(C2xC12).D4 - GroupNames

G = (C₂×C₁₂).D₄ order 192 = 2⁶·3

16^th non-split extension by C₂×C₁₂ of D₄ acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂×C₄).₄D₁₂, (C₂×C₁₂).₁₆D₄, C₄.₁₀D₄.S₃, (C₂×Q₈).₂₆D₆, (C₄×Dic₃).₂C₄, (C₂×Dic₆).₃C₄, (C₆×Q₈).₂C₂², C₆.₁₄(C₂³⋊C₄), C₃⋊₁(C₄².₃C₄), C₂².₁₅(D₆⋊C₄), Dic₃⋊Q₈.₁C₂, C₁₂.₁₀D₄.₁C₂, C₂.₁₅(C₂³.₆D₆), (C₂×C₄).₄(C₄×S₃), (C₂×C₁₂).₄(C₂×C₄), (C₂×C₄).₄(C₃⋊D₄), (C₂×C₆).₈(C₂²⋊C₄), (C₃×C₄.₁₀D₄).₁C₂, SmallGroup(192,37)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₂ — (C₂×C₁₂).D₄

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂×C₁₂ — C₆×Q₈ — Dic₃⋊Q₈ — (C₂×C₁₂).D₄

Lower central

C₃ — C₆ — C₂×C₆ — C₂×C₁₂ — (C₂×C₁₂).D₄

Upper central

C₁ — C₂ — C₂² — C₂×Q₈ — C₄.₁₀D₄

Generators and relations for (C₂×C₁₂).D₄
G = < a,b,c,d | a²=b¹²=1, c⁴=d²=b⁶, ab=ba, cac^-1=ab⁶, ad=da, cbc^-1=dbd^-1=ab⁵, dcd^-1=b⁹c³ >

Subgroups: 176 in 60 conjugacy classes, 21 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, Q₈, Dic₃, C₁₂, C₂×C₆, C₄², C₄⋊C₄, M₄(2), C₂×Q₈, C₂×Q₈, C₃⋊C₈, C₂₄, Dic₆, C₂×Dic₃, C₂×C₁₂, C₃×Q₈, C₄.₁₀D₄, C₄.₁₀D₄, C₄⋊Q₈, C₄.Dic₃, C₄×Dic₃, Dic₃⋊C₄, C₃×M₄(2), C₂×Dic₆, C₆×Q₈, C₄².₃C₄, C₁₂.₁₀D₄, C₃×C₄.₁₀D₄, Dic₃⋊Q₈, (C₂×C₁₂).D₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, D₆, C₂²⋊C₄, C₄×S₃, D₁₂, C₃⋊D₄, C₂³⋊C₄, D₆⋊C₄, C₄².₃C₄, C₂³.₆D₆, (C₂×C₁₂).D₄

