C2xD12:6C2^2 - GroupNames

G = C₂xD₁₂:₆C₂² order 192 = 2⁶·3

Direct product of C₂ and D₁₂:₆C₂²

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

Aliases: C₂xD₁₂:₆C₂², D₁₂:₇C₂³, C₁₂.₂₈C₂⁴, Dic₆:₆C₂³, C₃:C₈:₄C₂³, (C₂xD₄):₃₄D₆, C₆:₄(C₈:C₂²), (C₂²xD₄):₇S₃, D₄:S₃:₁₇C₂², C₁₂.₂₄₉(C₂xD₄), (C₂xC₁₂).₂₀₇D₄, (C₆xD₄):₄₂C₂², C₄.₂₈(S₃xC₂³), C₄oD₁₂:₁₉C₂², (C₂xD₁₂):₅₅C₂², D₄.S₃:₁₆C₂², D₄.₂₀(C₂²xS₃), (C₃xD₄).₂₀C₂³, (C₂²xC₆).₂₀₇D₄, C₆.₁₃₇(C₂²xD₄), (C₂²xC₄).₂₈₄D₆, (C₂xC₁₂).₅₃₇C₂³, (C₂xDic₆):₆₅C₂², C₂³.₉₉(C₃:D₄), C₄.Dic₃:₃₂C₂², (C₂²xC₁₂).₂₇₀C₂², (D₄xC₂xC₆):₃C₂, C₃:₅(C₂xC₈:C₂²), (C₂xD₄:S₃):₃₀C₂, (C₂xC₃:C₈):₂₀C₂², C₄.₂₁(C₂xC₃:D₄), (C₂xC₄oD₁₂):₂₈C₂, (C₂xD₄.S₃):₃₀C₂, (C₂xC₆).₅₇₇(C₂xD₄), (C₂xC₄).₉₂(C₃:D₄), (C₂xC₄.Dic₃):₂₆C₂, C₂.₁₀(C₂²xC₃:D₄), (C₂xC₄).₂₃₅(C₂²xS₃), C₂².₁₀₆(C₂xC₃:D₄), SmallGroup(192,1352)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — C₂xD₁₂:₆C₂²

Chief series

C₁ — C₃ — C₆ — C₁₂ — D₁₂ — C₂xD₁₂ — C₂xC₄oD₁₂ — C₂xD₁₂:₆C₂²

Lower central

C₃ — C₆ — C₁₂ — C₂xD₁₂:₆C₂²

Upper central

C₁ — C₂² — C₂²xC₄ — C₂²xD₄

Generators and relations for C₂xD₁₂:₆C₂²
G = < a,b,c,d,e | a²=b¹²=c²=d²=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, ebe=b⁷, dcd=b⁶c, ece=b³c, de=ed >

Subgroups: 744 in 298 conjugacy classes, 111 normal (25 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₆, C₈, C₂xC₄, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, C₂³, C₂³, Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₂xC₈, M₄(2), D₈, SD₁₆, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂⁴, C₃:C₈, Dic₆, Dic₆, C₄xS₃, D₁₂, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xD₄, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₂xM₄(2), C₂xD₈, C₂xSD₁₆, C₈:C₂², C₂²xD₄, C₂xC₄oD₄, C₂xC₃:C₈, C₄.Dic₃, D₄:S₃, D₄.S₃, C₂xDic₆, S₃xC₂xC₄, C₂xD₁₂, C₄oD₁₂, C₄oD₁₂, C₂xC₃:D₄, C₂²xC₁₂, C₆xD₄, C₆xD₄, C₂³xC₆, C₂xC₈:C₂², C₂xC₄.Dic₃, C₂xD₄:S₃, D₁₂:₆C₂², C₂xD₄.S₃, C₂xC₄oD₁₂, D₄xC₂xC₆, C₂xD₁₂:₆C₂²
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₂⁴, C₃:D₄, C₂²xS₃, C₈:C₂², C₂²xD₄, C₂xC₃:D₄, S₃xC₂³, C₂xC₈:C₂², D₁₂:₆C₂², C₂²xC₃:D₄, C₂xD₁₂:₆C₂²

Smallest permutation representation of C₂xD₁₂:₆C₂²
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

42 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	3	4A	4B	4C	4D	4E	4F	6A	···	6G	6H	···	6O	8A	8B	8C	8D	12A	12B	12C	12D
order	1	2	2	2	2	2	2	2	2	2	2	2	3	4	4	4	4	4	4	6	···	6	6	···	6	8	8	8	8	12	12	12	12
size	1	1	1	1	2	2	4	4	4	4	12	12	2	2	2	2	2	12	12	2	···	2	4	···	4	12	12	12	12	4	4	4	4

42 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

C₃:D₄

C₈:C₂²

D₁₂:₆C₂²

kernel

C₂xD₁₂:₆C₂²

C₂xC₄.Dic₃

C₂xD₄:S₃

D₁₂:₆C₂²

C₂xD₄.S₃

C₂xC₄oD₁₂

D₄xC₂xC₆

C₂²xD₄

C₂xC₁₂

C₂²xC₆

C₂²xC₄

C₂xD₄

C₂xC₄

C₂³

C₆

C₂

# reps

Matrix representation of C₂xD₁₂:₆C₂² ►in GL₆(F₇₃)

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	72	0
0	0	0	0	0	72

8	0	0	0	0	0
29	64	0	0	0	0
0	0	8	60	0	0
0	0	66	65	0	0
0	0	0	0	9	31
0	0	0	0	56	64

52	40	0	0	0	0
62	21	0	0	0	0
0	0	0	0	9	31
0	0	0	0	56	64
0	0	8	60	0	0
0	0	66	65	0	0

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	0	0	0	72

1	0	0	0	0	0
0	1	0	0	0	0
0	0	72	29	0	0
0	0	0	1	0	0
0	0	0	0	72	0
0	0	0	0	10	1

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[8,29,0,0,0,0,0,64,0,0,0,0,0,0,8,66,0,0,0,0,60,65,0,0,0,0,0,0,9,56,0,0,0,0,31,64],[52,62,0,0,0,0,40,21,0,0,0,0,0,0,0,0,8,66,0,0,0,0,60,65,0,0,9,56,0,0,0,0,31,64,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,29,1,0,0,0,0,0,0,72,10,0,0,0,0,0,1] >;

C₂xD₁₂:₆C₂² in GAP, Magma, Sage, TeX

C_2\times D_{12}\rtimes_6C_2^2

% in TeX

G:=Group("C2xD12:6C2^2");

// GroupNames label

G:=SmallGroup(192,1352);

// by ID

G=gap.SmallGroup(192,1352);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,297,1684,235,102,6278]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^12=c^2=d^2=e^2=1,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,e*b*e=b^7,d*c*d=b^6*c,e*c*e=b^3*c,d*e=e*d>;

// generators/relations

G = C2xD12:6C22 order 192 = 26·3

Direct product of C2 and D12:6C22

G = C₂xD₁₂:₆C₂² order 192 = 2⁶·3

Direct product of C₂ and D₁₂:₆C₂²