(C2xD4).D6 - GroupNames

G = (C₂×D₄).D₆ order 192 = 2⁶·3

2^nd non-split extension by C₂×D₄ of D₆ acting via D₆/C₃=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂×D₄).₂D₆, (C₂×C₁₂).₂D₄, (C₂×C₄).₂D₁₂, C₂³⋊C₄.₁S₃, C₂³.₇(C₄×S₃), (C₂²×C₆).₉D₄, C₆.D₄⋊₂C₄, C₆.₈(C₂³⋊C₄), (C₆×D₄).₂C₂², C₂².₉(D₆⋊C₄), C₁₂.D₄.₁C₂, (C₂²×Dic₃)⋊₁C₄, C₂³.₂(C₃⋊D₄), C₃⋊₁(C₂³.D₄), C₂³.₂₃D₆.₁C₂, C₂.₉(C₂³.₆D₆), (C₃×C₂³⋊C₄).₁C₂, (C₂²×C₆).₂(C₂×C₄), (C₂×C₆).₂(C₂²⋊C₄), SmallGroup(192,31)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₆ — (C₂×D₄).D₆

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×C₆ — C₆×D₄ — C₂³.₂₃D₆ — (C₂×D₄).D₆

Lower central

C₃ — C₆ — C₂×C₆ — C₂²×C₆ — (C₂×D₄).D₆

Upper central

C₁ — C₂ — C₂² — C₂×D₄ — C₂³⋊C₄

Generators and relations for (C₂×D₄).D₆
G = < a,b,c,d,e | a²=b⁴=c²=1, d⁶=ab²c, e²=ba=dbd^-1=ab, ece^-1=ac=ca, dad^-1=eae^-1=ab², cbc=ebe^-1=b^-1, dcd^-1=b²c, ede^-1=bcd⁵ >

Subgroups: 224 in 68 conjugacy classes, 21 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₄⋊C₄, M₄(2), C₂²×C₄, C₂×D₄, C₃⋊C₈, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₂²×C₆, C₂³⋊C₄, C₄.D₄, C₂².D₄, C₄.Dic₃, Dic₃⋊C₄, C₆.D₄, C₆.D₄, C₃×C₂²⋊C₄, C₂²×Dic₃, C₆×D₄, C₂³.D₄, C₁₂.D₄, C₃×C₂³⋊C₄, C₂³.₂₃D₆, (C₂×D₄).D₆
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, D₆, C₂²⋊C₄, C₄×S₃, D₁₂, C₃⋊D₄, C₂³⋊C₄, D₆⋊C₄, C₂³.D₄, C₂³.₆D₆, (C₂×D₄).D₆

