C2xD4oD12 - GroupNames

G = C₂xD₄oD₁₂ order 192 = 2⁶·3

Direct product of C₂ and D₄oD₁₂

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

Aliases: C₂xD₄oD₁₂, C₆.₁₂C₂⁵, D₆.₆C₂⁴, D₁₂:₁₂C₂³, C₁₂.₄₇C₂⁴, C₆:₂2₊¹⁺⁴, Dic₆:₁₄C₂³, Dic₃.₇C₂⁴, C₄oD₄:₂₀D₆, (C₂xD₄):₅₀D₆, (C₂xQ₈):₄₂D₆, (C₄xS₃):₂C₂³, (C₂xC₁₂):₇C₂³, D₄:₉(C₂²xS₃), (C₂²xC₄):₃₆D₆, C₃:D₄:₅C₂³, (C₂xC₆).₃C₂⁴, Q₈:₉(C₂²xS₃), (C₃xQ₈):₉C₂³, (C₆xD₄):₅₃C₂², (S₃xD₄):₁₂C₂², (C₃xD₄):₁₀C₂³, C₂.₁₃(S₃xC₂⁴), C₄.₆₂(S₃xC₂³), (C₆xQ₈):₄₆C₂², C₃:₂(C₂x2₊¹⁺⁴), C₄oD₁₂:₂₅C₂², (C₂xD₁₂):₆₄C₂², (C₂²xD₁₂):₂₄C₂, (C₂²xS₃):₅C₂³, (S₃xC₂³):₁₈C₂², (C₂²xC₁₂):₂₈C₂², Q₈:₃S₃:₁₃C₂², (C₂xDic₆):₇₈C₂², C₂².₅₆(S₃xC₂³), (C₂²xC₆).₂₄₈C₂³, C₂³.₂₂₄(C₂²xS₃), (C₂xDic₃).₂₉₉C₂³, (C₂xS₃xD₄):₂₈C₂, (C₂xC₄oD₄):₁₇S₃, (C₆xC₄oD₄):₁₄C₂, (S₃xC₂xC₄):₃₄C₂², (C₂xC₄):₆(C₂²xS₃), (C₂xC₄oD₁₂):₃₇C₂, (C₂xQ₈:₃S₃):₂₁C₂, (C₃xC₄oD₄):₂₀C₂², (C₂xC₃:D₄):₅₄C₂², SmallGroup(192,1521)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₂xD₄oD₁₂

Chief series

C₁ — C₃ — C₆ — D₆ — C₂²xS₃ — S₃xC₂³ — C₂xS₃xD₄ — C₂xD₄oD₁₂

Lower central

C₃ — C₆ — C₂xD₄oD₁₂

Upper central

C₁ — C₂² — C₂xC₄oD₄

Generators and relations for C₂xD₄oD₁₂
G = < a,b,c,d,e | a²=b⁴=c²=e²=1, d⁶=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede=b²d⁵ >

Subgroups: 2184 in 898 conjugacy classes, 447 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₆, C₂xC₄, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, Dic₃, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₂xQ₈, C₄oD₄, C₄oD₄, C₂⁴, Dic₆, C₄xS₃, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₂²xD₄, C₂xC₄oD₄, C₂xC₄oD₄, 2₊¹⁺⁴, C₂xDic₆, S₃xC₂xC₄, C₂xD₁₂, C₄oD₁₂, S₃xD₄, Q₈:₃S₃, C₂xC₃:D₄, C₂²xC₁₂, C₆xD₄, C₆xQ₈, C₃xC₄oD₄, S₃xC₂³, C₂x2₊¹⁺⁴, C₂²xD₁₂, C₂xC₄oD₁₂, C₂xS₃xD₄, C₂xQ₈:₃S₃, D₄oD₁₂, C₆xC₄oD₄, C₂xD₄oD₁₂
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₂⁴, C₂²xS₃, 2₊¹⁺⁴, C₂⁵, S₃xC₂³, C₂x2₊¹⁺⁴, D₄oD₁₂, S₃xC₂⁴, C₂xD₄oD₁₂

Smallest permutation representation of C₂xD₄oD₁₂
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

54 conjugacy classes

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

54 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

2₊¹⁺⁴

D₄oD₁₂

kernel

C₂xD₄oD₁₂

C₂²xD₁₂

C₂xC₄oD₁₂

C₂xS₃xD₄

C₂xQ₈:₃S₃

D₄oD₁₂

C₆xC₄oD₄

C₂xC₄oD₄

C₂²xC₄

C₂xD₄

C₂xQ₈

C₄oD₄

C₆

C₂

# reps

Matrix representation of C₂xD₄oD₁₂ ►in GL₆(F₁₃)

12	0	0	0	0	0
0	12	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	1

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

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

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

0	12	0	0	0	0
12	0	0	0	0	0
0	0	0	1	0	2
0	0	1	0	2	0
0	0	0	0	0	12
0	0	0	0	12	0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,6,3,7,0,0,7,3,6,10,0,0,0,0,3,7,0,0,0,0,6,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,6,3,7,0,0,7,3,6,10,0,0,7,12,3,7,0,0,1,6,6,10],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,7,3,0,0,0,0,10,10,0,0,0,0,0,0,7,3,0,0,0,0,10,10],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,2,0,12,0,0,2,0,12,0] >;

C₂xD₄oD₁₂ in GAP, Magma, Sage, TeX

C_2\times D_4\circ D_{12}

% in TeX

G:=Group("C2xD4oD12");

// GroupNames label

G:=SmallGroup(192,1521);

// by ID

G=gap.SmallGroup(192,1521);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,297,1684,102,6278]);

// Polycyclic

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

// generators/relations

G = C2xD4oD12 order 192 = 26·3

Direct product of C2 and D4oD12

G = C₂xD₄oD₁₂ order 192 = 2⁶·3

Direct product of C₂ and D₄oD₁₂