C2xC2^3:2D6 - GroupNames

G = C₂xC₂³:₂D₆ order 192 = 2⁶·3

Direct product of C₂ and C₂³:₂D₆

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

Aliases: C₂xC₂³:₂D₆, C₂⁴:₁₀D₆, C₆:₂C₂²wrC₂, (C₂xD₄):₃₅D₆, D₆:₁₂(C₂xD₄), (S₃xC₂⁴):₃C₂, (C₂xC₁₂):₉C₂³, (C₂²xD₄):₈S₃, (C₂²xC₄):₂₉D₆, (C₂²xC₆):₁₁D₄, D₆:C₄:₇₁C₂², (C₆xD₄):₅₅C₂², (C₂²xS₃):₁₅D₄, C₂³:₇(C₃:D₄), (C₂²xC₆):₅C₂³, C₂³:₄(C₂²xS₃), (C₂xC₆).₂₉₄C₂⁴, (C₂³xC₆):₁₂C₂², (C₂xDic₃):₃C₂³, C₂².₁₄₆(S₃xD₄), C₆.₁₄₁(C₂²xD₄), (S₃xC₂³):₂₁C₂², (C₂²xC₁₂):₄₃C₂², C₆.D₄:₆₀C₂², C₂².₃₀₇(S₃xC₂³), (C₂²xS₃).₂₃₈C₂³, (C₂²xDic₃):₃₂C₂², (D₄xC₂xC₆):₁₅C₂, (C₂xC₆):₇(C₂xD₄), C₃:₃(C₂xC₂²wrC₂), C₂.₁₀₁(C₂xS₃xD₄), (C₂xD₆:C₄):₄₁C₂, (C₂xC₄):₄(C₂²xS₃), C₂²:₄(C₂xC₃:D₄), (C₂²xC₃:D₄):₁₂C₂, (C₂xC₃:D₄):₄₃C₂², C₂.₁₄(C₂²xC₃:D₄), (C₂xC₆.D₄):₂₇C₂, SmallGroup(192,1358)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₂xC₂³:₂D₆

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂²xS₃ — S₃xC₂³ — S₃xC₂⁴ — C₂xC₂³:₂D₆

Lower central

C₃ — C₂xC₆ — C₂xC₂³:₂D₆

Upper central

C₁ — C₂³ — C₂²xD₄

Generators and relations for C₂xC₂³:₂D₆
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁶=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe^-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 1960 in 662 conjugacy classes, 143 normal (19 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, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂⁴, C₂⁴, C₂xDic₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₂²xC₆, C₂xC₂²:C₄, C₂²wrC₂, C₂²xD₄, C₂²xD₄, C₂⁵, D₆:C₄, C₆.D₄, C₂²xDic₃, C₂xC₃:D₄, C₂xC₃:D₄, C₂²xC₁₂, C₆xD₄, C₆xD₄, S₃xC₂³, S₃xC₂³, C₂³xC₆, C₂xC₂²wrC₂, C₂xD₆:C₄, C₂³:₂D₆, C₂xC₆.D₄, C₂²xC₃:D₄, D₄xC₂xC₆, S₃xC₂⁴, C₂xC₂³:₂D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₂⁴, C₃:D₄, C₂²xS₃, C₂²wrC₂, C₂²xD₄, S₃xD₄, C₂xC₃:D₄, S₃xC₂³, C₂xC₂²wrC₂, C₂³:₂D₆, C₂xS₃xD₄, C₂²xC₃:D₄, C₂xC₂³:₂D₆

Smallest permutation representation of C₂xC₂³:₂D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

C₃:D₄

S₃xD₄

kernel

C₂xC₂³:₂D₆

C₂xD₆:C₄

C₂³:₂D₆

C₂xC₆.D₄

C₂²xC₃:D₄

D₄xC₂xC₆

S₃xC₂⁴

C₂²xD₄

C₂²xS₃

C₂²xC₆

C₂²xC₄

C₂xD₄

C₂⁴

C₂³

C₂²

# reps

Matrix representation of C₂xC₂³:₂D₆ ►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

12	0	0	0	0
0	0	12	0	0
0	12	0	0	0
0	0	0	11	9
0	0	0	4	2

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

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

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

1	0	0	0	0
0	12	0	0	0
0	0	12	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],[12,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,11,4,0,0,0,9,2],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,12,1],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,1,0] >;

C₂xC₂³:₂D₆ in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes_2D_6

% in TeX

G:=Group("C2xC2^3:2D6");

// GroupNames label

G:=SmallGroup(192,1358);

// by ID

G=gap.SmallGroup(192,1358);

# by ID

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

// Polycyclic

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

// generators/relations

G = C2xC23:2D6 order 192 = 26·3

Direct product of C2 and C23:2D6

G = C₂xC₂³:₂D₆ order 192 = 2⁶·3

Direct product of C₂ and C₂³:₂D₆