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

G = 2_-¹⁺⁴.₂S₃ order 192 = 2⁶·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: 2_-¹⁺⁴.₂S₃, C₄○D₄.₂₈D₆, (C₃×D₄).₃₅D₄, (C₂×C₁₂).₂₃D₄, (C₂×Q₈).₉₇D₆, (C₃×Q₈).₃₅D₄, C₆.₈₅C₂²≀C₂, Q₈.₁₄D₆⋊₆C₂, C₁₂.₂₂₀(C₂×D₄), Dic₃⋊Q₈⋊₈C₂, C₃⋊₅(D₄.₁₀D₄), D₄.₁₇(C₃⋊D₄), (C₂×C₁₂).₂₄C₂³, Q₈.₂₄(C₃⋊D₄), Q₈⋊₃Dic₃⋊₁₄C₂, C₁₂.₁₀D₄⋊₁₂C₂, (C₆×Q₈).₁₀₀C₂², C₂.₁₉(C₂⁴⋊₄S₃), (C₄×Dic₃).₆₁C₂², C₄.Dic₃.₃₁C₂², (C₃×2_-¹⁺⁴).₁C₂, (C₂×Dic₆).₁₄₀C₂², (C₂×C₆).₄₇(C₂×D₄), C₄.₆₇(C₂×C₃⋊D₄), (C₂×C₄).₁₄(C₃⋊D₄), (C₂×C₄).₂₄(C₂²×S₃), C₂².₁₉(C₂×C₃⋊D₄), (C₃×C₄○D₄).₂₂C₂², SmallGroup(192,805)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₂ — 2_-¹⁺⁴.₂S₃

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂×C₁₂ — C₂×Dic₆ — Q₈.₁₄D₆ — 2_-¹⁺⁴.₂S₃

Lower central

C₃ — C₆ — C₂×C₁₂ — 2_-¹⁺⁴.₂S₃

Upper central

C₁ — C₂ — C₂×C₄ — 2_-¹⁺⁴

Generators and relations for 2_-¹⁺⁴.₂S₃
G = < a,b,c,d,e,f | a⁴=b²=e³=1, c²=d²=f²=a², bab=faf^-1=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf^-1=ab, dcd^-1=fcf^-1=a²c, ce=ec, de=ed, fdf^-1=a²cd, fef^-1=e^-1 >

Subgroups: 328 in 142 conjugacy classes, 43 normal (17 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, Dic₃, C₁₂, C₁₂, C₂×C₆, C₂×C₆, C₄², C₄⋊C₄, M₄(2), SD₁₆, Q₁₆, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₃⋊C₈, Dic₆, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×D₄, C₃×Q₈, C₃×Q₈, C₄.₁₀D₄, C₄≀C₂, C₄⋊Q₈, C₈.C₂², 2_-¹⁺⁴, C₄.Dic₃, C₄×Dic₃, Dic₃⋊C₄, D₄.S₃, C₃⋊Q₁₆, C₂×Dic₆, C₆×Q₈, C₆×Q₈, C₃×C₄○D₄, C₃×C₄○D₄, D₄.₁₀D₄, C₁₂.₁₀D₄, Q₈⋊₃Dic₃, Dic₃⋊Q₈, Q₈.₁₄D₆, C₃×2_-¹⁺⁴, 2_-¹⁺⁴.₂S₃
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₃⋊D₄, C₂²×S₃, C₂²≀C₂, C₂×C₃⋊D₄, D₄.₁₀D₄, C₂⁴⋊₄S₃, 2_-¹⁺⁴.₂S₃

Smallest permutation representation of 2_-¹⁺⁴.₂S₃
►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 33)(2 36)(3 35)(4 34)(5 31)(6 30)(7 29)(8 32)(9 27)(10 26)(11 25)(12 28)(13 43)(14 42)(15 41)(16 44)(17 39)(18 38)(19 37)(20 40)(21 46)(22 45)(23 48)(24 47)

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

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

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

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

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

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

33 conjugacy classes

class	1	2A	2B	2C	2D	3	4A	4B	4C	4D	4E	4F	4G	4H	4I	6A	6B	···	6F	8A	8B	12A	···	12J
order	1	2	2	2	2	3	4	4	4	4	4	4	4	4	4	6	6	···	6	8	8	12	···	12
size	1	1	2	4	4	2	2	2	4	4	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	D₄.₁₀D₄	2_-¹⁺⁴.₂S₃
kernel	2_-¹⁺⁴.₂S₃	C₁₂.₁₀D₄	Q₈⋊₃Dic₃	Dic₃⋊Q₈	Q₈.₁₄D₆	C₃×2_-¹⁺⁴	2_-¹⁺⁴	C₂×C₁₂	C₃×D₄	C₃×Q₈	C₂×Q₈	C₄○D₄	C₂×C₄	D₄	Q₈	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⁺⁴.₂S₃ ►in GL₆(𝔽₇₃)

72	0	0	0	0	0
0	72	0	0	0	0
0	0	0	1	0	0
0	0	72	0	0	0
0	0	72	72	1	2
0	0	0	1	72	72

72	0	0	0	0	0
0	1	0	0	0	0
0	0	72	72	1	2
0	0	0	0	1	0
0	0	0	1	0	0
0	0	0	72	1	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	72	0	0	0
0	0	1	1	72	71
0	0	72	0	1	1

72	0	0	0	0	0
0	72	0	0	0	0
0	0	41	41	15	64
0	0	17	17	24	39
0	0	56	41	0	0
0	0	17	0	56	15

64	0	0	0	0	0
0	8	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	8	0	0	0	0
64	0	0	0	0	0
0	0	41	17	0	0
0	0	17	32	0	0
0	0	41	41	15	64
0	0	0	32	17	58

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,72,0,0,0,1,0,72,1,0,0,0,0,1,72,0,0,0,0,2,72],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,72,0,1,72,0,0,1,1,0,1,0,0,2,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,1,72,0,0,1,0,1,0,0,0,0,0,72,1,0,0,0,0,71,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,41,17,56,17,0,0,41,17,41,0,0,0,15,24,0,56,0,0,64,39,0,15],[64,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,64,0,0,0,0,8,0,0,0,0,0,0,0,41,17,41,0,0,0,17,32,41,32,0,0,0,0,15,17,0,0,0,0,64,58] >;

2_-¹⁺⁴.₂S₃ in GAP, Magma, Sage, TeX

2_-^{1+4}._2S_3

% in TeX

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

// GroupNames label

G:=SmallGroup(192,805);

// by ID

G=gap.SmallGroup(192,805);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,184,570,1684,851,438,102,6278]);

// Polycyclic

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

// generators/relations

G = 2- 1+4.2S3 order 192 = 26·3

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

G = 2_-¹⁺⁴.₂S₃ order 192 = 2⁶·3

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