C3xC2^3.7D6 - GroupNames

G = C₃xC₂³.₇D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₇D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃xC₂³.₇D₆, C₆².₃₅D₄, (C₆xC₁₂):₆C₄, (C₂xC₁₂):₁C₁₂, (C₂xC₆²):₂C₄, (C₂²xC₆):₃C₁₂, (C₆xD₄).₂₄S₃, (C₆xD₄).₁₇C₆, (C₂xC₁₂):₁Dic₃, C₂³.₇(S₃xC₆), C₆.D₄:₂C₆, C₃²:₈(C₂³:C₄), C₆².₉₈(C₂xC₄), (C₂²xC₆):₃Dic₃, C₂³:₂(C₃xDic₃), (C₂²xC₆).₂₅D₆, C₂².₃(C₆xDic₃), (C₂xC₆²).₁₀C₂², C₆.₃₃(C₆.D₄), (C₂xC₄):(C₃xDic₃), C₃:₂(C₃xC₂³:C₄), (D₄xC₃xC₆).₁₁C₂, (C₂xC₆).₃(C₃xD₄), (C₂xD₄).₃(C₃xS₃), (C₂xC₆).₃₉(C₂xC₁₂), C₆.₁₅(C₃xC₂²:C₄), C₂².₂(C₃xC₃:D₄), (C₃xC₆.D₄):₄C₂, (C₂xC₆).₄₃(C₃:D₄), (C₂²xC₆).₁₇(C₂xC₆), (C₂xC₆).₂₄(C₂xDic₃), C₂.₅(C₃xC₆.D₄), (C₃xC₆).₆₆(C₂²:C₄), SmallGroup(288,268)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₃xC₂³.₇D₆

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂²xC₆ — C₂xC₆² — C₃xC₆.D₄ — C₃xC₂³.₇D₆

Lower central

C₃ — C₆ — C₂xC₆ — C₃xC₂³.₇D₆

Upper central

C₁ — C₆ — C₂²xC₆ — C₆xD₄

Generators and relations for C₃xC₂³.₇D₆
G = < a,b,c,d,e,f | a³=b²=c²=d²=e⁶=1, f²=cb=bc, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe^-1=fbf^-1=bd=db, fcf^-1=cd=dc, ce=ec, de=ed, df=fd, fef^-1=cde^-1 >

Subgroups: 330 in 131 conjugacy classes, 42 normal (30 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², C₂², C₆, C₆, C₂xC₄, C₂xC₄, D₄, C₂³, C₃², Dic₃, C₁₂, C₂xC₆, C₂xC₆, C₂xC₆, C₂²:C₄, C₂xD₄, C₃xC₆, C₃xC₆, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₂²xC₆, C₂²xC₆, C₂³:C₄, C₃xDic₃, C₃xC₁₂, C₆², C₆², C₆², C₆.D₄, C₃xC₂²:C₄, C₆xD₄, C₆xD₄, C₆xDic₃, C₆xC₁₂, D₄xC₃², C₂xC₆², C₂³.₇D₆, C₃xC₂³:C₄, C₃xC₆.D₄, D₄xC₃xC₆, C₃xC₂³.₇D₆
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₂xC₄, D₄, Dic₃, C₁₂, D₆, C₂xC₆, C₂²:C₄, C₃xS₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₃xD₄, C₂³:C₄, C₃xDic₃, S₃xC₆, C₆.D₄, C₃xC₂²:C₄, C₆xDic₃, C₃xC₃:D₄, C₂³.₇D₆, C₃xC₂³:C₄, C₃xC₆.D₄, C₃xC₂³.₇D₆

Permutation representations of C₃xC₂³.₇D₆
►On 24 points - transitive group 24T586

Generators in S₂₄

(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)

(2 16)(4 18)(6 14)(7 21)(9 23)(11 19)

(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)

(1 15)(2 16)(3 17)(4 18)(5 13)(6 14)(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)

(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)

(1 23)(2 8 16 22)(3 21)(4 12 18 20)(5 19)(6 10 14 24)(7 17)(9 15)(11 13)

G:=sub<Sym(24)| (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (2,16)(4,18)(6,14)(7,21)(9,23)(11,19), (7,21)(8,22)(9,23)(10,24)(11,19)(12,20), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,21)(8,22)(9,23)(10,24)(11,19)(12,20), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,23)(2,8,16,22)(3,21)(4,12,18,20)(5,19)(6,10,14,24)(7,17)(9,15)(11,13)>;

G:=Group( (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (2,16)(4,18)(6,14)(7,21)(9,23)(11,19), (7,21)(8,22)(9,23)(10,24)(11,19)(12,20), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,21)(8,22)(9,23)(10,24)(11,19)(12,20), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,23)(2,8,16,22)(3,21)(4,12,18,20)(5,19)(6,10,14,24)(7,17)(9,15)(11,13) );

G=PermutationGroup([[(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(2,16),(4,18),(6,14),(7,21),(9,23),(11,19)], [(7,21),(8,22),(9,23),(10,24),(11,19),(12,20)], [(1,15),(2,16),(3,17),(4,18),(5,13),(6,14),(7,21),(8,22),(9,23),(10,24),(11,19),(12,20)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,23),(2,8,16,22),(3,21),(4,12,18,20),(5,19),(6,10,14,24),(7,17),(9,15),(11,13)]])

