C3xC15:D4 - GroupNames

G = C₃xC₁₅:D₄ order 360 = 2³·3²·5

Direct product of C₃ and C₁₅:D₄

direct product, metabelian, supersoluble, monomial

Aliases: C₃xC₁₅:D₄, C₃₀.₃₁D₆, Dic₁₅:₄C₆, (C₃xC₁₅):₄D₄, (C₆xD₅):₄S₃, (S₃xC₆):₁D₅, (C₆xD₅):₁C₆, D₆:₁(C₃xD₅), C₁₅:₁(C₃xD₄), C₆.₄(C₆xD₅), (S₃xC₁₀):₁C₆, (S₃xC₃₀):₁C₂, D₁₀:₁(C₃xS₃), C₁₀.₄(S₃xC₆), C₃₀.₄(C₂xC₆), C₆.₃₁(S₃xD₅), (C₃xC₆).₁₆D₁₀, C₃²:₇(C₅:D₄), C₁₅:₁₀(C₃:D₄), (C₃xDic₁₅):₄C₂, (C₃xC₃₀).₄C₂², (D₅xC₃xC₆):₁C₂, C₅:₂(C₃xC₃:D₄), C₂.₄(C₃xS₃xD₅), C₃:₂(C₃xC₅:D₄), SmallGroup(360,61)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₀ — C₃xC₁₅:D₄

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₃xC₃₀ — D₅xC₃xC₆ — C₃xC₁₅:D₄

Lower central

C₁₅ — C₃₀ — C₃xC₁₅:D₄

Upper central

C₁ — C₆

Generators and relations for C₃xC₁₅:D₄
G = < a,b,c,d | a³=b¹⁵=c⁴=d²=1, ab=ba, ac=ca, ad=da, cbc^-1=b^-1, dbd=b⁴, dcd=c^-1 >

Subgroups: 276 in 74 conjugacy classes, 28 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₅, S₃, C₆, C₆, D₄, C₃², D₅, C₁₀, C₁₀, Dic₃, C₁₂, D₆, C₂xC₆, C₁₅, C₁₅, C₃xS₃, C₃xC₆, C₃xC₆, Dic₅, D₁₀, C₂xC₁₀, C₃:D₄, C₃xD₄, C₅xS₃, C₃xD₅, C₃₀, C₃₀, C₃xDic₃, S₃xC₆, C₆², C₅:D₄, C₃xC₁₅, C₃xDic₅, Dic₁₅, C₆xD₅, C₆xD₅, S₃xC₁₀, C₂xC₃₀, C₃xC₃:D₄, C₃²xD₅, S₃xC₁₅, C₃xC₃₀, C₁₅:D₄, C₃xC₅:D₄, C₃xDic₁₅, D₅xC₃xC₆, S₃xC₃₀, C₃xC₁₅:D₄
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, D₅, D₆, C₂xC₆, C₃xS₃, D₁₀, C₃:D₄, C₃xD₄, C₃xD₅, S₃xC₆, C₅:D₄, S₃xD₅, C₆xD₅, C₃xC₃:D₄, C₁₅:D₄, C₃xC₅:D₄, C₃xS₃xD₅, C₃xC₁₅:D₄

Smallest permutation representation of C₃xC₁₅:D₄
►On 60 points

Generators in S₆₀

(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)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(46 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 55)

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

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

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

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

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

63 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	4	5A	5B	6A	6B	6C	6D	6E	6F	6G	6H	···	6O	10A	10B	10C	10D	10E	10F	12A	12B	15A	15B	15C	15D	15E	···	15J	30A	30B	30C	30D	30E	···	30J	30K	···	30R
order	1	2	2	2	3	3	3	3	3	4	5	5	6	6	6	6	6	6	6	6	···	6	10	10	10	10	10	10	12	12	15	15	15	15	15	···	15	30	30	30	30	30	···	30	30	···	30
size	1	1	6	10	1	1	2	2	2	30	2	2	1	1	2	2	2	6	6	10	···	10	2	2	6	6	6	6	30	30	2	2	2	2	4	···	4	2	2	2	2	4	···	4	6	···	6

63 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+					+	+	+	+		+									+	-
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	S₃	D₄	D₅	D₆	C₃xS₃	D₁₀	C₃:D₄	C₃xD₄	C₃xD₅	S₃xC₆	C₅:D₄	C₆xD₅	C₃xC₃:D₄	C₃xC₅:D₄	S₃xD₅	C₁₅:D₄	C₃xS₃xD₅	C₃xC₁₅:D₄
kernel	C₃xC₁₅:D₄	C₃xDic₁₅	D₅xC₃xC₆	S₃xC₃₀	C₁₅:D₄	Dic₁₅	C₆xD₅	S₃xC₁₀	C₆xD₅	C₃xC₁₅	S₃xC₆	C₃₀	D₁₀	C₃xC₆	C₁₅	C₁₅	D₆	C₁₀	C₃²	C₆	C₅	C₃	C₆	C₃	C₂	C₁
# reps	1	1	1	1	2	2	2	2	1	1	2	1	2	2	2	2	4	2	4	4	4	8	2	2	4	4

Matrix representation of C₃xC₁₅:D₄ ►in GL₄(F₆₁) generated by

1	0	0	0
0	1	0	0
0	0	47	0
0	0	0	47

44	44	0	0
17	60	0	0
0	0	13	0
0	0	20	47

14	45	0	0
39	47	0	0
0	0	60	41
0	0	55	1

1	0	0	0
17	60	0	0
0	0	1	0
0	0	6	60

G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,47,0,0,0,0,47],[44,17,0,0,44,60,0,0,0,0,13,20,0,0,0,47],[14,39,0,0,45,47,0,0,0,0,60,55,0,0,41,1],[1,17,0,0,0,60,0,0,0,0,1,6,0,0,0,60] >;

C₃xC₁₅:D₄ in GAP, Magma, Sage, TeX

C_3\times C_{15}\rtimes D_4

% in TeX

G:=Group("C3xC15:D4");

// GroupNames label

G:=SmallGroup(360,61);

// by ID

G=gap.SmallGroup(360,61);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,169,730,10373]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^15=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^4,d*c*d=c^-1>;

// generators/relations

G = C3xC15:D4 order 360 = 23·32·5

Direct product of C3 and C15:D4

G = C₃xC₁₅:D₄ order 360 = 2³·3²·5

Direct product of C₃ and C₁₅:D₄