C2xD6.D6 - GroupNames

G = C₂xD₆.D₆ order 288 = 2⁵·3²

Direct product of C₂ and D₆.D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₂xD₆.D₆, C₆².₁₃₂C₂³, (C₄xS₃):₁₆D₆, C₆:₂(C₄oD₁₂), C₆.₇(S₃xC₂³), (C₃xC₆).₇C₂⁴, (C₂xC₁₂).₃₁₀D₆, (S₃xC₁₂):₁₄C₂², (S₃xC₆).₂₁C₂³, D₆.₁₉(C₂²xS₃), (C₂²xS₃).₇₁D₆, C₃:D₁₂:₁₉C₂², D₆:S₃:₁₉C₂², (C₆xC₁₂).₂₅₇C₂², (C₃xC₁₂).₁₅₀C₂³, C₁₂.₁₄₉(C₂²xS₃), (C₂xDic₃).₁₀₆D₆, C₃²:₂Q₈:₁₇C₂², C₃:Dic₃.₃₈C₂³, Dic₃.₁₉(C₂²xS₃), (C₃xDic₃).₂₀C₂³, (C₆xDic₃).₁₄₈C₂², (S₃xC₂xC₄):₁₅S₃, (S₃xC₂xC₁₂):₂C₂, C₄.₉₆(C₂xS₃²), (C₂xC₄).₁₄₃S₃², C₃:₂(C₂xC₄oD₁₂), C₃²:₃(C₂xC₄oD₄), (C₃xC₆):₃(C₄oD₄), C₂².₆₂(C₂xS₃²), C₂.₁₀(C₂²xS₃²), (C₄xC₃:S₃):₁₈C₂², (C₂xC₃:D₁₂):₂₃C₂, (C₂xD₆:S₃):₁₈C₂, (C₂xC₃²:₂Q₈):₁₉C₂, (C₂xC₃:S₃).₄₂C₂³, (S₃xC₂xC₆).₁₀₄C₂², (C₂xC₆).₁₄₉(C₂²xS₃), (C₂²xC₃:S₃).₁₀₁C₂², (C₂xC₃:Dic₃).₁₇₉C₂², (C₂xC₄xC₃:S₃):₂₄C₂, SmallGroup(288,948)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₂xD₆.D₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — S₃xC₆ — D₆:S₃ — C₂xD₆:S₃ — C₂xD₆.D₆

Lower central

C₃² — C₃xC₆ — C₂xD₆.D₆

Upper central

C₁ — C₂xC₄

Generators and relations for C₂xD₆.D₆
G = < a,b,c,d,e | a²=b⁶=c²=1, d⁶=e²=b³, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ece^-1=b³c, ede^-1=d⁵ >

Subgroups: 1234 in 355 conjugacy classes, 116 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₃², Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₂²xC₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, Dic₆, C₄xS₃, C₄xS₃, D₁₂, C₂xDic₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₂xC₄oD₄, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, S₃xC₆, S₃xC₆, C₂xC₃:S₃, C₂xC₃:S₃, C₆², C₂xDic₆, S₃xC₂xC₄, S₃xC₂xC₄, C₂xD₁₂, C₄oD₁₂, C₂xC₃:D₄, C₂²xC₁₂, D₆:S₃, C₃:D₁₂, C₃²:₂Q₈, S₃xC₁₂, C₆xDic₃, C₄xC₃:S₃, C₂xC₃:Dic₃, C₆xC₁₂, S₃xC₂xC₆, C₂²xC₃:S₃, C₂xC₄oD₁₂, D₆.D₆, C₂xD₆:S₃, C₂xC₃:D₁₂, C₂xC₃²:₂Q₈, S₃xC₂xC₁₂, C₂xC₄xC₃:S₃, C₂xD₆.D₆
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₄oD₄, C₂⁴, C₂²xS₃, C₂xC₄oD₄, S₃², C₄oD₁₂, S₃xC₂³, C₂xS₃², C₂xC₄oD₁₂, D₆.D₆, C₂²xS₃², C₂xD₆.D₆

Smallest permutation representation of C₂xD₆.D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

60 conjugacy classes

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

60 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

C₄oD₄

C₄oD₁₂

S₃²

C₂xS₃²

D₆.D₆

kernel

C₂xD₆.D₆

D₆.D₆

C₂xD₆:S₃

C₂xC₃:D₁₂

C₂xC₃²:₂Q₈

S₃xC₂xC₁₂

C₂xC₄xC₃:S₃

S₃xC₂xC₄

C₄xS₃

C₂xDic₃

C₂xC₁₂

C₂²xS₃

C₃xC₆

C₆

C₂xC₄

C₄

C₂²

C₂

# reps

Matrix representation of C₂xD₆.D₆ ►in GL₆(F₁₃)

12	0	0	0	0	0
0	12	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	12	0
0	0	0	0	0	12

1	0	0	0	0	0
0	1	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	12	12
0	0	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	12	0	0
0	0	12	0	0	0
0	0	0	0	1	1
0	0	0	0	0	12

1	12	0	0	0	0
1	0	0	0	0	0
0	0	5	0	0	0
0	0	0	5	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	0	12	0	0
0	0	1	0	0	0
0	0	0	0	12	0
0	0	0	0	0	12

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

C₂xD₆.D₆ in GAP, Magma, Sage, TeX

C_2\times D_6.D_6

% in TeX

G:=Group("C2xD6.D6");

// GroupNames label

G:=SmallGroup(288,948);

// by ID

G=gap.SmallGroup(288,948);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,80,1356,9414]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=c^2=1,d^6=e^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=d^5>;

// generators/relations

G = C2xD6.D6 order 288 = 25·32

Direct product of C2 and D6.D6

G = C₂xD₆.D₆ order 288 = 2⁵·3²

Direct product of C₂ and D₆.D₆