(C6xD4):C4 - GroupNames

G = (C₆xD₄):C₄ order 192 = 2⁶·3

1^st semidirect product of C₆xD₄ and C₄ acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₆xD₄):₁C₄, (C₂xC₆).₃D₈, C₆.₁₆C₄wrC₂, C₄:C₄:₁Dic₃, (C₂xD₄):₁Dic₃, C₄:D₄.₁S₃, (C₂xC₁₂).₂₂₉D₄, (C₂xC₆).₁₀SD₁₆, (C₂²xC₆).₄₄D₄, (C₂²xC₄).₇₅D₆, C₆.₂₀(C₂³:C₄), C₂².₂(D₄:S₃), C₃:₃(C₂².SD₁₆), C₆.₂₃(D₄:C₄), C₁₂.₅₅D₄:₂₈C₂, C₆.C₄²:₄₁C₂, C₂.₃(D₄:Dic₃), C₂³.₅₅(C₃:D₄), C₂².₂(D₄.S₃), C₂.₄(Q₈:₃Dic₃), C₂.₅(C₂³.₇D₆), (C₂²xC₁₂).₃₇₁C₂², C₂².₃₇(C₆.D₄), (C₃xC₄:C₄):₁C₄, (C₂xC₄).₇(C₂xDic₃), (C₂xC₁₂).₁₆₇(C₂xC₄), (C₃xC₄:D₄).₁₀C₂, (C₂xC₄).₁₆₃(C₃:D₄), (C₂xC₆).₉₅(C₂²:C₄), SmallGroup(192,96)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₂ — (C₆xD₄):C₄

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂²xC₆ — C₂²xC₁₂ — C₆.C₄² — (C₆xD₄):C₄

Lower central

C₃ — C₂xC₆ — C₂xC₁₂ — (C₆xD₄):C₄

Upper central

C₁ — C₂² — C₂²xC₄ — C₄:D₄

Generators and relations for (C₆xD₄):C₄
G = < a,b,c,d | a⁶=b⁴=c²=d⁴=1, ab=ba, ac=ca, dad^-1=a^-1, cbc=b^-1, dbd^-1=a³b, dcd^-1=bc >

Subgroups: 272 in 90 conjugacy classes, 31 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₆, C₆, C₈, C₂xC₄, C₂xC₄, D₄, C₂³, C₂³, Dic₃, C₁₂, C₂xC₆, C₂xC₆, C₂²:C₄, C₄:C₄, C₂xC₈, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₃:C₈, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₂²xC₆, C₂²xC₆, C₂.C₄², C₂²:C₈, C₄:D₄, C₂xC₃:C₈, C₃xC₂²:C₄, C₃xC₄:C₄, C₂²xDic₃, C₂²xC₁₂, C₆xD₄, C₆xD₄, C₂².SD₁₆, C₁₂.₅₅D₄, C₆.C₄², C₃xC₄:D₄, (C₆xD₄):C₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, Dic₃, D₆, C₂²:C₄, D₈, SD₁₆, C₂xDic₃, C₃:D₄, C₂³:C₄, D₄:C₄, C₄wrC₂, D₄:S₃, D₄.S₃, C₆.D₄, C₂².SD₁₆, D₄:Dic₃, C₂³.₇D₆, Q₈:₃Dic₃, (C₆xD₄):C₄

Smallest permutation representation of (C₆xD₄):C₄
►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 20 15 25)(2 21 16 26)(3 22 17 27)(4 23 18 28)(5 24 13 29)(6 19 14 30)(7 34 46 39)(8 35 47 40)(9 36 48 41)(10 31 43 42)(11 32 44 37)(12 33 45 38)

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

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+			+	+	+	-	+	-	+					+	+	-
image	C₁	C₂	C₂	C₂	C₄	C₄	S₃	D₄	D₄	Dic₃	D₆	Dic₃	D₈	SD₁₆	C₃:D₄	C₃:D₄	C₄wrC₂	C₂³:C₄	D₄:S₃	D₄.S₃	C₂³.₇D₆	Q₈:₃Dic₃
kernel	(C₆xD₄):C₄	C₁₂.₅₅D₄	C₆.C₄²	C₃xC₄:D₄	C₃xC₄:C₄	C₆xD₄	C₄:D₄	C₂xC₁₂	C₂²xC₆	C₄:C₄	C₂²xC₄	C₂xD₄	C₂xC₆	C₂xC₆	C₂xC₄	C₂³	C₆	C₆	C₂²	C₂²	C₂	C₂
# reps	1	1	1	1	2	2	1	1	1	1	1	1	2	2	2	2	4	1	1	1	2	2

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

72	1	0	0	0	0
72	0	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

72	0	0	0	0	0
0	72	0	0	0	0
0	0	27	3	0	0
0	0	0	46	0	0
0	0	0	0	46	0
0	0	0	0	0	27

43	60	0	0	0	0
13	30	0	0	0	0
0	0	22	25	0	0
0	0	42	51	0	0
0	0	0	0	0	46
0	0	0	0	27	0

0	1	0	0	0	0
1	0	0	0	0	0
0	0	1	0	0	0
0	0	55	72	0	0
0	0	0	0	1	0
0	0	0	0	0	27

G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,3,46,0,0,0,0,0,0,46,0,0,0,0,0,0,27],[43,13,0,0,0,0,60,30,0,0,0,0,0,0,22,42,0,0,0,0,25,51,0,0,0,0,0,0,0,27,0,0,0,0,46,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,55,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,27] >;

(C₆xD₄):C₄ in GAP, Magma, Sage, TeX

(C_6\times D_4)\rtimes C_4

% in TeX

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

// GroupNames label

G:=SmallGroup(192,96);

// by ID

G=gap.SmallGroup(192,96);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,219,1571,570,136,6278]);

// 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,c*b*c=b^-1,d*b*d^-1=a^3*b,d*c*d^-1=b*c>;

// generators/relations

G = (C6xD4):C4 order 192 = 26·3

1st semidirect product of C6xD4 and C4 acting faithfully

G = (C₆xD₄):C₄ order 192 = 2⁶·3

1^st semidirect product of C₆xD₄ and C₄ acting faithfully