M4(2).D6 - GroupNames

G = M₄(2).D₆ order 192 = 2⁶·3

12^nd non-split extension by M₄(2) of D₆ acting via D₆/C₃=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M₄(2).₁₂D₆, C₃:C₈.₃₀D₄, C₈:C₂²:₂S₃, D₄:D₆:₄C₂, C₄oD₄.₄₀D₆, (C₃xD₄).₁₁D₄, (C₂xD₄).₇₉D₆, C₄.₁₇₈(S₃xD₄), (C₃xQ₈).₁₁D₄, C₁₂.D₄:₉C₂, C₁₂.₁₉₅(C₂xD₄), D₄.₄(C₃:D₄), C₃:₅(D₄.₄D₄), D₄.Dic₃:₅C₂, C₁₂.₄₆D₄:₆C₂, C₁₂.₅₃D₄:₅C₂, (C₂xC₁₂).₁₄C₂³, Q₈.₁₁(C₃:D₄), C₆.₁₂₄(C₄:D₄), (C₆xD₄).₁₀₄C₂², (C₂xD₁₂).₁₂₉C₂², (C₃xM₄(2)).₉C₂², C₄.Dic₃.₂₄C₂², C₂.₃₀(C₂³.₁₄D₆), C₂².₁₃(D₄:₂S₃), (C₂xD₄:S₃):₂₂C₂, (C₃xC₈:C₂²):₆C₂, C₄.₅₁(C₂xC₃:D₄), (C₂xC₆).₃₆(C₄oD₄), (C₂xC₃:C₈).₁₇₀C₂², (C₂xC₄).₁₄(C₂²xS₃), (C₃xC₄oD₄).₁₂C₂², SmallGroup(192,758)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₂ — M₄(2).D₆

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂xC₁₂ — C₂xD₁₂ — D₄:D₆ — M₄(2).D₆

Lower central

C₃ — C₆ — C₂xC₁₂ — M₄(2).D₆

Upper central

C₁ — C₂ — C₂xC₄ — C₈:C₂²

Generators and relations for M₄(2).D₆
G = < a,b,c,d | a⁸=b²=c⁶=1, d²=a⁶, bab=a⁵, cac^-1=a^-1, dad^-1=a³b, cbc^-1=dbd^-1=a⁴b, dcd^-1=a⁶c^-1 >

Subgroups: 336 in 108 conjugacy classes, 37 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, C₂³, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₈, M₄(2), M₄(2), D₈, SD₁₆, C₂xD₄, C₂xD₄, C₄oD₄, C₃:C₈, C₃:C₈, C₂₄, D₁₂, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xD₄, C₃xQ₈, C₂²xS₃, C₂²xC₆, C₄.D₄, C₈.C₄, C₈oD₄, C₂xD₈, C₈:C₂², C₈:C₂², C₂xC₃:C₈, C₂xC₃:C₈, C₄.Dic₃, C₄.Dic₃, D₄:S₃, Q₈:₂S₃, C₃xM₄(2), C₃xD₈, C₃xSD₁₆, C₂xD₁₂, C₆xD₄, C₃xC₄oD₄, D₄.₄D₄, C₁₂.₅₃D₄, C₁₂.₄₆D₄, C₁₂.D₄, C₂xD₄:S₃, D₄.Dic₃, D₄:D₆, C₃xC₈:C₂², M₄(2).D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₄oD₄, C₃:D₄, C₂²xS₃, C₄:D₄, S₃xD₄, D₄:₂S₃, C₂xC₃:D₄, D₄.₄D₄, C₂³.₁₄D₆, M₄(2).D₆

