(C2xD4):43D6 - GroupNames

G = (C₂xD₄):₄₃D₆ order 192 = 2⁶·3

11^st semidirect product of C₂xD₄ and D₆ acting via D₆/C₆=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂xD₄):₄₃D₆, (C₂xC₁₂):₁₅D₄, (C₂xQ₈):₃₅D₆, D₆:₆(C₄oD₄), (C₂²xC₄):₃₂D₆, C₂³:₂D₆:₃₄C₂, D₆:₃D₄:₄₆C₂, D₆:₃Q₈:₄₇C₂, C₁₂.₂₆₅(C₂xD₄), (C₆xD₄):₄₇C₂², (C₆xQ₈):₃₉C₂², (C₂xC₆).₃₁₄C₂⁴, C₄:Dic₃:₈₀C₂², C₆.₁₆₄(C₂²xD₄), (C₂xC₁₂).₆₅₁C₂³, Dic₃:C₄:₇₆C₂², D₆:C₄.₁₅₉C₂², (C₂²xC₁₂):₄₂C₂², C₃:₈(C₂².₁₉C₂⁴), (C₄xDic₃):₆₀C₂², C₂³.₂₆D₆:₃₈C₂, C₂³.₂₃D₆:₃₄C₂, C₆.D₄:₆₅C₂², (C₂²xC₆).₂₄₀C₂³, C₂².₃₂₃(S₃xC₂³), C₂³.₁₄₇(C₂²xS₃), (C₂²xS₃).₂₄₃C₂³, (S₃xC₂³).₁₁₅C₂², (C₂xDic₃).₂₉₃C₂³, (C₂²xDic₃).₂₃₆C₂², (C₆xC₄oD₄):₆C₂, (S₃xC₂²xC₄):₇C₂, (C₂xC₄oD₄):₁₀S₃, (C₄xC₃:D₄):₆₀C₂, (C₂xC₆).₈₀(C₂xD₄), (C₂xC₄):₁₄(C₃:D₄), C₂.₁₀₃(S₃xC₄oD₄), C₆.₂₁₅(C₂xC₄oD₄), C₄.₁₀₀(C₂xC₃:D₄), (S₃xC₂xC₄).₂₆₃C₂², C₂.₃₇(C₂²xC₃:D₄), C₂².₂₃(C₂xC₃:D₄), (C₂xC₄).₆₃₉(C₂²xS₃), (C₂xC₃:D₄).₁₄₀C₂², SmallGroup(192,1387)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — (C₂xD₄):₄₃D₆

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂²xS₃ — S₃xC₂³ — S₃xC₂²xC₄ — (C₂xD₄):₄₃D₆

Lower central

C₃ — C₂xC₆ — (C₂xD₄):₄₃D₆

Upper central

C₁ — C₂xC₄ — C₂xC₄oD₄

Generators and relations for (C₂xD₄):₄₃D₆
G = < a,b,c,d,e | a²=b⁴=c²=d⁶=e²=1, ab=ba, ece=ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, dcd^-1=b²c, ede=d^-1 >

Subgroups: 808 in 330 conjugacy classes, 115 normal (25 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₆, C₂xC₄, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, C₂³, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, C₄:C₄, C₂²xC₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂⁴, C₄xS₃, C₂xDic₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₄²:C₂, C₄xD₄, C₂²wrC₂, C₄:D₄, C₂²:Q₈, C₂².D₄, C₂³xC₄, C₂xC₄oD₄, C₄xDic₃, Dic₃:C₄, C₄:Dic₃, D₆:C₄, C₆.D₄, S₃xC₂xC₄, S₃xC₂xC₄, C₂²xDic₃, C₂xC₃:D₄, C₂²xC₁₂, C₂²xC₁₂, C₆xD₄, C₆xD₄, C₆xQ₈, C₃xC₄oD₄, S₃xC₂³, C₂².₁₉C₂⁴, C₂³.₂₆D₆, C₄xC₃:D₄, C₂³.₂₃D₆, C₂³:₂D₆, D₆:₃D₄, D₆:₃Q₈, S₃xC₂²xC₄, C₆xC₄oD₄, (C₂xD₄):₄₃D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₄oD₄, C₂⁴, C₃:D₄, C₂²xS₃, C₂²xD₄, C₂xC₄oD₄, C₂xC₃:D₄, S₃xC₂³, C₂².₁₉C₂⁴, S₃xC₄oD₄, C₂²xC₃:D₄, (C₂xD₄):₄₃D₆

Smallest permutation representation of (C₂xD₄):₄₃D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

C₄oD₄

C₃:D₄

S₃xC₄oD₄

kernel

(C₂xD₄):₄₃D₆

C₂³.₂₆D₆

C₄xC₃:D₄

C₂³.₂₃D₆

C₂³:₂D₆

D₆:₃D₄

D₆:₃Q₈

S₃xC₂²xC₄

C₆xC₄oD₄

C₂xC₄oD₄

C₂xC₁₂

C₂²xC₄

C₂xD₄

C₂xQ₈

D₆

C₂xC₄

C₂

# reps

Matrix representation of (C₂xD₄):₄₃D₆ ►in GL₆(F₁₃)

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	12	0
0	0	0	0	0	12

5	0	0	0	0	0
0	8	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

0	5	0	0	0	0
8	0	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	1	10
0	0	0	0	0	12

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

12	0	0	0	0	0
0	12	0	0	0	0
0	0	12	12	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	5	12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,10,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,5,0,0,0,0,0,12] >;

(C₂xD₄):₄₃D₆ in GAP, Magma, Sage, TeX

(C_2\times D_4)\rtimes_{43}D_6

% in TeX

G:=Group("(C2xD4):43D6");

// GroupNames label

G:=SmallGroup(192,1387);

// by ID

G=gap.SmallGroup(192,1387);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,570,6278]);

// Polycyclic

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

// generators/relations

G = (C2xD4):43D6 order 192 = 26·3

11st semidirect product of C2xD4 and D6 acting via D6/C6=C2

G = (C₂xD₄):₄₃D₆ order 192 = 2⁶·3

11^st semidirect product of C₂xD₄ and D₆ acting via D₆/C₆=C₂