Character table of (C₂×D₄).D₆

class	1	2A	2B	2C	2D	3	4A	4B	4C	4D	4E	4F	6A	6B	6C	6D	6E	8A	8B	12A	12B	12C	12D	12E
size	1	1	2	4	4	2	4	8	8	12	12	24	2	4	4	4	8	24	24	8	8	8	8	8
ρ₁	1	1	1	1	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	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	-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	-1	1	-1	-1	linear of order 2
ρ₅	1	1	1	-1	1	1	-1	-i	i	1	1	-1	1	-1	-1	1	1	-i	i	i	i	-1	-i	-i	linear of order 4
ρ₆	1	1	1	-1	1	1	-1	-i	i	-1	-1	1	1	-1	-1	1	1	i	-i	i	i	-1	-i	-i	linear of order 4
ρ₇	1	1	1	-1	1	1	-1	i	-i	-1	-1	1	1	-1	-1	1	1	-i	i	-i	-i	-1	i	i	linear of order 4
ρ₈	1	1	1	-1	1	1	-1	i	-i	1	1	-1	1	-1	-1	1	1	i	-i	-i	-i	-1	i	i	linear of order 4
ρ₉	2	2	2	-2	-2	2	2	0	0	0	0	0	2	-2	-2	2	-2	0	0	0	0	2	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	2	-1	2	2	2	0	0	0	-1	-1	-1	-1	-1	0	0	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₁	2	2	2	2	-2	2	-2	0	0	0	0	0	2	2	2	2	-2	0	0	0	0	-2	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	2	-1	2	-2	-2	0	0	0	-1	-1	-1	-1	-1	0	0	1	1	-1	1	1	orthogonal lifted from D₆
ρ₁₃	2	2	2	-2	-2	-1	2	0	0	0	0	0	-1	1	1	-1	1	0	0	-√3	√3	-1	√3	-√3	orthogonal lifted from D₁₂
ρ₁₄	2	2	2	-2	-2	-1	2	0	0	0	0	0	-1	1	1	-1	1	0	0	√3	-√3	-1	-√3	√3	orthogonal lifted from D₁₂
ρ₁₅	2	2	2	-2	2	-1	-2	2i	-2i	0	0	0	-1	1	1	-1	-1	0	0	i	i	1	-i	-i	complex lifted from C₄×S₃
ρ₁₆	2	2	2	-2	2	-1	-2	-2i	2i	0	0	0	-1	1	1	-1	-1	0	0	-i	-i	1	i	i	complex lifted from C₄×S₃
ρ₁₇	2	2	2	2	-2	-1	-2	0	0	0	0	0	-1	-1	-1	-1	1	0	0	√-3	-√-3	1	√-3	-√-3	complex lifted from C₃⋊D₄
ρ₁₈	2	2	2	2	-2	-1	-2	0	0	0	0	0	-1	-1	-1	-1	1	0	0	-√-3	√-3	1	-√-3	√-3	complex lifted from C₃⋊D₄
ρ₁₉	4	4	-4	0	0	4	0	0	0	0	0	0	4	0	0	-4	0	0	0	0	0	0	0	0	orthogonal lifted from C₂³⋊C₄
ρ₂₀	4	4	-4	0	0	-2	0	0	0	0	0	0	-2	-2√-3	2√-3	2	0	0	0	0	0	0	0	0	complex lifted from C₂³.₆D₆
ρ₂₁	4	4	-4	0	0	-2	0	0	0	0	0	0	-2	2√-3	-2√-3	2	0	0	0	0	0	0	0	0	complex lifted from C₂³.₆D₆
ρ₂₂	4	-4	0	0	0	4	0	0	0	-2i	2i	0	-4	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₂³.D₄
ρ₂₃	4	-4	0	0	0	4	0	0	0	2i	-2i	0	-4	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₂³.D₄
ρ₂₄	8	-8	0	0	0	-4	0	0	0	0	0	0	4	0	0	0	0	0	0	0	0	0	0	0	symplectic faithful, Schur index 2

Smallest permutation representation of (C₂×D₄).D₆
►On 48 points

Generators in S₄₈

(2 29)(4 31)(6 33)(8 35)(10 25)(12 27)(14 38)(16 40)(18 42)(20 44)(22 46)(24 48)

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

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

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

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

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

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

Matrix representation of (C₂×D₄).D₆ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	72	0
0	0	0	0	0	72

1	0	0	0	0	0
0	1	0	0	0	0
0	0	46	0	0	0
0	0	0	27	0	0
0	0	0	0	27	0
0	0	0	0	0	46

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	46	0	0
0	0	27	0	0	0
0	0	0	0	0	27
0	0	0	0	46	0

0	72	0	0	0	0
1	72	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	27	0	0
0	0	46	0	0	0

1	72	0	0	0	0
0	72	0	0	0	0
0	0	0	0	46	0
0	0	0	0	0	46
0	0	1	0	0	0
0	0	0	72	0	0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,46],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,0,0,0,0,46,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,46,0,0,0,0,0,0,46,0,0] >;

(C₂×D₄).D₆ in GAP, Magma, Sage, TeX

(C_2\times D_4).D_6

% in TeX

G:=Group("(C2xD4).D6");

// GroupNames label

G:=SmallGroup(192,31);

// by ID

G=gap.SmallGroup(192,31);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,422,346,297,851,6278]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^6=a*b^2*c,e^2=b*a=d*b*d^-1=a*b,e*c*e^-1=a*c=c*a,d*a*d^-1=e*a*e^-1=a*b^2,c*b*c=e*b*e^-1=b^-1,d*c*d^-1=b^2*c,e*d*e^-1=b*c*d^5>;

// generators/relations

Export

Character table of (C₂×D₄).D₆ in TeX

G = (C2×D4).D6 order 192 = 26·3

2nd non-split extension by C2×D4 of D6 acting via D6/C3=C22

G = (C₂×D₄).D₆ order 192 = 2⁶·3

2^nd non-split extension by C₂×D₄ of D₆ acting via D₆/C₃=C₂²