C15xC3:D4 - GroupNames

G = C₁₅xC₃:D₄ order 360 = 2³·3²·5

Direct product of C₁₅ and C₃:D₄

direct product, metabelian, supersoluble, monomial

Aliases: C₁₅xC₃:D₄, D₆:₂C₃₀, Dic₃:C₃₀, C₆²:₂C₁₀, C₃₀.₇₁D₆, (C₂xC₃₀):₅S₃, (C₂xC₆):₄C₃₀, (C₆xC₃₀):₈C₂, C₃:₂(D₄xC₁₅), C₁₅:₉(C₃xD₄), (C₃xC₁₅):₂₅D₄, (S₃xC₆):₄C₁₀, (S₃xC₁₀):₅C₆, C₃²:₆(C₅xD₄), (C₂xC₃₀):₁₀C₆, C₂.₅(S₃xC₃₀), C₆.₅(C₂xC₃₀), (S₃xC₃₀):₁₀C₂, C₁₀.₁₇(S₃xC₆), C₆.₂₁(S₃xC₁₀), C₃₀.₂₈(C₂xC₆), C₂²:₃(S₃xC₁₅), (C₅xDic₃):₄C₆, (C₃xDic₃):₄C₁₀, (Dic₃xC₁₅):₁₀C₂, (C₃xC₃₀).₅₁C₂², (C₂xC₆):₃(C₅xS₃), (C₂xC₁₀):₅(C₃xS₃), (C₃xC₆).₁₀(C₂xC₁₀), SmallGroup(360,99)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₁₅xC₃:D₄

Chief series

C₁ — C₃ — C₆ — C₃₀ — C₃xC₃₀ — S₃xC₃₀ — C₁₅xC₃:D₄

Lower central

C₃ — C₆ — C₁₅xC₃:D₄

Upper central

C₁ — C₃₀ — C₂xC₃₀

Generators and relations for C₁₅xC₃:D₄
G = < a,b,c,d | a¹⁵=b³=c⁴=d²=1, ab=ba, ac=ca, ad=da, cbc^-1=dbd=b^-1, dcd=c^-1 >

Subgroups: 140 in 74 conjugacy classes, 36 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², C₅, S₃, C₆, C₆, D₄, C₃², C₁₀, C₁₀, Dic₃, C₁₂, D₆, C₂xC₆, C₂xC₆, C₁₅, C₁₅, C₃xS₃, C₃xC₆, C₃xC₆, C₂₀, C₂xC₁₀, C₂xC₁₀, C₃:D₄, C₃xD₄, C₅xS₃, C₃₀, C₃₀, C₃xDic₃, S₃xC₆, C₆², C₅xD₄, C₃xC₁₅, C₅xDic₃, C₆₀, S₃xC₁₀, C₂xC₃₀, C₂xC₃₀, C₃xC₃:D₄, S₃xC₁₅, C₃xC₃₀, C₃xC₃₀, C₅xC₃:D₄, D₄xC₁₅, Dic₃xC₁₅, S₃xC₃₀, C₆xC₃₀, C₁₅xC₃:D₄
Quotients: C₁, C₂, C₃, C₂², C₅, S₃, C₆, D₄, C₁₀, D₆, C₂xC₆, C₁₅, C₃xS₃, C₂xC₁₀, C₃:D₄, C₃xD₄, C₅xS₃, C₃₀, S₃xC₆, C₅xD₄, S₃xC₁₀, C₂xC₃₀, C₃xC₃:D₄, S₃xC₁₅, C₅xC₃:D₄, D₄xC₁₅, S₃xC₃₀, C₁₅xC₃:D₄

Smallest permutation representation of C₁₅xC₃:D₄
►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 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(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 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)

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

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

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

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

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

135 conjugacy classes

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

135 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
type	+	+	+	+													+	+	+
image	C₁	C₂	C₂	C₂	C₃	C₅	C₆	C₆	C₆	C₁₀	C₁₀	C₁₀	C₁₅	C₃₀	C₃₀	C₃₀	S₃	D₄	D₆	C₃xS₃	C₃:D₄	C₃xD₄	C₅xS₃	S₃xC₆	C₅xD₄	S₃xC₁₀	C₃xC₃:D₄	S₃xC₁₅	C₅xC₃:D₄	D₄xC₁₅	S₃xC₃₀	C₁₅xC₃:D₄
kernel	C₁₅xC₃:D₄	Dic₃xC₁₅	S₃xC₃₀	C₆xC₃₀	C₅xC₃:D₄	C₃xC₃:D₄	C₅xDic₃	S₃xC₁₀	C₂xC₃₀	C₃xDic₃	S₃xC₆	C₆²	C₃:D₄	Dic₃	D₆	C₂xC₆	C₂xC₃₀	C₃xC₁₅	C₃₀	C₂xC₁₀	C₁₅	C₁₅	C₂xC₆	C₁₀	C₃²	C₆	C₅	C₂²	C₃	C₃	C₂	C₁
# reps	1	1	1	1	2	4	2	2	2	4	4	4	8	8	8	8	1	1	1	2	2	2	4	2	4	4	4	8	8	8	8	16

Matrix representation of C₁₅xC₃:D₄ ►in GL₂(F₃₁) generated by

28	0
0	28

1	13
26	29

20	4
16	11

30	18
0	1

G:=sub<GL(2,GF(31))| [28,0,0,28],[1,26,13,29],[20,16,4,11],[30,0,18,1] >;

C₁₅xC₃:D₄ in GAP, Magma, Sage, TeX

C_{15}\times C_3\rtimes D_4

% in TeX

G:=Group("C15xC3:D4");

// GroupNames label

G:=SmallGroup(360,99);

// by ID

G=gap.SmallGroup(360,99);

# by ID

G:=PCGroup([6,-2,-2,-3,-5,-2,-3,745,8645]);

// Polycyclic

G:=Group<a,b,c,d|a^15=b^3=c^4=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 = C15xC3:D4 order 360 = 23·32·5

Direct product of C15 and C3:D4

G = C₁₅xC₃:D₄ order 360 = 2³·3²·5

Direct product of C₁₅ and C₃:D₄