S3xM5(2) - GroupNames

G = S₃xM₅(2) order 192 = 2⁶·3

Direct product of S₃ and M₅(2)

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

Aliases: S₃xM₅(2), C₁₆:₆D₆, C₄₈:₇C₂², C₂₄.₇₄C₂³, (S₃xC₁₆):₇C₂, (C₄xS₃).₁C₈, (S₃xC₈).₄C₄, C₈.₃₅(C₄xS₃), C₄.₁₅(S₃xC₈), D₆.C₈:₅C₂, D₆.₆(C₂xC₈), C₃:₂(C₂xM₅(2)), C₃:C₁₆:₁₁C₂², C₁₂.₁₂(C₂xC₈), C₂₄.₄₄(C₂xC₄), (C₂xC₈).₂₇₂D₆, C₂².₇(S₃xC₈), (C₃xM₅(2)):₅C₂, (C₂²xS₃).₄C₈, C₈.₆₀(C₂²xS₃), C₆.₁₅(C₂²xC₈), Dic₃.₇(C₂xC₈), (C₂xDic₃).₆C₈, C₁₂.C₈:₁₃C₂, (S₃xC₈).₁₈C₂², (C₂xC₂₄).₂₇₃C₂², C₁₂.₁₃₁(C₂²xC₄), (C₂xC₃:C₈).₈C₄, (S₃xC₂xC₄).₈C₄, C₂.₁₆(S₃xC₂xC₈), C₃:C₈.₂₂(C₂xC₄), (S₃xC₂xC₈).₁₈C₂, C₄.₁₀₅(S₃xC₂xC₄), (C₂xC₆).₅(C₂xC₈), (C₄xS₃).₃₄(C₂xC₄), (C₂xC₁₂).₇₂(C₂xC₄), (C₂xC₄).₁₄₈(C₄xS₃), SmallGroup(192,465)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — S₃xM₅(2)

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂₄ — S₃xC₈ — S₃xC₂xC₈ — S₃xM₅(2)

Lower central

C₃ — C₆ — S₃xM₅(2)

Upper central

C₁ — C₈ — M₅(2)

Generators and relations for S₃xM₅(2)
G = < a,b,c,d | a³=b²=c¹⁶=d²=1, bab=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c⁹ >

Subgroups: 168 in 90 conjugacy classes, 53 normal (41 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, S₃, C₆, C₆, C₈, C₈, C₂xC₄, C₂xC₄, C₂³, Dic₃, C₁₂, D₆, D₆, C₂xC₆, C₁₆, C₁₆, C₂xC₈, C₂xC₈, C₂²xC₄, C₃:C₈, C₂₄, C₄xS₃, C₂xDic₃, C₂xC₁₂, C₂²xS₃, C₂xC₁₆, M₅(2), M₅(2), C₂²xC₈, C₃:C₁₆, C₄₈, S₃xC₈, C₂xC₃:C₈, C₂xC₂₄, S₃xC₂xC₄, C₂xM₅(2), S₃xC₁₆, D₆.C₈, C₁₂.C₈, C₃xM₅(2), S₃xC₂xC₈, S₃xM₅(2)
Quotients: C₁, C₂, C₄, C₂², S₃, C₈, C₂xC₄, C₂³, D₆, C₂xC₈, C₂²xC₄, C₄xS₃, C₂²xS₃, M₅(2), C₂²xC₈, S₃xC₈, S₃xC₂xC₄, C₂xM₅(2), S₃xC₂xC₈, S₃xM₅(2)

Smallest permutation representation of S₃xM₅(2)
►On 48 points

Generators in S₄₈

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

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

(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)

(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)

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

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

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

60 conjugacy classes

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	4D	4E	4F	6A	6B	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L	12A	12B	12C	16A	···	16H	16I	···	16P	24A	24B	24C	24D	24E	24F	48A	···	48H
order	1	2	2	2	2	2	3	4	4	4	4	4	4	6	6	8	8	8	8	8	8	8	8	8	8	8	8	12	12	12	16	···	16	16	···	16	24	24	24	24	24	24	48	···	48
size	1	1	2	3	3	6	2	1	1	2	3	3	6	2	4	1	1	1	1	2	2	3	3	3	3	6	6	2	2	4	2	···	2	6	···	6	2	2	2	2	4	4	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	2	4
type	+	+	+	+	+	+							+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₈	C₈	C₈	S₃	D₆	D₆	C₄xS₃	C₄xS₃	M₅(2)	S₃xC₈	S₃xC₈	S₃xM₅(2)
kernel	S₃xM₅(2)	S₃xC₁₆	D₆.C₈	C₁₂.C₈	C₃xM₅(2)	S₃xC₂xC₈	S₃xC₈	C₂xC₃:C₈	S₃xC₂xC₄	C₄xS₃	C₂xDic₃	C₂²xS₃	M₅(2)	C₁₆	C₂xC₈	C₈	C₂xC₄	S₃	C₄	C₂²	C₁
# reps	1	2	2	1	1	1	4	2	2	8	4	4	1	2	1	2	2	8	4	4	4

Matrix representation of S₃xM₅(2) ►in GL₄(F₉₇) generated by

0	96	0	0
1	96	0	0
0	0	1	0
0	0	0	1

1	96	0	0
0	96	0	0
0	0	96	0
0	0	0	96

64	0	0	0
0	64	0	0
0	0	47	3
0	0	24	50

1	0	0	0
0	1	0	0
0	0	1	0
0	0	1	96

G:=sub<GL(4,GF(97))| [0,1,0,0,96,96,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,96,96,0,0,0,0,96,0,0,0,0,96],[64,0,0,0,0,64,0,0,0,0,47,24,0,0,3,50],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,96] >;

S₃xM₅(2) in GAP, Magma, Sage, TeX

S_3\times M_5(2)

% in TeX

G:=Group("S3xM5(2)");

// GroupNames label

G:=SmallGroup(192,465);

// by ID

G=gap.SmallGroup(192,465);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,58,80,102,6278]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^16=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^9>;

// generators/relations

G = S3xM5(2) order 192 = 26·3

Direct product of S3 and M5(2)

G = S₃xM₅(2) order 192 = 2⁶·3

Direct product of S₃ and M₅(2)