(C6xD4):6C4 - GroupNames

G = (C₆xD₄):₆C₄ order 192 = 2⁶·3

2^nd semidirect product of C₆xD₄ and C₄ acting via C₄/C₂=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₆xD₄):₆C₄, (C₂xD₄):₄Dic₃, (C₂xD₄).₁₉₆D₆, (C₂xC₁₂).₁₈₉D₄, C₁₂.₂₀₃(C₂xD₄), D₄.₅(C₂xDic₃), (C₂²xD₄).₃S₃, D₄:Dic₃:₃₇C₂, C₁₂.₈₀(C₂²xC₄), C₄:Dic₃:₆₈C₂², (C₂²xC₄).₁₆₃D₆, (C₂²xC₆).₁₉₅D₄, C₆.₁₀₁(C₈:C₂²), C₁₂.₃₁(C₂²:C₄), (C₂xC₁₂).₄₇₀C₂³, C₂.₅(D₁₂:₆C₂²), C₄.₈(C₆.D₄), (C₆xD₄).₂₃₈C₂², C₂³.₉₁(C₃:D₄), C₃:₄(C₂³.₃₇D₄), C₄.₁₀(C₂²xDic₃), C₂³.₂₆D₆:₁₈C₂, (C₂²xC₁₂).₁₉₅C₂², C₂².₂₀(C₆.D₄), (D₄xC₂xC₆).₂C₂, (C₂xC₃:C₈):₉C₂², C₄.₈₉(C₂xC₃:D₄), (C₃xD₄).₂₂(C₂xC₄), (C₂xC₆).₅₅₂(C₂xD₄), C₆.₇₂(C₂xC₂²:C₄), (C₂xC₁₂).₁₁₆(C₂xC₄), (C₂xC₄.Dic₃):₁₇C₂, (C₂xC₄).₂₃(C₂xDic₃), C₂.₈(C₂xC₆.D₄), C₂².₉₀(C₂xC₃:D₄), (C₂xC₄).₁₉₆(C₃:D₄), (C₂xC₄).₅₅₇(C₂²xS₃), (C₂xC₆).₁₁₁(C₂²:C₄), SmallGroup(192,774)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — (C₆xD₄):₆C₄

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂xC₁₂ — C₄:Dic₃ — C₂³.₂₆D₆ — (C₆xD₄):₆C₄

Lower central

C₃ — C₆ — C₁₂ — (C₆xD₄):₆C₄

Upper central

C₁ — C₂² — C₂²xC₄ — C₂²xD₄

Generators and relations for (C₆xD₄):₆C₄
G = < a,b,c,d | a⁶=b⁴=c²=d⁴=1, ab=ba, ac=ca, dad^-1=a^-1b², cbc=dbd^-1=b^-1, dcd^-1=b^-1c >

Subgroups: 456 in 190 conjugacy classes, 71 normal (23 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₄, C₂², C₂², C₂², C₆, C₆, C₆, C₈, C₂xC₄, C₂xC₄, C₂xC₄, D₄, D₄, C₂³, C₂³, Dic₃, C₁₂, C₁₂, C₂xC₆, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, C₄:C₄, C₂xC₈, M₄(2), C₂²xC₄, C₂xD₄, C₂xD₄, C₂⁴, C₃:C₈, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xD₄, C₂²xC₆, C₂²xC₆, D₄:C₄, C₄²:C₂, C₂xM₄(2), C₂²xD₄, C₂xC₃:C₈, C₄.Dic₃, C₄xDic₃, C₄:Dic₃, C₆.D₄, C₂²xC₁₂, C₆xD₄, C₆xD₄, C₂³xC₆, C₂³.₃₇D₄, D₄:Dic₃, C₂xC₄.Dic₃, C₂³.₂₆D₆, D₄xC₂xC₆, (C₆xD₄):₆C₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, C₂³, Dic₃, D₆, C₂²:C₄, C₂²xC₄, C₂xD₄, C₂xDic₃, C₃:D₄, C₂²xS₃, C₂xC₂²:C₄, C₈:C₂², C₆.D₄, C₂²xDic₃, C₂xC₃:D₄, C₂³.₃₇D₄, D₁₂:₆C₂², C₂xC₆.D₄, (C₆xD₄):₆C₄

Smallest permutation representation of (C₆xD₄):₆C₄
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+		+	+	+	+	-	+			+
image	C₁	C₂	C₂	C₂	C₂	C₄	S₃	D₄	D₄	D₆	Dic₃	D₆	C₃:D₄	C₃:D₄	C₈:C₂²	D₁₂:₆C₂²
kernel	(C₆xD₄):₆C₄	D₄:Dic₃	C₂xC₄.Dic₃	C₂³.₂₆D₆	D₄xC₂xC₆	C₆xD₄	C₂²xD₄	C₂xC₁₂	C₂²xC₆	C₂²xC₄	C₂xD₄	C₂xD₄	C₂xC₄	C₂³	C₆	C₂
# reps	1	4	1	1	1	8	1	3	1	1	4	2	6	2	2	4

Matrix representation of (C₆xD₄):₆C₄ ►in GL₆(F₇₃)

65	0	0	0	0	0
0	9	0	0	0	0
0	0	72	0	0	0
0	0	0	72	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	1	29	0	0
0	0	10	72	0	0
0	0	0	0	72	44
0	0	0	0	63	1

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

0	28	0	0	0	0
13	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	0	0

G:=sub<GL(6,GF(73))| [65,0,0,0,0,0,0,9,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,10,0,0,0,0,29,72,0,0,0,0,0,0,72,63,0,0,0,0,44,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,29,72,0,0,0,0,0,0,1,10,0,0,0,0,0,72],[0,13,0,0,0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C₆xD₄):₆C₄ in GAP, Magma, Sage, TeX

(C_6\times D_4)\rtimes_6C_4

% in TeX

G:=Group("(C6xD4):6C4");

// GroupNames label

G:=SmallGroup(192,774);

// by ID

G=gap.SmallGroup(192,774);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,422,387,1684,438,102,6278]);

// Polycyclic

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

// generators/relations

G = (C6xD4):6C4 order 192 = 26·3

2nd semidirect product of C6xD4 and C4 acting via C4/C2=C2

G = (C₆xD₄):₆C₄ order 192 = 2⁶·3

2^nd semidirect product of C₆xD₄ and C₄ acting via C₄/C₂=C₂