C2xC4^2:C6

direct product, metabelian, soluble, monomial

Aliases: C₂×C₄²⋊C₆, C₂⁴.₄A₄, C₄²⋊₂(C₂×C₆), (C₂×C₄²)⋊₁C₆, C₄²⋊C₃⋊₃C₂², C₂³.₃(C₂×A₄), C₄²⋊₂C₂⋊₂C₆, C₂².₃(C₂²×A₄), (C₂×C₄²⋊₂C₂)⋊C₃, (C₂×C₄²⋊C₃)⋊₁C₂, SmallGroup(192,1001)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄² — C₂×C₄²⋊C₆

Chief series

C₁ — C₂² — C₄² — C₄²⋊C₃ — C₄²⋊C₆ — C₂×C₄²⋊C₆

Lower central

C₄² — C₂×C₄²⋊C₆

Upper central

C₁ — C₂

Subgroups: 282 in 66 conjugacy classes, 17 normal (11 characteristic)
C₁, C₂, C₂^[×4], C₃, C₄^[×4], C₂², C₂²^[×6], C₆^[×3], C₂×C₄^[×8], C₂³, C₂³^[×2], C₂³^[×2], A₄, C₂×C₆, C₄², C₄², C₂²⋊C₄^[×4], C₄⋊C₄^[×4], C₂²×C₄^[×2], C₂⁴, C₂×A₄^[×3], C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊₂C₂^[×2], C₄²⋊₂C₂^[×2], C₄²⋊C₃, C₂²×A₄, C₂×C₄²⋊₂C₂, C₂×C₄²⋊C₃, C₄²⋊C₆^[×2], C₂×C₄²⋊C₆

Quotients:
C₁, C₂^[×3], C₃, C₂², C₆^[×3], A₄, C₂×C₆, C₂×A₄^[×3], C₂²×A₄, C₄²⋊C₆, C₂×C₄²⋊C₆

Generators and relations
G = < a,b,c,d | a²=b⁴=c⁴=d⁶=1, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=c^-1, dcd^-1=b^-1c >

Permutation representations
►On 24 points - transitive group 24T290

Generators in S₂₄

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

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

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

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

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

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

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

G:=TransitiveGroup(24,290);

►On 24 points - transitive group 24T300

Generators in S₂₄

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

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

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

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

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

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

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

G:=TransitiveGroup(24,300);

►On 24 points - transitive group 24T306

Generators in S₂₄

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

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

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

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

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

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

G:=TransitiveGroup(24,306);

Matrix representation ►G ⊆ 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	12	0
0	0	0	0	0	12

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

8	0	0	0	0	0
0	12	0	0	0	0
0	0	8	0	0	0
0	0	0	8	0	0
0	8	0	0	1	0
4	5	5	0	0	5

7	7	12	0	0	11
5	0	0	11	0	0
0	5	0	0	11	0
11	11	10	0	0	8
12	0	0	8	0	0
3	2	3	6	1	6

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,1,0,12,0,5,0,0,0,8,0,0,1,0,0,12,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12],[8,0,0,0,0,4,0,12,0,0,8,5,0,0,8,0,0,5,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,5],[7,5,0,11,12,3,7,0,5,11,0,2,12,0,0,10,0,3,0,11,0,0,8,6,0,0,11,0,0,1,11,0,0,8,0,6] >;

