S3xES-(2,2)

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

Aliases: S₃×2_-⁽¹⁺⁴⁾, C₆.₁₇C₂⁵, C₁₂.₅₂C₂⁴, D₆.₁₅C₂⁴, D₁₂.₃₈C₂³, Dic₃.₁₂C₂⁴, Dic₆.₃₉C₂³, C₄○D₄⋊₁₅D₆, (C₂×Q₈)⋊₂₇D₆, Q₈○D₁₂⋊₁₂C₂, (C₂×C₆).₈C₂⁴, C₄.₄₉(S₃×C₂³), C₂.₁₈(S₃×C₂⁴), (C₆×Q₈)⋊₂₄C₂², (S₃×Q₈)⋊₁₇C₂², (S₃×D₄).₈C₂², C₃⋊D₄.₄C₂³, C₃⋊₃(C₂×2_-⁽¹⁺⁴⁾), C₄○D₁₂⋊₁₄C₂², (C₄×S₃).₂₁C₂³, Q₈.₁₅D₆⋊₇C₂, (C₃×D₄).₃₂C₂³, D₄.₃₂(C₂²×S₃), C₂².₅(S₃×C₂³), Q₈.₄₃(C₂²×S₃), (C₃×Q₈).₃₃C₂³, D₄⋊₂S₃⋊₁₇C₂², (C₂×C₁₂).₁₂₃C₂³, Q₈⋊₃S₃⋊₁₆C₂², (C₂×Dic₆)⋊₄₅C₂², (C₃×2_-⁽¹⁺⁴⁾)⋊₄C₂, (C₂²×S₃).₂₅₁C₂³, (C₂×Dic₃).₁₇₀C₂³, (C₂×S₃×Q₈)⋊₂₂C₂, (S₃×C₄○D₄)⋊₉C₂, (C₃×C₄○D₄)⋊₁₂C₂², (S₃×C₂×C₄).₁₇₄C₂², (C₂×C₄).₁₀₇(C₂²×S₃), SmallGroup(192,1526)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — S₃×2_-⁽¹⁺⁴⁾

Chief series

C₁ — C₃ — C₆ — D₆ — C₂²×S₃ — S₃×C₂×C₄ — C₂×S₃×Q₈ — S₃×2_-⁽¹⁺⁴⁾

Lower central

C₃ — C₆ — S₃×2_-⁽¹⁺⁴⁾

Upper central

C₁ — C₂ — 2_-⁽¹⁺⁴⁾

Subgroups: 1512 in 794 conjugacy classes, 445 normal (8 characteristic)
C₁, C₂, C₂^[×12], C₃, C₄^[×10], C₄^[×10], C₂²^[×5], C₂²^[×16], S₃^[×2], S₃^[×5], C₆, C₆^[×5], C₂×C₄^[×15], C₂×C₄^[×55], D₄^[×10], D₄^[×30], Q₈^[×10], Q₈^[×30], C₂³^[×5], Dic₃^[×10], C₁₂^[×10], D₆, D₆^[×5], D₆^[×10], C₂×C₆^[×5], C₂²×C₄^[×15], C₂×D₄^[×10], C₂×Q₈^[×5], C₂×Q₈^[×45], C₄○D₄^[×10], C₄○D₄^[×70], Dic₆^[×30], C₄×S₃^[×40], D₁₂^[×10], C₂×Dic₃^[×15], C₃⋊D₄^[×20], C₂×C₁₂^[×15], C₃×D₄^[×10], C₃×Q₈^[×10], C₂²×S₃^[×5], C₂²×Q₈^[×5], C₂×C₄○D₄^[×10], 2_-⁽¹⁺⁴⁾, 2_-⁽¹⁺⁴⁾^[×15], C₂×Dic₆^[×15], S₃×C₂×C₄^[×15], C₄○D₁₂^[×30], S₃×D₄^[×10], D₄⋊₂S₃^[×30], S₃×Q₈^[×30], Q₈⋊₃S₃^[×10], C₆×Q₈^[×5], C₃×C₄○D₄^[×10], C₂×2_-⁽¹⁺⁴⁾, C₂×S₃×Q₈^[×5], Q₈.₁₅D₆^[×5], S₃×C₄○D₄^[×10], Q₈○D₁₂^[×10], C₃×2_-⁽¹⁺⁴⁾, S₃×2_-⁽¹⁺⁴⁾