Character table of M₄(2).D₆

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	6A	6B	6C	6D	6E	8A	8B	8C	8D	8E	8F	8G	12A	12B	12C	24A	24B
size	1	1	2	4	8	24	2	2	2	4	2	4	8	8	8	6	6	8	12	12	12	24	4	4	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	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	-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	-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	1	-1	-1	linear of order 2
ρ₅	1	1	1	-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
ρ₆	1	1	1	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
ρ₇	1	1	1	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
ρ₈	1	1	1	-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
ρ₉	2	2	2	2	2	0	-1	2	2	2	-1	-1	-1	-1	-1	0	0	2	0	0	0	0	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₀	2	2	2	2	-2	0	-1	2	2	2	-1	-1	1	1	-1	0	0	-2	0	0	0	0	-1	-1	-1	1	1	orthogonal lifted from D₆
ρ₁₁	2	2	2	-2	2	0	-1	2	2	-2	-1	-1	-1	-1	1	0	0	-2	0	0	0	0	-1	-1	1	1	1	orthogonal lifted from D₆
ρ₁₂	2	2	-2	0	0	0	2	2	-2	0	2	-2	0	0	0	-2	-2	0	0	2	0	0	2	-2	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	-2	0	0	0	2	2	-2	0	2	-2	0	0	0	2	2	0	0	-2	0	0	2	-2	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	-2	-2	0	0	2	-2	2	2	2	-2	0	0	-2	0	0	0	0	0	0	0	-2	2	2	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	2	-2	-2	0	-1	2	2	-2	-1	-1	1	1	1	0	0	2	0	0	0	0	-1	-1	1	-1	-1	orthogonal lifted from D₆
ρ₁₆	2	2	-2	2	0	0	2	-2	2	-2	2	-2	0	0	2	0	0	0	0	0	0	0	-2	2	-2	0	0	orthogonal lifted from D₄
ρ₁₇	2	2	-2	2	0	0	-1	-2	2	-2	-1	1	√-3	-√-3	-1	0	0	0	0	0	0	0	1	-1	1	√-3	-√-3	complex lifted from C₃:D₄
ρ₁₈	2	2	-2	-2	0	0	-1	-2	2	2	-1	1	-√-3	√-3	1	0	0	0	0	0	0	0	1	-1	-1	√-3	-√-3	complex lifted from C₃:D₄
ρ₁₉	2	2	-2	-2	0	0	-1	-2	2	2	-1	1	√-3	-√-3	1	0	0	0	0	0	0	0	1	-1	-1	-√-3	√-3	complex lifted from C₃:D₄
ρ₂₀	2	2	-2	2	0	0	-1	-2	2	-2	-1	1	-√-3	√-3	-1	0	0	0	0	0	0	0	1	-1	1	-√-3	√-3	complex lifted from C₃:D₄
ρ₂₁	2	2	2	0	0	0	2	-2	-2	0	2	2	0	0	0	0	0	0	-2i	0	2i	0	-2	-2	0	0	0	complex lifted from C₄oD₄
ρ₂₂	2	2	2	0	0	0	2	-2	-2	0	2	2	0	0	0	0	0	0	2i	0	-2i	0	-2	-2	0	0	0	complex lifted from C₄oD₄
ρ₂₃	4	4	-4	0	0	0	-2	4	-4	0	-2	2	0	0	0	0	0	0	0	0	0	0	-2	2	0	0	0	orthogonal lifted from S₃xD₄
ρ₂₄	4	-4	0	0	0	0	4	0	0	0	-4	0	0	0	0	-2√2	2√2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄.₄D₄
ρ₂₅	4	-4	0	0	0	0	4	0	0	0	-4	0	0	0	0	2√2	-2√2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄.₄D₄
ρ₂₆	4	4	4	0	0	0	-2	-4	-4	0	-2	-2	0	0	0	0	0	0	0	0	0	0	2	2	0	0	0	symplectic lifted from D₄:₂S₃, Schur index 2
ρ₂₇	8	-8	0	0	0	0	-4	0	0	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal faithful, Schur index 2

Smallest permutation representation of M₄(2).D₆
►On 48 points

Generators in S₄₈

(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 5)(3 7)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)

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

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

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

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

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

Matrix representation of M₄(2).D₆ ►in GL₆(F₇₃)

0	72	0	0	0	0
72	0	0	0	0	0
0	0	40	21	0	32
0	0	40	21	41	32
0	0	16	16	0	0
0	0	15	33	21	12

72	0	0	0	0	0
0	72	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	1	0
0	0	39	15	0	1

36	28	0	0	0	0
28	36	0	0	0	0
0	0	0	0	1	0
0	0	34	58	1	71
0	0	1	0	0	0
0	0	1	39	46	15

37	28	0	0	0	0
45	36	0	0	0	0
0	0	34	58	1	71
0	0	0	0	1	0
0	0	1	0	0	0
0	0	31	1	46	39

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,72,0,0,0,0,0,0,0,40,40,16,15,0,0,21,21,16,33,0,0,0,41,0,21,0,0,32,32,0,12],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,39,0,0,0,72,0,15,0,0,0,0,1,0,0,0,0,0,0,1],[36,28,0,0,0,0,28,36,0,0,0,0,0,0,0,34,1,1,0,0,0,58,0,39,0,0,1,1,0,46,0,0,0,71,0,15],[37,45,0,0,0,0,28,36,0,0,0,0,0,0,34,0,1,31,0,0,58,0,0,1,0,0,1,1,0,46,0,0,71,0,0,39] >;

M₄(2).D₆ in GAP, Magma, Sage, TeX

M_4(2).D_6

% in TeX

G:=Group("M4(2).D6");

// GroupNames label

G:=SmallGroup(192,758);

// by ID

G=gap.SmallGroup(192,758);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,219,1123,297,136,1684,851,438,102,6278]);

// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=a^6,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^-1>;

// generators/relations

Export

Character table of M₄(2).D₆ in TeX

G = M4(2).D6 order 192 = 26·3

12nd non-split extension by M4(2) of D6 acting via D6/C3=C22

G = M₄(2).D₆ order 192 = 2⁶·3

12^nd non-split extension by M₄(2) of D₆ acting via D₆/C₃=C₂²