C₂, (C₃²×D₅).₁C₂², (C₃×C₃⋊F₅)⋊₁C₂, (C₃×C₁₅)⋊₃(C₂×C₄), (C₃×D₅).(C₂×C₆), (C₃×S₃×D₅).₂C₂, SmallGroup(360,126)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₅ — C₃×S₃×F₅

Chief series

C₁ — C₅ — C₁₅ — C₃×D₅ — C₃²×D₅ — C₃²×F₅ — C₃×S₃×F₅

Lower central

C₁₅ — C₃×S₃×F₅

Upper central

C₁ — C₃

Generators and relations for C₃×S₃×F₅
G = < a,b,c,d,e | a³=b³=c²=d⁵=e⁴=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d³ >

Subgroups: 292 in 70 conjugacy classes, 28 normal (all characteristic)
C₁, C₂, C₃, C₃, C₄, C₂², C₅, S₃, S₃, C₆, C₂×C₄, C₃², D₅, D₅, C₁₀, Dic₃, C₁₂, D₆, C₂×C₆, C₁₅, C₁₅, C₃×S₃, C₃×S₃, C₃×C₆, F₅, F₅, D₁₀, C₄×S₃, C₂×C₁₂, C₅×S₃, C₃×D₅, C₃×D₅, D₁₅, C₃₀, C₃×Dic₃, C₃×C₁₂, S₃×C₆, C₂×F₅, C₃×C₁₅, C₃×F₅, C₃×F₅, C₃⋊F₅, S₃×D₅, C₆×D₅, S₃×C₁₂, C₃²×D₅, S₃×C₁₅, C₃×D₁₅, S₃×F₅, C₆×F₅, C₃²×F₅, C₃×C₃⋊F₅, C₃×S₃×D₅, C₃×S₃×F₅
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₂×C₄, C₁₂, D₆, C₂×C₆, C₃×S₃, F₅, C₄×S₃, C₂×C₁₂, S₃×C₆, C₂×F₅, C₃×F₅, S₃×C₁₂, S₃×F₅, C₆×F₅, C₃×S₃×F₅

Permutation representations of C₃×S₃×F₅
►On 30 points - transitive group 30T91

Generators in S₃₀

(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)

(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)

(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)

(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)

(2 3 5 4)(7 8 10 9)(12 13 15 14)(17 18 20 19)(22 23 25 24)(27 28 30 29)

G:=sub<Sym(30)| (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30), (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), (2,3,5,4)(7,8,10,9)(12,13,15,14)(17,18,20,19)(22,23,25,24)(27,28,30,29)>;

G:=Group( (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30), (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), (2,3,5,4)(7,8,10,9)(12,13,15,14)(17,18,20,19)(22,23,25,24)(27,28,30,29) );

G=PermutationGroup([[(1,6,11),(2,7,12),(3,8,13),(4,9,14),(5,10,15),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(1,6,11),(2,7,12),(3,8,13),(4,9,14),(5,10,15),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30)], [(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)], [(2,3,5,4),(7,8,10,9),(12,13,15,14),(17,18,20,19),(22,23,25,24),(27,28,30,29)]])

G:=TransitiveGroup(30,91);

45 conjugacy classes

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

45 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4	4	4	8	8
type	+	+	+	+									+	+					+	+			+
image	C₁	C₂	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₆	C₁₂	C₁₂	S₃	D₆	C₃×S₃	C₄×S₃	S₃×C₆	S₃×C₁₂	F₅	C₂×F₅	C₃×F₅	C₆×F₅	S₃×F₅	C₃×S₃×F₅
kernel	C₃×S₃×F₅	C₃²×F₅	C₃×C₃⋊F₅	C₃×S₃×D₅	S₃×F₅	S₃×C₁₅	C₃×D₁₅	C₃×F₅	C₃⋊F₅	S₃×D₅	C₅×S₃	D₁₅	C₃×F₅	C₃×D₅	F₅	C₁₅	D₅	C₅	C₃×S₃	C₃²	S₃	C₃	C₃	C₁
# reps	1	1	1	1	2	2	2	2	2	2	4	4	1	1	2	2	2	4	1	1	2	2	1	2

Matrix representation of C₃×S₃×F₅ ►in GL₆(𝔽₆₁)

47	0	0	0	0	0
0	47	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

47	0	0	0	0	0
19	13	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

60	5	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	0	1

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

1	0	0	0	0	0
0	1	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(61))| [47,0,0,0,0,0,0,47,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[47,19,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,5,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,60,60,60,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

C₃×S₃×F₅ in GAP, Magma, Sage, TeX

C_3\times S_3\times F_5

% in TeX

G:=Group("C3xS3xF5");

// GroupNames label

G:=SmallGroup(360,126);

// by ID

G=gap.SmallGroup(360,126);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,72,730,5189,887]);

// Polycyclic

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

// generators/relations

G = C3×S3×F5 order 360 = 23·32·5

Direct product of C3, S3 and F5

G = C₃×S₃×F₅ order 360 = 2³·3²·5

Direct product of C₃, S₃ and F₅