C2xDic3.D6 - GroupNames

G = C₂×Dic₃.D₆ order 288 = 2⁵·3²

Direct product of C₂ and Dic₃.D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₂×Dic₃.D₆, Dic₆⋊₂₁D₆, C₆².₁₃₁C₂³, C₆⋊₁(S₃×Q₈), C₆.₆(S₃×C₂³), (C₃×C₆).₆C₂⁴, (C₂×Dic₆)⋊₁₄S₃, (C₆×Dic₆)⋊₂₀C₂, (C₂×C₁₂).₁₆₈D₆, C₃²⋊₃(C₂²×Q₈), (C₂×Dic₃).₈₈D₆, (C₆×C₁₂).₁₆₁C₂², (C₃×C₁₂).₁₁₅C₂³, C₁₂.₁₀₈(C₂²×S₃), (C₃×Dic₆)⋊₂₆C₂², C₃²⋊₂Q₈⋊₁₀C₂², C₃⋊Dic₃.₃₇C₂³, (C₃×Dic₃).₅C₂³, Dic₃.₄(C₂²×S₃), C₆.D₆.₈C₂², (C₆×Dic₃).₄₆C₂², C₃⋊₁(C₂×S₃×Q₈), C₄.₇₉(C₂×S₃²), C₃⋊S₃⋊₂(C₂×Q₈), (C₃×C₆)⋊₃(C₂×Q₈), (C₂×C₄).₁₂₃S₃², (C₂×C₃⋊S₃)⋊₁₀Q₈, C₂.₉(C₂²×S₃²), C₂².₆₁(C₂×S₃²), (C₂×C₃²⋊₂Q₈)⋊₁₅C₂, (C₄×C₃⋊S₃).₇₆C₂², (C₂×C₃⋊S₃).₄₁C₂³, (C₂×C₆.D₆).₅C₂, (C₂×C₆).₁₄₈(C₂²×S₃), (C₂²×C₃⋊S₃).₁₀₀C₂², (C₂×C₃⋊Dic₃).₁₇₈C₂², (C₂×C₄×C₃⋊S₃).₉C₂, SmallGroup(288,947)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₆ — C₂×Dic₃.D₆

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃×Dic₃ — C₆.D₆ — C₂×C₆.D₆ — C₂×Dic₃.D₆

Lower central

C₃² — C₃×C₆ — C₂×Dic₃.D₆

Upper central

C₁ — C₂² — C₂×C₄

Generators and relations for C₂×Dic₃.D₆
G = < a,b,c,d,e | a²=b⁶=1, c²=d⁶=e²=b³, ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=b^-1, bd=db, be=eb, dcd^-1=ece^-1=b³c, ede^-1=b³d⁵ >

Subgroups: 1106 in 339 conjugacy classes, 124 normal (12 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, C₂×C₄, Q₈, C₂³, C₃², Dic₃, Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, C₂×C₆, C₂²×C₄, C₂×Q₈, C₃⋊S₃, C₃×C₆, C₃×C₆, Dic₆, Dic₆, C₄×S₃, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₃×Q₈, C₂²×S₃, C₂²×Q₈, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₂×C₃⋊S₃, C₆², C₂×Dic₆, C₂×Dic₆, S₃×C₂×C₄, S₃×Q₈, C₆×Q₈, C₆.D₆, C₃²⋊₂Q₈, C₃×Dic₆, C₆×Dic₃, C₄×C₃⋊S₃, C₂×C₃⋊Dic₃, C₆×C₁₂, C₂²×C₃⋊S₃, C₂×S₃×Q₈, Dic₃.D₆, C₂×C₆.D₆, C₂×C₃²⋊₂Q₈, C₆×Dic₆, C₂×C₄×C₃⋊S₃, C₂×Dic₃.D₆
Quotients: C₁, C₂, C₂², S₃, Q₈, C₂³, D₆, C₂×Q₈, C₂⁴, C₂²×S₃, C₂²×Q₈, S₃², S₃×Q₈, S₃×C₂³, C₂×S₃², C₂×S₃×Q₈, Dic₃.D₆, C₂²×S₃², C₂×Dic₃.D₆

Smallest permutation representation of C₂×Dic₃.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 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)

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

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

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+	+	+	+	-	+	+	+	+	-	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	Q₈	D₆	D₆	D₆	S₃²	S₃×Q₈	C₂×S₃²	C₂×S₃²	Dic₃.D₆
kernel	C₂×Dic₃.D₆	Dic₃.D₆	C₂×C₆.D₆	C₂×C₃²⋊₂Q₈	C₆×Dic₆	C₂×C₄×C₃⋊S₃	C₂×Dic₆	C₂×C₃⋊S₃	Dic₆	C₂×Dic₃	C₂×C₁₂	C₂×C₄	C₆	C₄	C₂²	C₂
# reps	1	8	2	2	2	1	2	4	8	4	2	1	4	2	1	4

Matrix representation of C₂×Dic₃.D₆ ►in GL₆(𝔽₁₃)

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

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

12	12	0	0	0	0
0	1	0	0	0	0
0	0	5	11	0	0
0	0	0	8	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

C₂×Dic₃.D₆ in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_3.D_6

% in TeX

G:=Group("C2xDic3.D6");

// GroupNames label

G:=SmallGroup(288,947);

// by ID

G=gap.SmallGroup(288,947);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,675,346,80,1356,9414]);

// Polycyclic

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

// generators/relations

G = C2×Dic3.D6 order 288 = 25·32

Direct product of C2 and Dic3.D6

G = C₂×Dic₃.D₆ order 288 = 2⁵·3²

Direct product of C₂ and Dic₃.D₆