Quotients:
C₁, C₂^[×31], C₂²^[×155], S₃, C₂³^[×155], D₆^[×15], C₂⁴^[×31], C₂²×S₃^[×35], 2_-⁽¹⁺⁴⁾^[×2], C₂⁵, S₃×C₂³^[×15], C₂×2_-⁽¹⁺⁴⁾, S₃×C₂⁴, S₃×2_-⁽¹⁺⁴⁾

Generators and relations
G = < a,b,c,d,e,f | a³=b²=c⁴=d²=1, e²=f²=c², bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c^-1, ce=ec, cf=fc, de=ed, df=fd, fef^-1=c²e >

Smallest permutation representation
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₁₃)

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

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	8	0	10	0
0	0	8	1	0	11
0	0	8	0	5	0
0	0	10	0	1	12

12	0	0	0	0	0
0	12	0	0	0	0
0	0	8	0	0	0
0	0	1	5	0	0
0	0	0	0	8	0
0	0	2	0	8	5

1	0	0	0	0	0
0	1	0	0	0	0
0	0	8	2	0	0
0	0	0	5	0	0
0	0	12	9	8	10
0	0	12	8	0	5

G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,5,1,0,0,0,5,0,5,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,8,8,10,0,0,0,1,0,0,0,0,10,0,5,1,0,0,0,11,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,1,0,2,0,0,0,5,0,0,0,0,0,0,8,8,0,0,0,0,0,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,12,12,0,0,2,5,9,8,0,0,0,0,8,0,0,0,0,0,10,5] >;

51 conjugacy classes

class	1	2A	2B	···	2F	2G	2H	2I	···	2M	3	4A	···	4J	4K	···	4T	6A	6B	···	6F	12A	···	12J
order	1	2	2	···	2	2	2	2	···	2	3	4	···	4	4	···	4	6	6	···	6	12	···	12
size	1	1	2	···	2	3	3	6	···	6	2	2	···	2	6	···	6	2	4	···	4	4	···	4

51 irreducible representations

dim	1	1	1	1	1	1	2	2	2	4	8
type	+	+	+	+	+	+	+	+	+	-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₆	D₆	2_-⁽¹⁺⁴⁾	S₃×2_-⁽¹⁺⁴⁾
kernel	S₃×2_-⁽¹⁺⁴⁾	C₂×S₃×Q₈	Q₈.₁₅D₆	S₃×C₄○D₄	Q₈○D₁₂	C₃×2_-⁽¹⁺⁴⁾	2_-⁽¹⁺⁴⁾	C₂×Q₈	C₄○D₄	S₃	C₁
# reps	1	5	5	10	10	1	1	5	10	2	1

In GAP, Magma, Sage, TeX

S_3\times 2_-^{(1+4)}

% in TeX

G:=Group("S3xES-(2,2)");

// GroupNames label

G:=SmallGroup(192,1526);

// by ID

G=gap.SmallGroup(192,1526);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,297,136,851,6278]);

// Polycyclic

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

// generators/relations

G = S₃×2_-⁽¹⁺⁴⁾ order 192 = 2⁶·3

Direct product of S₃ and 2_-⁽¹⁺⁴⁾

G = S3×2- (1+4) order 192 = 26·3

Direct product of S3 and 2- (1+4)

G = S₃×2_-⁽¹⁺⁴⁾ order 192 = 2⁶·3

Direct product of S₃ and 2_-⁽¹⁺⁴⁾