A4xD15 - GroupNames

G = A₄×D₁₅ order 360 = 2³·3²·5

Direct product of A₄ and D₁₅

direct product, metabelian, soluble, monomial, A-group

Aliases: A₄×D₁₅, C₅⋊(S₃×A₄), C₃⋊(D₅×A₄), (C₅×A₄)⋊₂S₃, C₁₅⋊₁(C₂×A₄), (C₂×C₃₀)⋊₁C₆, (C₃×A₄)⋊₃D₅, (A₄×C₁₅)⋊₃C₂, (C₂²×D₁₅)⋊C₃, C₂²⋊₂(C₃×D₁₅), (C₂×C₆)⋊(C₃×D₅), (C₂×C₁₀)⋊₃(C₃×S₃), SmallGroup(360,144)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₃₀ — A₄×D₁₅

Chief series

C₁ — C₅ — C₁₅ — C₂×C₃₀ — A₄×C₁₅ — A₄×D₁₅

Lower central

C₂×C₃₀ — A₄×D₁₅

Upper central

C₁

Generators and relations for A₄×D₁₅
G = < a,b,c,d,e | a²=b²=c³=d¹⁵=e²=1, cac^-1=ab=ba, ad=da, ae=ea, cbc^-1=a, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

C₁

₃C₂

₁₅C₂

₄₅C₂

C₃

₄C₃

₈C₃

C₅

C₂²

₄₅C₂²

₃C₆

₅S₃

₁₅S₃

₆₀C₆

₄C₃²

₃C₁₀

₃D₅

₉D₅

C₁₅

₄C₁₅

₈C₁₅

₁₅C₂³

A₄

C₂×C₆

₂A₄

₁₅D₆

₂₀C₃×S₃

C₂×C₁₀

₉D₁₀

D₁₅

₃C₃₀

₃D₁₅

₁₂C₃×D₅

₄C₃×C₁₅

₅C₂²×S₃

₁₅C₂×A₄

C₃×A₄

₃C₂²×D₅

C₂×C₃₀

C₅×A₄

₂C₅×A₄

₃D₃₀

₄C₃×D₁₅

₅S₃×A₄

C₂²×D₁₅

₃D₅×A₄

A₄×C₁₅

A₄×D₁₅

Smallest permutation representation of A₄×D₁₅
►On 60 points

Generators in S₆₀

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

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

(16 38 53)(17 39 54)(18 40 55)(19 41 56)(20 42 57)(21 43 58)(22 44 59)(23 45 60)(24 31 46)(25 32 47)(26 33 48)(27 34 49)(28 35 50)(29 36 51)(30 37 52)

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

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

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

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

36 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	5A	5B	6A	6B	6C	10A	10B	15A	15B	15C	15D	15E	···	15P	30A	30B	30C	30D
order	1	2	2	2	3	3	3	3	3	5	5	6	6	6	10	10	15	15	15	15	15	···	15	30	30	30	30
size	1	3	15	45	2	4	4	8	8	2	2	6	60	60	6	6	2	2	2	2	8	···	8	6	6	6	6

36 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	3	3	6	6	6
type	+	+			+	+		+			+	+	+	+	+
image	C₁	C₂	C₃	C₆	S₃	D₅	C₃×S₃	D₁₅	C₃×D₅	C₃×D₁₅	A₄	C₂×A₄	S₃×A₄	D₅×A₄	A₄×D₁₅
kernel	A₄×D₁₅	A₄×C₁₅	C₂²×D₁₅	C₂×C₃₀	C₅×A₄	C₃×A₄	C₂×C₁₀	A₄	C₂×C₆	C₂²	D₁₅	C₁₅	C₅	C₃	C₁
# reps	1	1	2	2	1	2	2	4	4	8	1	1	1	2	4

Matrix representation of A₄×D₁₅ ►in GL₅(𝔽₃₁)

1	0	0	0	0
0	1	0	0	0
0	0	0	1	0
0	0	1	0	0
0	0	30	30	30

1	0	0	0	0
0	1	0	0	0
0	0	0	0	1
0	0	30	30	30
0	0	1	0	0

5	0	0	0	0
0	5	0	0	0
0	0	1	0	0
0	0	30	30	30
0	0	0	1	0

11	9	0	0	0
6	5	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

14	20	0	0	0
29	17	0	0	0
0	0	30	0	0
0	0	0	30	0
0	0	0	0	30

G:=sub<GL(5,GF(31))| [1,0,0,0,0,0,1,0,0,0,0,0,0,1,30,0,0,1,0,30,0,0,0,0,30],[1,0,0,0,0,0,1,0,0,0,0,0,0,30,1,0,0,0,30,0,0,0,1,30,0],[5,0,0,0,0,0,5,0,0,0,0,0,1,30,0,0,0,0,30,1,0,0,0,30,0],[11,6,0,0,0,9,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[14,29,0,0,0,20,17,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,30] >;

A₄×D₁₅ in GAP, Magma, Sage, TeX

A_4\times D_{15}

% in TeX

G:=Group("A4xD15");

// GroupNames label

G:=SmallGroup(360,144);

// by ID

G=gap.SmallGroup(360,144);

# by ID

G:=PCGroup([6,-2,-3,-2,2,-3,-5,170,81,1444,10373]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of A₄×D₁₅ in TeX

G = A4×D15 order 360 = 23·32·5

Direct product of A4 and D15

G = A₄×D₁₅ order 360 = 2³·3²·5

Direct product of A₄ and D₁₅