Character table of (C₂×C₁₂).D₄

class	1	2A	2B	3	4A	4B	4C	4D	4E	4F	6A	6B	8A	8B	8C	8D	12A	12B	12C	12D	24A	24B	24C	24D
size	1	1	2	2	4	4	4	12	12	24	2	4	8	8	24	24	4	4	8	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	-1	-1	1	1	1	i	-i	i	-i	-1	-1	-1	1	i	i	-i	-i	linear of order 4
ρ₆	1	1	1	1	-1	1	-1	1	1	-1	1	1	i	-i	-i	i	-1	-1	-1	1	i	i	-i	-i	linear of order 4
ρ₇	1	1	1	1	-1	1	-1	1	1	-1	1	1	-i	i	i	-i	-1	-1	-1	1	-i	-i	i	i	linear of order 4
ρ₈	1	1	1	1	-1	1	-1	-1	-1	1	1	1	-i	i	-i	i	-1	-1	-1	1	-i	-i	i	i	linear of order 4
ρ₉	2	2	2	-1	2	2	2	0	0	0	-1	-1	-2	-2	0	0	-1	-1	-1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₁₀	2	2	2	2	-2	-2	2	0	0	0	2	2	0	0	0	0	-2	-2	2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	2	-2	-2	0	0	0	2	2	0	0	0	0	2	2	-2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	-1	2	2	2	0	0	0	-1	-1	2	2	0	0	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₃	2	2	2	-1	-2	-2	2	0	0	0	-1	-1	0	0	0	0	1	1	-1	1	√3	-√3	√3	-√3	orthogonal lifted from D₁₂
ρ₁₄	2	2	2	-1	-2	-2	2	0	0	0	-1	-1	0	0	0	0	1	1	-1	1	-√3	√3	-√3	√3	orthogonal lifted from D₁₂
ρ₁₅	2	2	2	-1	-2	2	-2	0	0	0	-1	-1	-2i	2i	0	0	1	1	1	-1	i	i	-i	-i	complex lifted from C₄×S₃
ρ₁₆	2	2	2	-1	-2	2	-2	0	0	0	-1	-1	2i	-2i	0	0	1	1	1	-1	-i	-i	i	i	complex lifted from C₄×S₃
ρ₁₇	2	2	2	-1	2	-2	-2	0	0	0	-1	-1	0	0	0	0	-1	-1	1	1	-√-3	√-3	√-3	-√-3	complex lifted from C₃⋊D₄
ρ₁₈	2	2	2	-1	2	-2	-2	0	0	0	-1	-1	0	0	0	0	-1	-1	1	1	√-3	-√-3	-√-3	√-3	complex lifted from C₃⋊D₄
ρ₁₉	4	4	-4	4	0	0	0	0	0	0	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂³⋊C₄
ρ₂₀	4	-4	0	4	0	0	0	2	-2	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₄².₃C₄, Schur index 2
ρ₂₁	4	-4	0	4	0	0	0	-2	2	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₄².₃C₄, Schur index 2
ρ₂₂	4	4	-4	-2	0	0	0	0	0	0	-2	2	0	0	0	0	2√-3	-2√-3	0	0	0	0	0	0	complex lifted from C₂³.₆D₆
ρ₂₃	4	4	-4	-2	0	0	0	0	0	0	-2	2	0	0	0	0	-2√-3	2√-3	0	0	0	0	0	0	complex lifted from C₂³.₆D₆
ρ₂₄	8	-8	0	-4	0	0	0	0	0	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic faithful, Schur index 2

Smallest permutation representation of (C₂×C₁₂).D₄
►On 48 points

Generators in S₄₈

(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)

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

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

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

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

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

Matrix representation of (C₂×C₁₂).D₄ ►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	58	58	0	72

1	72	0	0	0	0
1	0	0	0	0	0
0	0	50	57	0	0
0	0	24	23	0	0
0	0	21	21	50	32
0	0	19	47	61	23

59	7	0	0	0	0
66	14	0	0	0	0
0	0	69	69	37	53
0	0	44	44	9	1
0	0	72	0	0	0
0	0	65	29	20	33

0	72	0	0	0	0
72	0	0	0	0	0
0	0	50	57	0	0
0	0	24	23	0	0
0	0	69	69	37	53
0	0	7	35	32	36

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,58,0,0,0,1,0,58,0,0,0,0,72,0,0,0,0,0,0,72],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,50,24,21,19,0,0,57,23,21,47,0,0,0,0,50,61,0,0,0,0,32,23],[59,66,0,0,0,0,7,14,0,0,0,0,0,0,69,44,72,65,0,0,69,44,0,29,0,0,37,9,0,20,0,0,53,1,0,33],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,50,24,69,7,0,0,57,23,69,35,0,0,0,0,37,32,0,0,0,0,53,36] >;

(C₂×C₁₂).D₄ in GAP, Magma, Sage, TeX

(C_2\times C_{12}).D_4

% in TeX

G:=Group("(C2xC12).D4");

// GroupNames label

G:=SmallGroup(192,37);

// by ID

G=gap.SmallGroup(192,37);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,422,184,1123,794,297,136,851,6278]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=1,c^4=d^2=b^6,a*b=b*a,c*a*c^-1=a*b^6,a*d=d*a,c*b*c^-1=d*b*d^-1=a*b^5,d*c*d^-1=b^9*c^3>;

// generators/relations

Export

Character table of (C₂×C₁₂).D₄ in TeX

G = (C2×C12).D4 order 192 = 26·3

16th non-split extension by C2×C12 of D4 acting faithfully

G = (C₂×C₁₂).D₄ order 192 = 2⁶·3

16^th non-split extension by C₂×C₁₂ of D₄ acting faithfully