C6xC3:F5 - GroupNames

G = C₆xC₃:F₅ order 360 = 2³·3²·5

Direct product of C₆ and C₃:F₅

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

Aliases: C₆xC₃:F₅, C₃₀:₁C₁₂, C₃₀:₂Dic₃, C₆:(C₃xF₅), C₅:(C₆xDic₃), (C₃xC₆):₂F₅, C₃:₂(C₆xF₅), (C₃xC₃₀):₃C₄, D₅:(C₃xDic₃), D₁₀.(C₃xS₃), C₁₀:(C₃xDic₃), C₁₅:₂(C₂xC₁₂), C₃²:₇(C₂xF₅), (C₃xD₅):₂C₁₂, (C₃xD₅).₉D₆, D₅.₂(S₃xC₆), (C₆xD₅).₈S₃, (C₆xD₅).₄C₆, (C₃xD₅):₃Dic₃, C₁₅:₃(C₂xDic₃), (C₃²xD₅):₆C₄, (C₃²xD₅).₆C₂², (C₃xC₁₅):₉(C₂xC₄), (D₅xC₃xC₆).₄C₂, (C₃xD₅).₂(C₂xC₆), SmallGroup(360,146)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₅ — C₆xC₃:F₅

Chief series

C₁ — C₅ — C₁₅ — C₃xD₅ — C₃²xD₅ — C₃xC₃:F₅ — C₆xC₃:F₅

Lower central

C₁₅ — C₆xC₃:F₅

Upper central

C₁ — C₆

Generators and relations for C₆xC₃:F₅
G = < a,b,c,d | a⁶=b³=c⁵=d⁴=1, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b^-1, dcd^-1=c³ >

Subgroups: 276 in 74 conjugacy classes, 34 normal (30 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₅, C₆, C₆, C₂xC₄, C₃², D₅, C₁₀, Dic₃, C₁₂, C₂xC₆, C₁₅, C₁₅, C₃xC₆, C₃xC₆, F₅, D₁₀, C₂xDic₃, C₂xC₁₂, C₃xD₅, C₃xD₅, C₃₀, C₃₀, C₃xDic₃, C₆², C₂xF₅, C₃xC₁₅, C₃xF₅, C₃:F₅, C₆xD₅, C₆xD₅, C₆xDic₃, C₃²xD₅, C₃xC₃₀, C₆xF₅, C₂xC₃:F₅, C₃xC₃:F₅, D₅xC₃xC₆, C₆xC₃:F₅
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₂xC₄, Dic₃, C₁₂, D₆, C₂xC₆, C₃xS₃, F₅, C₂xDic₃, C₂xC₁₂, C₃xDic₃, S₃xC₆, C₂xF₅, C₃xF₅, C₃:F₅, C₆xDic₃, C₆xF₅, C₂xC₃:F₅, C₃xC₃:F₅, C₆xC₃:F₅

Smallest permutation representation of C₆xC₃:F₅
►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 3 5)(2 4 6)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 59 57)(56 60 58)

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

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

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

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

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

54 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	6J	···	6O	10	12A	···	12H	15A	···	15H	30A	···	30H
order	1	2	2	2	3	3	3	3	3	4	4	4	4	5	6	6	6	6	6	6	6	6	6	6	···	6	10	12	···	12	15	···	15	30	···	30
size	1	1	5	5	1	1	2	2	2	15	15	15	15	4	1	1	2	2	2	5	5	5	5	10	···	10	4	15	···	15	4	···	4	4	···	4

54 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4
type	+	+	+								+	-	+	-					+	+
image	C₁	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₁₂	C₁₂	S₃	Dic₃	D₆	Dic₃	C₃xS₃	C₃xDic₃	S₃xC₆	C₃xDic₃	F₅	C₂xF₅	C₃xF₅	C₃:F₅	C₆xF₅	C₂xC₃:F₅	C₃xC₃:F₅	C₆xC₃:F₅
kernel	C₆xC₃:F₅	C₃xC₃:F₅	D₅xC₃xC₆	C₂xC₃:F₅	C₃²xD₅	C₃xC₃₀	C₃:F₅	C₆xD₅	C₃xD₅	C₃₀	C₆xD₅	C₃xD₅	C₃xD₅	C₃₀	D₁₀	D₅	D₅	C₁₀	C₃xC₆	C₃²	C₆	C₆	C₃	C₃	C₂	C₁
# reps	1	2	1	2	2	2	4	2	4	4	1	1	1	1	2	2	2	2	1	1	2	2	2	2	4	4

Matrix representation of C₆xC₃:F₅ ►in GL₄(F₆₁) generated by

48	0	0	0
0	48	0	0
0	0	48	0
0	0	0	48

47	0	0	0
0	47	0	0
0	0	13	0
0	0	0	13

0	1	0	0
60	17	0	0
0	0	44	44
0	0	17	60

0	0	1	0
0	0	17	60
60	0	0	0
0	60	0	0

G:=sub<GL(4,GF(61))| [48,0,0,0,0,48,0,0,0,0,48,0,0,0,0,48],[47,0,0,0,0,47,0,0,0,0,13,0,0,0,0,13],[0,60,0,0,1,17,0,0,0,0,44,17,0,0,44,60],[0,0,60,0,0,0,0,60,1,17,0,0,0,60,0,0] >;

C₆xC₃:F₅ in GAP, Magma, Sage, TeX

C_6\times C_3\rtimes F_5

% in TeX

G:=Group("C6xC3:F5");

// GroupNames label

G:=SmallGroup(360,146);

// by ID

G=gap.SmallGroup(360,146);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,72,1444,7781,887]);

// Polycyclic

G:=Group<a,b,c,d|a^6=b^3=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;

// generators/relations

G = C6xC3:F5 order 360 = 23·32·5

Direct product of C6 and C3:F5

G = C₆xC₃:F₅ order 360 = 2³·3²·5

Direct product of C₆ and C₃:F₅