ES+(2,2).4S3 - GroupNames

G = 2₊¹⁺⁴.₄S₃ order 192 = 2⁶·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: 2₊¹⁺⁴.₄S₃, C₄○D₄.₂₆D₆, (C₂×D₄).₈₃D₆, (C₃×D₄).₃₃D₄, (C₃×Q₈).₃₃D₄, C₆.₈₁C₂²≀C₂, Q₈.₁₄D₆⋊₅C₂, C₁₂.₂₁₈(C₂×D₄), C₃⋊₅(D₄.₉D₄), (C₂²×C₆).₂₅D₄, C₂³.₁₂D₆⋊₈C₂, C₁₂.D₄⋊₁₂C₂, D₄.₁₅(C₃⋊D₄), (C₂×C₁₂).₂₂C₂³, Q₈.₂₂(C₃⋊D₄), (C₄×Dic₃)⋊₉C₂², Q₈⋊₃Dic₃⋊₁₂C₂, (C₂×Dic₆)⋊₁₆C₂², (C₆×D₄).₁₀₈C₂², C₂³.₁₃(C₃⋊D₄), C₄.Dic₃⋊₁₁C₂², C₂.₁₅(C₂⁴⋊₄S₃), (C₃×2₊¹⁺⁴).₁C₂, (C₂×C₆).₄₃(C₂×D₄), C₄.₆₅(C₂×C₃⋊D₄), (C₂×C₄).₂₂(C₂²×S₃), C₂².₁₅(C₂×C₃⋊D₄), (C₃×C₄○D₄).₂₀C₂², SmallGroup(192,801)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₂ — 2₊¹⁺⁴.₄S₃

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂×C₁₂ — C₂×Dic₆ — Q₈.₁₄D₆ — 2₊¹⁺⁴.₄S₃

Lower central

C₃ — C₆ — C₂×C₁₂ — 2₊¹⁺⁴.₄S₃

Upper central

C₁ — C₂ — C₂×C₄ — 2₊¹⁺⁴

Generators and relations for 2₊¹⁺⁴.₄S₃
G = < a,b,c,d,e,f | a⁴=b²=d²=e³=1, c²=f²=a², bab=faf^-1=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf^-1=ab, dcd=fcf^-1=a²c, ce=ec, de=ed, fdf^-1=cd, fef^-1=e^-1 >

Subgroups: 392 in 152 conjugacy classes, 43 normal (17 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, Dic₃, C₁₂, C₁₂, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, M₄(2), SD₁₆, Q₁₆, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, C₃⋊C₈, Dic₆, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×D₄, C₃×Q₈, C₂²×C₆, C₂²×C₆, C₄.D₄, C₄≀C₂, C₄.₄D₄, C₈.C₂², 2₊¹⁺⁴, C₄.Dic₃, C₄×Dic₃, D₄.S₃, C₃⋊Q₁₆, C₆.D₄, C₂×Dic₆, C₆×D₄, C₆×D₄, C₃×C₄○D₄, C₃×C₄○D₄, D₄.₉D₄, C₁₂.D₄, Q₈⋊₃Dic₃, C₂³.₁₂D₆, Q₈.₁₄D₆, C₃×2₊¹⁺⁴, 2₊¹⁺⁴.₄S₃
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₃⋊D₄, C₂²×S₃, C₂²≀C₂, C₂×C₃⋊D₄, D₄.₉D₄, C₂⁴⋊₄S₃, 2₊¹⁺⁴.₄S₃

Smallest permutation representation of 2₊¹⁺⁴.₄S₃
►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 45)(2 48)(3 47)(4 46)(5 19)(6 18)(7 17)(8 20)(9 15)(10 14)(11 13)(12 16)(21 34)(22 33)(23 36)(24 35)(25 43)(26 42)(27 41)(28 44)(29 39)(30 38)(31 37)(32 40)

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

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

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

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

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

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

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

33 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	3	4A	4B	4C	4D	4E	4F	4G	6A	6B	···	6J	8A	8B	12A	···	12F
order	1	2	2	2	2	2	2	3	4	4	4	4	4	4	4	6	6	···	6	8	8	12	···	12
size	1	1	2	4	4	4	4	2	2	2	4	4	12	12	24	2	4	···	4	24	24	4	···	4

33 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	8
type	+	+	+	+	+	+	+	+	+	+	+	+					-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₄	D₄	D₆	D₆	C₃⋊D₄	C₃⋊D₄	C₃⋊D₄	D₄.₉D₄	2₊¹⁺⁴.₄S₃
kernel	2₊¹⁺⁴.₄S₃	C₁₂.D₄	Q₈⋊₃Dic₃	C₂³.₁₂D₆	Q₈.₁₄D₆	C₃×2₊¹⁺⁴	2₊¹⁺⁴	C₃×D₄	C₃×Q₈	C₂²×C₆	C₂×D₄	C₄○D₄	D₄	Q₈	C₂³	C₃	C₁
# reps	1	1	2	1	2	1	1	2	2	2	1	2	4	4	4	2	1

Matrix representation of 2₊¹⁺⁴.₄S₃ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	46	0	0	0
0	0	0	27	0	0
0	0	0	0	46	0
0	0	0	0	0	27

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

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

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	0	27	0
0	0	0	0	0	46
0	0	46	0	0	0
0	0	0	27	0	0

36	45	0	0	0	0
45	36	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

36	45	0	0	0	0
28	37	0	0	0	0
0	0	0	72	0	0
0	0	1	0	0	0
0	0	0	0	0	46
0	0	0	0	46	0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,46,0,0,0,0,0,0,27],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,46,0,0,0,0,27,0,0,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,46],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,27,0,0,0,0,0,0,46,0,0],[36,45,0,0,0,0,45,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,28,0,0,0,0,45,37,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,46,0,0,0,0,46,0] >;

2₊¹⁺⁴.₄S₃ in GAP, Magma, Sage, TeX

2_+^{1+4}._4S_3

% in TeX

G:=Group("ES+(2,2).4S3");

// GroupNames label

G:=SmallGroup(192,801);

// by ID

G=gap.SmallGroup(192,801);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,570,1684,851,438,102,6278]);

// 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=f*a*f^-1=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,f*b*f^-1=a*b,d*c*d=f*c*f^-1=a^2*c,c*e=e*c,d*e=e*d,f*d*f^-1=c*d,f*e*f^-1=e^-1>;

// generators/relations

G = 2+ 1+4.4S3 order 192 = 26·3

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

G = 2₊¹⁺⁴.₄S₃ order 192 = 2⁶·3

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