C2xD6:D4 - GroupNames

G = C₂xD₆:D₄ order 192 = 2⁶·3

Direct product of C₂ and D₆:D₄

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂xD₆:D₄, C₂³:₆D₁₂, C₂⁴.₆₅D₆, C₆:₁C₂²wrC₂, D₆:₁₀(C₂xD₄), (C₂²xC₆):₉D₄, (S₃xC₂⁴):₁C₂, (C₂xC₁₂):₃C₂³, C₂²:C₄:₃₇D₆, C₂²:₄(C₂xD₁₂), (C₂²xC₄):₁₀D₆, C₆.₆(C₂²xD₄), D₆:C₄:₄₄C₂², (C₂²xD₁₂):₅C₂, (C₂²xS₃):₁₃D₄, (C₂xC₆).₃₁C₂⁴, C₂.₈(C₂²xD₁₂), (C₂xD₁₂):₄₃C₂², (C₂²xS₃):₁C₂³, (S₃xC₂³):₃C₂², (C₂²xC₁₂):₇C₂², (C₂xDic₃):₁C₂³, C₂².₁₂₅(S₃xD₄), C₂².₇₀(S₃xC₂³), (C₂³xC₆).₅₇C₂², (C₂²xC₆).₁₂₃C₂³, C₂³.₁₅₆(C₂²xS₃), (C₂²xDic₃):₆C₂², C₂.₈(C₂xS₃xD₄), (C₂xC₆):₄(C₂xD₄), C₃:₁(C₂xC₂²wrC₂), (C₂xD₆:C₄):₁₆C₂, (C₂xC₄):₃(C₂²xS₃), (C₂xC₂²:C₄):₁₀S₃, (C₆xC₂²:C₄):₁₃C₂, (C₂²xC₃:D₄):₃C₂, (C₂xC₃:D₄):₃₄C₂², (C₃xC₂²:C₄):₄₆C₂², SmallGroup(192,1046)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₂xD₆:D₄

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂²xS₃ — S₃xC₂³ — S₃xC₂⁴ — C₂xD₆:D₄

Lower central

C₃ — C₂xC₆ — C₂xD₆:D₄

Upper central

C₁ — C₂³ — C₂xC₂²:C₄

Generators and relations for C₂xD₆:D₄
G = < a,b,c,d,e | a²=b⁶=c²=d⁴=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b^-1, bd=db, dcd^-1=b³c, ece=bc, ede=d^-1 >

Subgroups: 2088 in 662 conjugacy classes, 143 normal (17 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₆, C₂xC₄, C₂xC₄, D₄, C₂³, C₂³, C₂³, Dic₃, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₂²:C₄, C₂²:C₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂⁴, C₂⁴, D₁₂, C₂xDic₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₂²xC₆, C₂xC₂²:C₄, C₂xC₂²:C₄, C₂²wrC₂, C₂²xD₄, C₂⁵, D₆:C₄, C₃xC₂²:C₄, C₂xD₁₂, C₂xD₁₂, C₂²xDic₃, C₂xC₃:D₄, C₂xC₃:D₄, C₂²xC₁₂, S₃xC₂³, S₃xC₂³, S₃xC₂³, C₂³xC₆, C₂xC₂²wrC₂, D₆:D₄, C₂xD₆:C₄, C₆xC₂²:C₄, C₂²xD₁₂, C₂²xC₃:D₄, S₃xC₂⁴, C₂xD₆:D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₂⁴, D₁₂, C₂²xS₃, C₂²wrC₂, C₂²xD₄, C₂xD₁₂, S₃xD₄, S₃xC₂³, C₂xC₂²wrC₂, D₆:D₄, C₂²xD₁₂, C₂xS₃xD₄, C₂xD₆:D₄

Smallest permutation representation of C₂xD₆:D₄
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

48 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	···	2S	2T	2U	3	4A	4B	4C	4D	4E	4F	6A	···	6G	6H	6I	6J	6K	12A	···	12H
order	1	2	···	2	2	2	2	2	2	···	2	2	2	3	4	4	4	4	4	4	6	···	6	6	6	6	6	12	···	12
size	1	1	···	1	2	2	2	2	6	···	6	12	12	2	4	4	4	4	12	12	2	···	2	4	4	4	4	4	···	4

48 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

D₁₂

S₃xD₄

kernel

C₂xD₆:D₄

D₆:D₄

C₂xD₆:C₄

C₆xC₂²:C₄

C₂²xD₁₂

C₂²xC₃:D₄

S₃xC₂⁴

C₂xC₂²:C₄

C₂²xS₃

C₂²xC₆

C₂²:C₄

C₂²xC₄

C₂⁴

C₂³

C₂²

# reps

Matrix representation of C₂xD₆:D₄ ►in GL₅(F₁₃)

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

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

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

1	0	0	0	0
0	3	6	0	0
0	7	10	0	0
0	0	0	0	12
0	0	0	1	0

1	0	0	0	0
0	3	6	0	0
0	3	10	0	0
0	0	0	0	1
0	0	0	1	0

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

C₂xD₆:D₄ in GAP, Magma, Sage, TeX

C_2\times D_6\rtimes D_4

% in TeX

G:=Group("C2xD6:D4");

// GroupNames label

G:=SmallGroup(192,1046);

// by ID

G=gap.SmallGroup(192,1046);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,297,80,6278]);

// Polycyclic

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

// generators/relations

G = C2xD6:D4 order 192 = 26·3

Direct product of C2 and D6:D4

G = C₂xD₆:D₄ order 192 = 2⁶·3

Direct product of C₂ and D₆:D₄