C15xD15 - GroupNames

G = C₁₅xD₁₅ order 450 = 2·3²·5²

Direct product of C₁₅ and D₁₅

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

Aliases: C₁₅xD₁₅, C₁₅ wr C₂, C₁₅:₁C₃₀, C₁₅²:₃C₂, C₅:(S₃xC₁₅), C₃:(D₅xC₁₅), C₁₅:₂(C₅xS₃), (C₅xC₁₅):₅S₃, (C₅xC₁₅):₆C₆, (C₃xC₁₅):₄D₅, C₁₅:₃(C₃xD₅), C₅²:₄(C₃xS₃), C₃²:₁(C₅xD₅), (C₃xC₁₅):₂C₁₀, SmallGroup(450,29)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₅ — C₁₅xD₁₅

Chief series

C₁ — C₅ — C₁₅ — C₅xC₁₅ — C₁₅² — C₁₅xD₁₅

Lower central

C₁₅ — C₁₅xD₁₅

Upper central

C₁ — C₁₅

Generators and relations for C₁₅xD₁₅
G = < a,b,c | a¹⁵=b¹⁵=c²=1, ab=ba, ac=ca, cbc=b^-1 >

Subgroups: 144 in 48 conjugacy classes, 20 normal (all characteristic)
Quotients: C₁, C₂, C₃, C₅, S₃, C₆, D₅, C₁₀, C₁₅, C₃xS₃, C₅xS₃, C₃xD₅, D₁₅, C₃₀, C₅xD₅, S₃xC₁₅, C₃xD₁₅, D₅xC₁₅, C₅xD₁₅, C₁₅xD₁₅

C₁

₁₅C₂

C₃

₂C₃

C₅

₂C₅

₅S₃

₁₅C₆

C₃²

₃D₅

₁₅C₁₀

C₁₅

₂C₁₅

C₅²

₅C₃xS₃

D₁₅

₃C₃xD₅

₅C₅xS₃

₁₅C₃₀

C₃xC₁₅

₂C₃xC₁₅

₃C₅xD₅

C₅xC₁₅

₂C₅xC₁₅

C₃xD₁₅

₅S₃xC₁₅

C₅xD₁₅

₃D₅xC₁₅

C₁₅²

C₁₅xD₁₅

Permutation representations of C₁₅xD₁₅
►On 30 points - transitive group 30T104

Generators in S₃₀

(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)

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

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

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

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

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

G:=TransitiveGroup(30,104);

135 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	5A	5B	5C	5D	5E	···	5N	6A	6B	10A	10B	10C	10D	15A	···	15H	15I	···	15CV	30A	···	30H
order	1	2	3	3	3	3	3	5	5	5	5	5	···	5	6	6	10	10	10	10	15	···	15	15	···	15	30	···	30
size	1	15	1	1	2	2	2	1	1	1	1	2	···	2	15	15	15	15	15	15	1	···	1	2	···	2	15	···	15

135 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2
type	+	+							+	+				+
image	C₁	C₂	C₃	C₅	C₆	C₁₀	C₁₅	C₃₀	S₃	D₅	C₃xS₃	C₅xS₃	C₃xD₅	D₁₅	C₅xD₅	S₃xC₁₅	C₃xD₁₅	D₅xC₁₅	C₅xD₁₅	C₁₅xD₁₅
kernel	C₁₅xD₁₅	C₁₅²	C₅xD₁₅	C₃xD₁₅	C₅xC₁₅	C₃xC₁₅	D₁₅	C₁₅	C₅xC₁₅	C₃xC₁₅	C₅²	C₁₅	C₁₅	C₁₅	C₃²	C₅	C₅	C₃	C₃	C₁
# reps	1	1	2	4	2	4	8	8	1	2	2	4	4	4	8	8	8	16	16	32

Matrix representation of C₁₅xD₁₅ ►in GL₂(F₃₁) generated by

10	0
0	10

14	0
12	20

11	21
12	20

G:=sub<GL(2,GF(31))| [10,0,0,10],[14,12,0,20],[11,12,21,20] >;

C₁₅xD₁₅ in GAP, Magma, Sage, TeX

C_{15}\times D_{15}

% in TeX

G:=Group("C15xD15");

// GroupNames label

G:=SmallGroup(450,29);

// by ID

G=gap.SmallGroup(450,29);

# by ID

G:=PCGroup([5,-2,-3,-5,-3,-5,1203,9004]);

// Polycyclic

G:=Group<a,b,c|a^15=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;

// generators/relations

Export

Subgroup lattice of C₁₅xD₁₅ in TeX

G = C15xD15 order 450 = 2·32·52

Direct product of C15 and D15

G = C₁₅xD₁₅ order 450 = 2·3²·5²

Direct product of C₁₅ and D₁₅