Dic15:S3 - GroupNames

G = Dic₁₅⋊S₃ order 360 = 2³·3²·5

3^rd semidirect product of Dic₁₅ and S₃ acting via S₃/C₃=C₂

metabelian, supersoluble, monomial, A-group

Aliases: C₃₀.₁₈D₆, Dic₁₅⋊₃S₃, C₁₀.₉S₃², C₁₅⋊₉(C₄×S₃), C₃⋊S₃⋊₁Dic₅, C₃⋊₂(S₃×Dic₅), C₆.₂₄(S₃×D₅), (C₃×C₆).₉D₁₀, C₅⋊₃(C₆.D₆), C₃²⋊₃(C₂×Dic₅), C₂.₂(D₁₅⋊S₃), (C₃×Dic₁₅)⋊₉C₂, (C₃×C₃₀).₂₃C₂², (C₅×C₃⋊S₃)⋊₅C₄, (C₃×C₁₅)⋊₂₃(C₂×C₄), (C₂×C₃⋊S₃).₁D₅, (C₁₀×C₃⋊S₃).₂C₂, SmallGroup(360,85)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₁₅ — Dic₁₅⋊S₃

Chief series

C₁ — C₅ — C₁₅ — C₃×C₁₅ — C₃×C₃₀ — C₃×Dic₁₅ — Dic₁₅⋊S₃

Lower central

C₃×C₁₅ — Dic₁₅⋊S₃

Upper central

C₁ — C₂

Generators and relations for Dic₁₅⋊S₃
G = < a,b,c,d | a³⁰=c³=d²=1, b²=a¹⁵, bab^-1=a^-1, ac=ca, dad=a¹¹, bc=cb, bd=db, dcd=c^-1 >

Subgroups: 348 in 74 conjugacy classes, 27 normal (11 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₅, S₃, C₆, C₆, C₂×C₄, C₃², C₁₀, C₁₀, Dic₃, C₁₂, D₆, C₁₅, C₁₅, C₃⋊S₃, C₃×C₆, Dic₅, C₂×C₁₀, C₄×S₃, C₅×S₃, C₃₀, C₃₀, C₃×Dic₃, C₂×C₃⋊S₃, C₂×Dic₅, C₃×C₁₅, C₃×Dic₅, Dic₁₅, S₃×C₁₀, C₆.D₆, C₅×C₃⋊S₃, C₃×C₃₀, S₃×Dic₅, C₃×Dic₁₅, C₁₀×C₃⋊S₃, Dic₁₅⋊S₃
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₅, D₆, Dic₅, D₁₀, C₄×S₃, S₃², C₂×Dic₅, S₃×D₅, C₆.D₆, S₃×Dic₅, D₁₅⋊S₃, Dic₁₅⋊S₃

Smallest permutation representation of Dic₁₅⋊S₃
►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 41 16 56)(2 40 17 55)(3 39 18 54)(4 38 19 53)(5 37 20 52)(6 36 21 51)(7 35 22 50)(8 34 23 49)(9 33 24 48)(10 32 25 47)(11 31 26 46)(12 60 27 45)(13 59 28 44)(14 58 29 43)(15 57 30 42)

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

(2 12)(3 23)(5 15)(6 26)(8 18)(9 29)(11 21)(14 24)(17 27)(20 30)(31 51)(33 43)(34 54)(36 46)(37 57)(39 49)(40 60)(42 52)(45 55)(48 58)

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

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

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

42 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	4A	4B	4C	4D	5A	5B	6A	6B	6C	10A	10B	10C	10D	10E	10F	12A	12B	12C	12D	15A	···	15H	30A	···	30H
order	1	2	2	2	3	3	3	4	4	4	4	5	5	6	6	6	10	10	10	10	10	10	12	12	12	12	15	···	15	30	···	30
size	1	1	9	9	2	2	4	15	15	15	15	2	2	2	2	4	2	2	18	18	18	18	30	30	30	30	4	···	4	4	···	4

42 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+		+	+	+	-	+		+	+	+	-
image	C₁	C₂	C₂	C₄	S₃	D₅	D₆	Dic₅	D₁₀	C₄×S₃	S₃²	S₃×D₅	C₆.D₆	S₃×Dic₅	D₁₅⋊S₃	Dic₁₅⋊S₃
kernel	Dic₁₅⋊S₃	C₃×Dic₁₅	C₁₀×C₃⋊S₃	C₅×C₃⋊S₃	Dic₁₅	C₂×C₃⋊S₃	C₃₀	C₃⋊S₃	C₃×C₆	C₁₅	C₁₀	C₆	C₅	C₃	C₂	C₁
# reps	1	2	1	4	2	2	2	4	2	4	1	4	1	4	4	4

Matrix representation of Dic₁₅⋊S₃ ►in GL₆(𝔽₆₁)

0	60	0	0	0	0
1	18	0	0	0	0
0	0	1	1	0	0
0	0	60	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

6	38	0	0	0	0
52	55	0	0	0	0
0	0	11	0	0	0
0	0	50	50	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

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

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,18,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,52,0,0,0,0,38,55,0,0,0,0,0,0,11,50,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,60] >;

Dic₁₅⋊S₃ in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes S_3

% in TeX

G:=Group("Dic15:S3");

// GroupNames label

G:=SmallGroup(360,85);

// by ID

G=gap.SmallGroup(360,85);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,387,201,730,10373]);

// Polycyclic

G:=Group<a,b,c,d|a^30=c^3=d^2=1,b^2=a^15,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^11,b*c=c*b,b*d=d*b,d*c*d=c^-1>;

// generators/relations

G = Dic15⋊S3 order 360 = 23·32·5

3rd semidirect product of Dic15 and S3 acting via S3/C3=C2

G = Dic₁₅⋊S₃ order 360 = 2³·3²·5

3^rd semidirect product of Dic₁₅ and S₃ acting via S₃/C₃=C₂