Character table of C₂×C₄²⋊C₆

class	1	2A	2B	2C	2D	2E	3A	3B	4A	4B	4C	4D	4E	4F	6A	6B	6C	6D	6E	6F
size	1	1	3	3	4	4	16	16	6	6	6	6	12	12	16	16	16	16	16	16
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	-1	1	1	-1	1	1	-1	1	1	-1	-1	1	1	-1	-1	-1	1	-1	linear of order 2
ρ₃	1	1	1	1	-1	-1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	-1	-1	linear of order 2
ρ₄	1	-1	-1	1	-1	1	1	1	-1	1	1	-1	1	-1	-1	1	-1	-1	-1	1	linear of order 2
ρ₅	1	-1	-1	1	1	-1	ζ₃²	ζ₃	-1	1	1	-1	-1	1	ζ₃	ζ₆⁵	ζ₆	ζ₆⁵	ζ₃²	ζ₆	linear of order 6
ρ₆	1	-1	-1	1	-1	1	ζ₃²	ζ₃	-1	1	1	-1	1	-1	ζ₆⁵	ζ₃	ζ₆	ζ₆⁵	ζ₆	ζ₃²	linear of order 6
ρ₇	1	1	1	1	-1	-1	ζ₃	ζ₃²	1	1	1	1	-1	-1	ζ₆	ζ₆	ζ₃	ζ₃²	ζ₆⁵	ζ₆⁵	linear of order 6
ρ₈	1	-1	-1	1	-1	1	ζ₃	ζ₃²	-1	1	1	-1	1	-1	ζ₆	ζ₃²	ζ₆⁵	ζ₆	ζ₆⁵	ζ₃	linear of order 6
ρ₉	1	1	1	1	1	1	ζ₃	ζ₃²	1	1	1	1	1	1	ζ₃²	ζ₃²	ζ₃	ζ₃²	ζ₃	ζ₃	linear of order 3
ρ₁₀	1	1	1	1	1	1	ζ₃²	ζ₃	1	1	1	1	1	1	ζ₃	ζ₃	ζ₃²	ζ₃	ζ₃²	ζ₃²	linear of order 3
ρ₁₁	1	-1	-1	1	1	-1	ζ₃	ζ₃²	-1	1	1	-1	-1	1	ζ₃²	ζ₆	ζ₆⁵	ζ₆	ζ₃	ζ₆⁵	linear of order 6
ρ₁₂	1	1	1	1	-1	-1	ζ₃²	ζ₃	1	1	1	1	-1	-1	ζ₆⁵	ζ₆⁵	ζ₃²	ζ₃	ζ₆	ζ₆	linear of order 6
ρ₁₃	3	3	3	3	3	3	0	0	-1	-1	-1	-1	-1	-1	0	0	0	0	0	0	orthogonal lifted from A₄
ρ₁₄	3	-3	-3	3	3	-3	0	0	1	-1	-1	1	1	-1	0	0	0	0	0	0	orthogonal lifted from C₂×A₄
ρ₁₅	3	3	3	3	-3	-3	0	0	-1	-1	-1	-1	1	1	0	0	0	0	0	0	orthogonal lifted from C₂×A₄
ρ₁₆	3	-3	-3	3	-3	3	0	0	1	-1	-1	1	-1	1	0	0	0	0	0	0	orthogonal lifted from C₂×A₄
ρ₁₇	6	6	-2	-2	0	0	0	0	2i	2i	2i	2i	0	0	0	0	0	0	0	0	complex lifted from C₄²⋊C₆
ρ₁₈	6	-6	2	-2	0	0	0	0	2i	2i	2i	2i	0	0	0	0	0	0	0	0	complex faithful
ρ₁₉	6	6	-2	-2	0	0	0	0	2i	2i	2i	2i	0	0	0	0	0	0	0	0	complex lifted from C₄²⋊C₆
ρ₂₀	6	-6	2	-2	0	0	0	0	2i	2i	2i	2i	0	0	0	0	0	0	0	0	complex faithful

In GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes C_6

% in TeX

G:=Group("C2xC4^2:C6");

// GroupNames label

G:=SmallGroup(192,1001);

// by ID

G=gap.SmallGroup(192,1001);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,1683,185,360,4204,1173,102,1027,1784]);

// Polycyclic

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

// generators/relations

G = C₂×C₄²⋊C₆ order 192 = 2⁶·3

Direct product of C₂ and C₄²⋊C₆

G = C2×C42⋊C6 order 192 = 26·3

Direct product of C2 and C42⋊C6

G = C₂×C₄²⋊C₆ order 192 = 2⁶·3

Direct product of C₂ and C₄²⋊C₆