ES+(2,2).5S3 - GroupNames

G = 2₊¹⁺⁴.₅S₃ order 192 = 2⁶·3

2^nd non-split extension by 2₊¹⁺⁴ of S₃ acting via S₃/C₃=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: 2₊¹⁺⁴.₅S₃, (C₂×D₄).₈₄D₆, (C₂×C₁₂).₂₁D₄, C₆.₈₂C₂²≀C₂, (C₂²×C₆).₂₆D₄, C₂³.₇D₆⋊₉C₂, C₃⋊₃(C₂³.₇D₄), (C₂²×C₆).₅C₂³, (C₆×D₄).₁₇₈C₂², C₆.D₄⋊₆C₂², C₂³.₁₄(C₃⋊D₄), C₂³.₁₅(C₂²×S₃), C₂.₁₆(C₂⁴⋊₄S₃), C₂³.₂₃D₆⋊₁₇C₂, (C₂²×Dic₃)⋊₃C₂², (C₃×2₊¹⁺⁴).₂C₂, (C₂×C₆).₄₄(C₂×D₄), (C₂×C₄).₁₂(C₃⋊D₄), C₂².₁₆(C₂×C₃⋊D₄), SmallGroup(192,802)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₆ — 2₊¹⁺⁴.₅S₃

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×C₆ — C₂²×Dic₃ — C₂³.₂₃D₆ — 2₊¹⁺⁴.₅S₃

Lower central

C₃ — C₆ — C₂²×C₆ — 2₊¹⁺⁴.₅S₃

Upper central

C₁ — C₂ — C₂³ — 2₊¹⁺⁴

Generators and relations for 2₊¹⁺⁴.₅S₃
G = < a,b,c,d,e,f | a⁴=b²=d²=e³=1, c²=f²=a², bab=a^-1, ac=ca, ad=da, ae=ea, faf^-1=a^-1cd, fcf^-1=bc=cb, fdf^-1=bd=db, be=eb, bf=fb, dcd=a²c, ce=ec, de=ed, fef^-1=e^-1 >

Subgroups: 424 in 160 conjugacy classes, 43 normal (10 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₆, C₆, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₄○D₄, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×Q₈, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂³⋊C₄, C₂².D₄, 2₊¹⁺⁴, Dic₃⋊C₄, C₆.D₄, C₆.D₄, C₂²×Dic₃, C₆×D₄, C₆×D₄, C₃×C₄○D₄, C₂³.₇D₄, C₂³.₇D₆, C₂³.₂₃D₆, C₃×2₊¹⁺⁴, 2₊¹⁺⁴.₅S₃
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₃⋊D₄, C₂²×S₃, C₂²≀C₂, C₂×C₃⋊D₄, C₂³.₇D₄, C₂⁴⋊₄S₃, 2₊¹⁺⁴.₅S₃

Smallest permutation representation of 2₊¹⁺⁴.₅S₃
►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 3)(5 7)(10 12)(14 16)(18 20)(21 23)(25 27)(30 32)(33 35)(37 39)(42 44)(45 47)

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

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

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

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

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

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

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

33 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3	4A	4B	4C	4D	4E	4F	4G	4H	6A	6B	···	6J	12A	···	12F
order	1	2	2	2	2	2	2	2	3	4	4	4	4	4	4	4	4	6	6	···	6	12	···	12
size	1	1	2	2	2	4	4	4	2	4	4	4	12	12	24	24	24	2	4	···	4	4	···	4

33 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	4	8
type	+	+	+	+	+	+	+	+				-
image	C₁	C₂	C₂	C₂	S₃	D₄	D₄	D₆	C₃⋊D₄	C₃⋊D₄	C₂³.₇D₄	2₊¹⁺⁴.₅S₃
kernel	2₊¹⁺⁴.₅S₃	C₂³.₇D₆	C₂³.₂₃D₆	C₃×2₊¹⁺⁴	2₊¹⁺⁴	C₂×C₁₂	C₂²×C₆	C₂×D₄	C₂×C₄	C₂³	C₃	C₁
# reps	1	3	3	1	1	3	3	3	6	6	2	1

Matrix representation of 2₊¹⁺⁴.₅S₃ ►in GL₆(𝔽₁₃)

12	0	0	0	0	0
0	12	0	0	0	0
0	0	1	1	12	11
0	0	0	0	1	0
0	0	0	12	0	0
0	0	1	1	0	12

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

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

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

3	0	0	0	0	0
0	9	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	9	0	0	0	0
3	0	0	0	0	0
0	0	0	8	0	0
0	0	8	0	0	0
0	0	0	0	8	0
0	0	0	0	0	8

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,1,0,12,1,0,0,12,1,0,0,0,0,11,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,12,0,0,0,12,0,12,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,5,0,0,0,0,5,8,0,0,0,0,0,8],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,8,0,0,0,5,0,8,5,0,0,0,0,5,8,0,0,0,0,10,8],[3,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,3,0,0,0,0,9,0,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;

2₊¹⁺⁴.₅S₃ in GAP, Magma, Sage, TeX

2_+^{1+4}._5S_3

% in TeX

G:=Group("ES+(2,2).5S3");

// GroupNames label

G:=SmallGroup(192,802);

// by ID

G=gap.SmallGroup(192,802);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,570,438,6278]);

// Polycyclic

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

// generators/relations

G = 2+ 1+4.5S3 order 192 = 26·3

2nd non-split extension by 2+ 1+4 of S3 acting via S3/C3=C2

G = 2₊¹⁺⁴.₅S₃ order 192 = 2⁶·3

2^nd non-split extension by 2₊¹⁺⁴ of S₃ acting via S₃/C₃=C₂