S3xC8oD4 - GroupNames

G = S₃xC₈oD₄ order 192 = 2⁶·3

Direct product of S₃ and C₈oD₄

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

Aliases: S₃xC₈oD₄, M₄(2):₂₇D₆, C₁₂.₇₁C₂⁴, C₂₄.₅₃C₂³, (C₂xC₈):₃₀D₆, (S₃xD₄).₂C₄, (S₃xQ₈).₂C₄, C₈oD₁₂:₁₆C₂, C₄oD₄.₅₈D₆, D₄.₁₂(C₄xS₃), C₃:C₈.₃₆C₂³, Q₈.₁₈(C₄xS₃), (S₃xC₈):₂₀C₂², D₁₂.C₄:₁₄C₂, (C₂xC₂₄):₃₁C₂², D₁₂.₂₀(C₂xC₄), D₄:₂S₃.₂C₄, C₆.₃₄(C₂³xC₄), C₄.₇₀(S₃xC₂³), C₈.₆₆(C₂²xS₃), Q₈:₃S₃.₂C₄, C₈:S₃:₂₀C₂², (S₃xM₄(2)):₁₂C₂, D₄.Dic₃:₁₄C₂, (C₄xS₃).₄₁C₂³, C₁₂.₃₈(C₂²xC₄), Dic₆.₂₁(C₂xC₄), D₆.₁₅(C₂²xC₄), (C₂xC₁₂).₅₁₃C₂³, C₄oD₁₂.₅₁C₂², C₄.Dic₃:₂₆C₂², (C₃xM₄(2)):₃₂C₂², Dic₃.₁₅(C₂²xC₄), C₃:₃(C₂xC₈oD₄), (S₃xC₂xC₈):₃₀C₂, C₄.₃₈(S₃xC₂xC₄), C₂².₄(S₃xC₂xC₄), (C₃xC₈oD₄):₁₃C₂, (C₂xC₃:C₈):₃₄C₂², (S₃xC₄oD₄).₅C₂, C₃:D₄.₁(C₂xC₄), C₂.₃₅(S₃xC₂²xC₄), (C₄xS₃).₁₈(C₂xC₄), (C₃xD₄).₁₆(C₂xC₄), (C₂xC₆).₄(C₂²xC₄), (C₃xQ₈).₁₇(C₂xC₄), (S₃xC₂xC₄).₂₅₄C₂², (C₂²xS₃).₄₇(C₂xC₄), (C₂xC₄).₆₀₆(C₂²xS₃), (C₂xDic₃).₇₃(C₂xC₄), (C₃xC₄oD₄).₄₃C₂², SmallGroup(192,1308)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — S₃xC₈oD₄

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₄xS₃ — S₃xC₂xC₄ — S₃xC₄oD₄ — S₃xC₈oD₄

Lower central

C₃ — C₆ — S₃xC₈oD₄

Upper central

C₁ — C₈ — C₈oD₄

Generators and relations for S₃xC₈oD₄
G = < a,b,c,d,e | a³=b²=c⁸=e²=1, d²=c⁴, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c⁴d >

Subgroups: 512 in 266 conjugacy classes, 149 normal (24 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₄, C₂², C₂², S₃, S₃, C₆, C₆, C₈, C₈, C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₆, D₆, C₂xC₆, C₂xC₈, C₂xC₈, M₄(2), M₄(2), C₂²xC₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₄oD₄, C₃:C₈, C₃:C₈, C₂₄, C₂₄, Dic₆, C₄xS₃, C₄xS₃, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₂²xS₃, C₂²xC₈, C₂xM₄(2), C₈oD₄, C₈oD₄, C₂xC₄oD₄, S₃xC₈, S₃xC₈, C₈:S₃, C₂xC₃:C₈, C₄.Dic₃, C₂xC₂₄, C₃xM₄(2), S₃xC₂xC₄, C₄oD₁₂, S₃xD₄, D₄:₂S₃, S₃xQ₈, Q₈:₃S₃, C₃xC₄oD₄, C₂xC₈oD₄, S₃xC₂xC₈, C₈oD₁₂, S₃xM₄(2), D₁₂.C₄, D₄.Dic₃, C₃xC₈oD₄, S₃xC₄oD₄, S₃xC₈oD₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, C₂³, D₆, C₂²xC₄, C₂⁴, C₄xS₃, C₂²xS₃, C₈oD₄, C₂³xC₄, S₃xC₂xC₄, S₃xC₂³, C₂xC₈oD₄, S₃xC₂²xC₄, S₃xC₈oD₄

Smallest permutation representation of S₃xC₈oD₄
►On 48 points

Generators in S₄₈

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

(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 33)

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

(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(41 45)(42 46)(43 47)(44 48)

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

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

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

60 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	3	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	6A	6B	6C	6D	8A	8B	8C	8D	8E	···	8J	8K	8L	8M	8N	8O	···	8T	12A	12B	12C	12D	12E	24A	24B	24C	24D	24E	···	24J
order	1	2	2	2	2	2	2	2	2	2	3	4	4	4	4	4	4	4	4	4	4	6	6	6	6	8	8	8	8	8	···	8	8	8	8	8	8	···	8	12	12	12	12	12	24	24	24	24	24	···	24
size	1	1	2	2	2	3	3	6	6	6	2	1	1	2	2	2	3	3	6	6	6	2	4	4	4	1	1	1	1	2	···	2	3	3	3	3	6	···	6	2	2	4	4	4	2	2	2	2	4	···	4

60 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	4
type	+	+	+	+	+	+	+	+					+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	S₃	D₆	D₆	D₆	C₄xS₃	C₄xS₃	C₈oD₄	S₃xC₈oD₄
kernel	S₃xC₈oD₄	S₃xC₂xC₈	C₈oD₁₂	S₃xM₄(2)	D₁₂.C₄	D₄.Dic₃	C₃xC₈oD₄	S₃xC₄oD₄	S₃xD₄	D₄:₂S₃	S₃xQ₈	Q₈:₃S₃	C₈oD₄	C₂xC₈	M₄(2)	C₄oD₄	D₄	Q₈	S₃	C₁
# reps	1	3	3	3	3	1	1	1	6	6	2	2	1	3	3	1	6	2	8	4

Matrix representation of S₃xC₈oD₄ ►in GL₄(F₇₃) generated by

0	72	0	0
1	72	0	0
0	0	1	0
0	0	0	1

1	72	0	0
0	72	0	0
0	0	1	0
0	0	0	1

27	0	0	0
0	27	0	0
0	0	22	0
0	0	0	22

72	0	0	0
0	72	0	0
0	0	0	72
0	0	1	0

1	0	0	0
0	1	0	0
0	0	1	0
0	0	0	72

G:=sub<GL(4,GF(73))| [0,1,0,0,72,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,72,72,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,0,27,0,0,0,0,22,0,0,0,0,22],[72,0,0,0,0,72,0,0,0,0,0,1,0,0,72,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,72] >;

S₃xC₈oD₄ in GAP, Magma, Sage, TeX

S_3\times C_8\circ D_4

% in TeX

G:=Group("S3xC8oD4");

// GroupNames label

G:=SmallGroup(192,1308);

// by ID

G=gap.SmallGroup(192,1308);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,80,102,6278]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^8=e^2=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^4*d>;

// generators/relations

G = S3xC8oD4 order 192 = 26·3

Direct product of S3 and C8oD4

G = S₃xC₈oD₄ order 192 = 2⁶·3

Direct product of S₃ and C₈oD₄