D12:15D6 - GroupNames

G = D₁₂:₁₅D₆ order 288 = 2⁵·3²

9^th semidirect product of D₁₂ and D₆ acting via D₆/S₃=C₂

Aliases: D₁₂:₁₅D₆, Q₈:₇S₃², (C₄xS₃):₁₀D₆, (C₃xQ₈):₁₂D₆, Q₈:₃S₃:₉S₃, D₆:D₆:₁₃C₂, C₆.₂₆(S₃xC₂³), (C₃xC₆).₂₆C₂⁴, D₁₂:₅S₃:₁₂C₂, (S₃xC₁₂):₁₀C₂², D₆:S₃:₆C₂², (C₃xD₁₂):₁₇C₂², (S₃xC₆).₁₄C₂³, C₁₂.₃₈(C₂²xS₃), (C₃xC₁₂).₃₈C₂³, D₆.₁₄(C₂²xS₃), (S₃xDic₃):₁₅C₂², C₃:Dic₃.₂₆C₂³, (Q₈xC₃²):₁₁C₂², C₃²:₄Q₈:₁₂C₂², Dic₃.₂₇(C₂²xS₃), (C₃xDic₃).₂₇C₂³, C₆.D₆.₁₅C₂², (C₄xS₃²):₈C₂, C₄.₃₈(C₂xS₃²), C₃:₆(S₃xC₄oD₄), (Q₈xC₃:S₃):₈C₂, C₃:S₃:₃(C₄oD₄), C₂.₂₈(C₂²xS₃²), C₃²:₁₀(C₂xC₄oD₄), (C₂xS₃²).₁₄C₂², (C₃xQ₈:₃S₃):₉C₂, (C₄xC₃:S₃).₄₇C₂², (C₂xC₃:S₃).₄₈C₂³, SmallGroup(288,967)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — D₁₂:₁₅D₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — S₃xC₆ — C₂xS₃² — C₄xS₃² — D₁₂:₁₅D₆

Lower central

C₃² — C₃xC₆ — D₁₂:₁₅D₆

Upper central

C₁ — C₂ — Q₈

Generators and relations for D₁₂:₁₅D₆
G = < a,b,c,d | a¹²=b²=c⁶=d²=1, bab=a^-1, cac^-1=a⁵, ad=da, cbc^-1=a¹⁰b, dbd=a⁶b, dcd=c^-1 >

Subgroups: 1250 in 347 conjugacy classes, 110 normal (10 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₂xC₄, D₄, Q₈, Q₈, C₂³, C₃², Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂xC₆, C₂²xC₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₃xS₃, C₃:S₃, C₃xC₆, Dic₆, C₄xS₃, C₄xS₃, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₃xQ₈, C₂²xS₃, C₂xC₄oD₄, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, S₃², S₃xC₆, C₂xC₃:S₃, S₃xC₂xC₄, C₄oD₁₂, S₃xD₄, D₄:₂S₃, S₃xQ₈, Q₈:₃S₃, C₃xC₄oD₄, S₃xDic₃, C₆.D₆, D₆:S₃, S₃xC₁₂, C₃xD₁₂, C₃²:₄Q₈, C₄xC₃:S₃, Q₈xC₃², C₂xS₃², S₃xC₄oD₄, D₁₂:₅S₃, C₄xS₃², D₆:D₆, C₃xQ₈:₃S₃, Q₈xC₃:S₃, D₁₂:₁₅D₆
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₄oD₄, C₂⁴, C₂²xS₃, C₂xC₄oD₄, S₃², S₃xC₂³, C₂xS₃², S₃xC₄oD₄, C₂²xS₃², D₁₂:₁₅D₆

Smallest permutation representation of D₁₂:₁₅D₆
►On 48 points

Generators in S₄₈

(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)

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

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

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

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

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

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

45 conjugacy classes

class	1	2A	2B	···	2G	2H	2I	3A	3B	3C	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	6A	6B	6C	6D	···	6I	12A	···	12F	12G	12H	12I	12J	12K	12L	12M
order	1	2	2	···	2	2	2	3	3	3	4	4	4	4	4	4	4	4	4	4	6	6	6	6	···	6	12	···	12	12	12	12	12	12	12	12
size	1	1	6	···	6	9	9	2	2	4	2	2	2	3	3	3	3	18	18	18	2	2	4	12	···	12	4	···	4	6	6	6	6	8	8	8

45 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

C₄oD₄

S₃²

C₂xS₃²

S₃xC₄oD₄

D₁₂:₁₅D₆

kernel

D₁₂:₁₅D₆

D₁₂:₅S₃

C₄xS₃²

D₆:D₆

C₃xQ₈:₃S₃

Q₈xC₃:S₃

Q₈:₃S₃

C₄xS₃

D₁₂

C₃xQ₈

C₃:S₃

Q₈

C₄

C₃

C₁

# reps

Matrix representation of D₁₂:₁₅D₆ ►in GL₆(F₁₃)

12	1	0	0	0	0
12	0	0	0	0	0
0	0	12	11	0	0
0	0	1	1	0	0
0	0	0	0	12	0
0	0	0	0	0	12

12	0	0	0	0	0
12	1	0	0	0	0
0	0	1	2	0	0
0	0	0	12	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
1	0	0	0	0	0
0	0	8	3	0	0
0	0	5	5	0	0
0	0	0	0	0	1
0	0	0	0	12	12

1	0	0	0	0	0
0	1	0	0	0	0
0	0	5	10	0	0
0	0	8	8	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,11,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,5,0,0,0,0,3,5,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,8,0,0,0,0,10,8,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D₁₂:₁₅D₆ in GAP, Magma, Sage, TeX

D_{12}\rtimes_{15}D_6

% in TeX

G:=Group("D12:15D6");

// GroupNames label

G:=SmallGroup(288,967);

// by ID

G=gap.SmallGroup(288,967);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,346,185,80,1356,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=a^5,a*d=d*a,c*b*c^-1=a^10*b,d*b*d=a^6*b,d*c*d=c^-1>;

// generators/relations

G = D12:15D6 order 288 = 25·32

9th semidirect product of D12 and D6 acting via D6/S3=C2

G = D₁₂:₁₅D₆ order 288 = 2⁵·3²

9^th semidirect product of D₁₂ and D₆ acting via D₆/S₃=C₂