G:=TransitiveGroup(24,586);

►On 24 points - transitive group 24T589

Generators in S₂₄

(1 3 2)(4 5 6)(7 9 8)(10 11 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)

(1 14)(2 18)(3 16)(4 24)(5 22)(6 20)(7 13)(8 17)(9 15)(10 19)(11 23)(12 21)

(1 8)(2 9)(3 7)(13 16)(14 17)(15 18)

(1 8)(2 9)(3 7)(4 12)(5 10)(6 11)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)

(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)

(1 10 17 19)(2 12 15 21)(3 11 13 23)(4 18 24 9)(5 14 22 8)(6 16 20 7)

G:=sub<Sym(24)| (1,3,2)(4,5,6)(7,9,8)(10,11,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,14)(2,18)(3,16)(4,24)(5,22)(6,20)(7,13)(8,17)(9,15)(10,19)(11,23)(12,21), (1,8)(2,9)(3,7)(13,16)(14,17)(15,18), (1,8)(2,9)(3,7)(4,12)(5,10)(6,11)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,10,17,19)(2,12,15,21)(3,11,13,23)(4,18,24,9)(5,14,22,8)(6,16,20,7)>;

G:=Group( (1,3,2)(4,5,6)(7,9,8)(10,11,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,14)(2,18)(3,16)(4,24)(5,22)(6,20)(7,13)(8,17)(9,15)(10,19)(11,23)(12,21), (1,8)(2,9)(3,7)(13,16)(14,17)(15,18), (1,8)(2,9)(3,7)(4,12)(5,10)(6,11)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,10,17,19)(2,12,15,21)(3,11,13,23)(4,18,24,9)(5,14,22,8)(6,16,20,7) );

G=PermutationGroup([[(1,3,2),(4,5,6),(7,9,8),(10,11,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(1,14),(2,18),(3,16),(4,24),(5,22),(6,20),(7,13),(8,17),(9,15),(10,19),(11,23),(12,21)], [(1,8),(2,9),(3,7),(13,16),(14,17),(15,18)], [(1,8),(2,9),(3,7),(4,12),(5,10),(6,11),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,10,17,19),(2,12,15,21),(3,11,13,23),(4,18,24,9),(5,14,22,8),(6,16,20,7)]])

G:=TransitiveGroup(24,589);

63 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	6A	6B	6C	···	6Q	6R	···	6AE	12A	···	12H	12I	···	12P
order	1	2	2	2	2	2	3	3	3	3	3	4	4	4	4	4	6	6	6	···	6	6	···	6	12	···	12	12	···	12
size	1	1	2	2	2	4	1	1	2	2	2	4	12	12	12	12	1	1	2	···	2	4	···	4	4	···	4	12	···	12

63 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+								+	+	-	-	+								+
image	C₁	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₁₂	C₁₂	S₃	D₄	Dic₃	Dic₃	D₆	C₃xS₃	C₃:D₄	C₃xD₄	C₃xDic₃	C₃xDic₃	S₃xC₆	C₃xC₃:D₄	C₂³:C₄	C₂³.₇D₆	C₃xC₂³:C₄	C₃xC₂³.₇D₆
kernel	C₃xC₂³.₇D₆	C₃xC₆.D₄	D₄xC₃xC₆	C₂³.₇D₆	C₆xC₁₂	C₂xC₆²	C₆.D₄	C₆xD₄	C₂xC₁₂	C₂²xC₆	C₆xD₄	C₆²	C₂xC₁₂	C₂²xC₆	C₂²xC₆	C₂xD₄	C₂xC₆	C₂xC₆	C₂xC₄	C₂³	C₂³	C₂²	C₃²	C₃	C₃	C₁
# reps	1	2	1	2	2	2	4	2	4	4	1	2	1	1	1	2	4	4	2	2	2	8	1	2	2	4

Matrix representation of C₃xC₂³.₇D₆ ►in GL₄(F₇) generated by

2	0	0	0
0	2	0	0
0	0	2	0
0	0	0	2

5	1	0	3
5	1	3	6
0	0	1	0
2	5	6	0

0	1	4	5
1	0	3	5
0	0	1	0
0	0	0	6

6	0	0	0
0	6	0	0
0	0	6	0
0	0	0	6

2	4	4	3
3	1	2	5
2	2	2	3
0	0	0	2

0	5	6	3
5	5	0	2
2	5	6	6
2	2	5	3

G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[5,5,0,2,1,1,0,5,0,3,1,6,3,6,0,0],[0,1,0,0,1,0,0,0,4,3,1,0,5,5,0,6],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[2,3,2,0,4,1,2,0,4,2,2,0,3,5,3,2],[0,5,2,2,5,5,5,2,6,0,6,5,3,2,6,3] >;

C₃xC₂³.₇D₆ in GAP, Magma, Sage, TeX

C_3\times C_2^3._7D_6

% in TeX

G:=Group("C3xC2^3.7D6");

// GroupNames label

G:=SmallGroup(288,268);

// by ID

G=gap.SmallGroup(288,268);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,850,2524,9414]);

// Polycyclic

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

// generators/relations

G = C3xC23.7D6 order 288 = 25·32

Direct product of C3 and C23.7D6

G = C₃xC₂³.₇D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₇D₆