C2xC6^2:C4 - GroupNames

G = C₂xC₆²:C₄ order 288 = 2⁵·3²

Direct product of C₂ and C₆²:C₄

direct product, metabelian, soluble, monomial

Aliases: C₂xC₆²:C₄, C₆²:₂(C₂xC₄), (C₂xC₆²):₅C₄, C₂³:₂(C₃²:C₄), (C₂²xC₃:S₃):₈C₄, (C₂xC₃:S₃).₆₉D₄, C₃:S₃.₁₂(C₂xD₄), C₂²:₂(C₂xC₃²:C₄), C₃²:₄(C₂xC₂²:C₄), C₃:S₃:₄(C₂²:C₄), (C₃xC₆):₂(C₂²:C₄), (C₂³xC₃:S₃).₅C₂, (C₂²xC₃²:C₄):₄C₂, (C₂xC₃²:C₄):₃C₂², (C₂xC₃:S₃).₃₉C₂³, (C₃xC₆).₃₆(C₂²xC₄), C₂.₁₃(C₂²xC₃²:C₄), (C₂²xC₃:S₃).₉₈C₂², (C₂xC₃:S₃):₁₈(C₂xC₄), SmallGroup(288,941)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₂xC₆²:C₄

Chief series

C₁ — C₃² — C₃:S₃ — C₂xC₃:S₃ — C₂xC₃²:C₄ — C₂²xC₃²:C₄ — C₂xC₆²:C₄

Lower central

C₃² — C₃xC₆ — C₂xC₆²:C₄

Upper central

C₁ — C₂² — C₂³

Generators and relations for C₂xC₆²:C₄
G = < a,b,c,d | a²=b⁶=c⁶=d⁴=1, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b^-1c, dcd^-1=b⁴c >

Subgroups: 1488 in 266 conjugacy classes, 46 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₂xC₄, C₂³, C₂³, C₃², D₆, C₂xC₆, C₂²:C₄, C₂²xC₄, C₂⁴, C₃:S₃, C₃:S₃, C₃xC₆, C₃xC₆, C₃xC₆, C₂²xS₃, C₂²xC₆, C₂xC₂²:C₄, C₃²:C₄, C₂xC₃:S₃, C₂xC₃:S₃, C₂xC₃:S₃, C₆², C₆², C₆², S₃xC₂³, C₂xC₃²:C₄, C₂xC₃²:C₄, C₂²xC₃:S₃, C₂²xC₃:S₃, C₂²xC₃:S₃, C₂xC₆², C₆²:C₄, C₂²xC₃²:C₄, C₂³xC₃:S₃, C₂xC₆²:C₄
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, C₂²:C₄, C₂²xC₄, C₂xD₄, C₂xC₂²:C₄, C₃²:C₄, C₂xC₃²:C₄, C₆²:C₄, C₂²xC₃²:C₄, C₂xC₆²:C₄

Permutation representations of C₂xC₆²:C₄
►On 24 points - transitive group 24T674

Generators in S₂₄

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

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

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

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

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

G:=TransitiveGroup(24,674);

►On 24 points - transitive group 24T675

Generators in S₂₄

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

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

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

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

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

G:=TransitiveGroup(24,675);

36 conjugacy classes

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

36 irreducible representations

dim	1	1	1	1	1	1	2	4	4	4
type	+	+	+	+			+	+	+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	C₃²:C₄	C₂xC₃²:C₄	C₆²:C₄
kernel	C₂xC₆²:C₄	C₆²:C₄	C₂²xC₃²:C₄	C₂³xC₃:S₃	C₂²xC₃:S₃	C₂xC₆²	C₂xC₃:S₃	C₂³	C₂²	C₂
# reps	1	4	2	1	6	2	4	2	6	8

Matrix representation of C₂xC₆²:C₄ ►in GL₆(F₁₃)

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	0
0	0	0	0	0	12

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

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	0	1
0	0	0	0	12	1

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

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

C₂xC₆²:C₄ in GAP, Magma, Sage, TeX

C_2\times C_6^2\rtimes C_4

% in TeX

G:=Group("C2xC6^2:C4");

// GroupNames label

G:=SmallGroup(288,941);

// by ID

G=gap.SmallGroup(288,941);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,422,9413,362,12550,1203]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^6=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=b^4*c>;

// generators/relations

G = C2xC62:C4 order 288 = 25·32

Direct product of C2 and C62:C4

G = C₂xC₆²:C₄ order 288 = 2⁵·3²

Direct product of C₂ and C₆²:C₄