C5xC3:D12 - GroupNames

G = C₅xC₃:D₁₂ order 360 = 2³·3²·5

Direct product of C₅ and C₃:D₁₂

direct product, metabelian, supersoluble, monomial

Aliases: C₅xC₃:D₁₂, C₁₅:₉D₁₂, C₃₀.₃₈D₆, C₁₀.₁₆S₃², D₆:₂(C₅xS₃), C₃:₂(C₅xD₁₂), (C₃xC₁₅):₁₆D₄, Dic₃:(C₅xS₃), (S₃xC₃₀):₈C₂, (S₃xC₆):₂C₁₀, (S₃xC₁₀):₅S₃, C₃²:₃(C₅xD₄), C₆.₄(S₃xC₁₀), C₁₅:₉(C₃:D₄), (C₅xDic₃):₄S₃, (C₃xDic₃):₁C₁₀, (Dic₃xC₁₅):₅C₂, (C₃xC₃₀).₃₀C₂², C₂.₄(C₅xS₃²), C₃:₁(C₅xC₃:D₄), (C₁₀xC₃:S₃):₆C₂, (C₂xC₃:S₃):₁C₁₀, (C₃xC₆).₄(C₂xC₁₀), SmallGroup(360,75)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₅xC₃:D₁₂

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃xC₃₀ — S₃xC₃₀ — C₅xC₃:D₁₂

Lower central

C₃² — C₃xC₆ — C₅xC₃:D₁₂

Upper central

C₁ — C₁₀

Generators and relations for C₅xC₃:D₁₂
G = < a,b,c,d | a⁵=b³=c¹²=d²=1, ab=ba, ac=ca, ad=da, cbc^-1=dbd=b^-1, dcd=c^-1 >

Subgroups: 252 in 74 conjugacy classes, 28 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₅, S₃, C₆, C₆, D₄, C₃², C₁₀, C₁₀, Dic₃, C₁₂, D₆, D₆, C₂xC₆, C₁₅, C₁₅, C₃xS₃, C₃:S₃, C₃xC₆, C₂₀, C₂xC₁₀, D₁₂, C₃:D₄, C₅xS₃, C₃₀, C₃₀, C₃xDic₃, S₃xC₆, C₂xC₃:S₃, C₅xD₄, C₃xC₁₅, C₅xDic₃, C₆₀, S₃xC₁₀, S₃xC₁₀, C₂xC₃₀, C₃:D₁₂, S₃xC₁₅, C₅xC₃:S₃, C₃xC₃₀, C₅xD₁₂, C₅xC₃:D₄, Dic₃xC₁₅, S₃xC₃₀, C₁₀xC₃:S₃, C₅xC₃:D₁₂
Quotients: C₁, C₂, C₂², C₅, S₃, D₄, C₁₀, D₆, C₂xC₁₀, D₁₂, C₃:D₄, C₅xS₃, S₃², C₅xD₄, S₃xC₁₀, C₃:D₁₂, C₅xD₁₂, C₅xC₃:D₄, C₅xS₃², C₅xC₃:D₁₂

Smallest permutation representation of C₅xC₃:D₁₂
►On 60 points

Generators in S₆₀

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

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

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

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

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

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

75 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	4	5A	5B	5C	5D	6A	6B	6C	6D	6E	10A	10B	10C	10D	10E	10F	10G	10H	10I	10J	10K	10L	12A	12B	15A	···	15H	15I	15J	15K	15L	20A	20B	20C	20D	30A	···	30H	30I	30J	30K	30L	30M	···	30T	60A	···	60H
order	1	2	2	2	3	3	3	4	5	5	5	5	6	6	6	6	6	10	10	10	10	10	10	10	10	10	10	10	10	12	12	15	···	15	15	15	15	15	20	20	20	20	30	···	30	30	30	30	30	30	···	30	60	···	60
size	1	1	6	18	2	2	4	6	1	1	1	1	2	2	4	6	6	1	1	1	1	6	6	6	6	18	18	18	18	6	6	2	···	2	4	4	4	4	6	6	6	6	2	···	2	4	4	4	4	6	···	6	6	···	6

75 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+					+	+	+	+	+								+	+
image	C₁	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	S₃	S₃	D₄	D₆	D₁₂	C₃:D₄	C₅xS₃	C₅xS₃	C₅xD₄	S₃xC₁₀	C₅xD₁₂	C₅xC₃:D₄	S₃²	C₃:D₁₂	C₅xS₃²	C₅xC₃:D₁₂
kernel	C₅xC₃:D₁₂	Dic₃xC₁₅	S₃xC₃₀	C₁₀xC₃:S₃	C₃:D₁₂	C₃xDic₃	S₃xC₆	C₂xC₃:S₃	C₅xDic₃	S₃xC₁₀	C₃xC₁₅	C₃₀	C₁₅	C₁₅	Dic₃	D₆	C₃²	C₆	C₃	C₃	C₁₀	C₅	C₂	C₁
# reps	1	1	1	1	4	4	4	4	1	1	1	2	2	2	4	4	4	8	8	8	1	1	4	4

Matrix representation of C₅xC₃:D₁₂ ►in GL₆(F₆₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	58	0	0	0
0	0	0	58	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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	60
0	0	0	0	1	60

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

60	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	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,58,0,0,0,0,0,0,58,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[60,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₅xC₃:D₁₂ in GAP, Magma, Sage, TeX

C_5\times C_3\rtimes D_{12}

% in TeX

G:=Group("C5xC3:D12");

// GroupNames label

G:=SmallGroup(360,75);

// by ID

G=gap.SmallGroup(360,75);

# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,-3,265,127,1210,8645]);

// Polycyclic

G:=Group<a,b,c,d|a^5=b^3=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;

// generators/relations

G = C5xC3:D12 order 360 = 23·32·5

Direct product of C5 and C3:D12

G = C₅xC₃:D₁₂ order 360 = 2³·3²·5

Direct product of C₅ and C₃:D₁₂