S3xC2^2:Q8 - GroupNames

G = S₃×C₂²⋊Q₈ order 192 = 2⁶·3

Direct product of S₃ and C₂²⋊Q₈

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

Aliases: S₃×C₂²⋊Q₈, C₄⋊C₄⋊₂₅D₆, D₆⋊₃(C₂×Q₈), (C₂×Q₈)⋊₁₇D₆, C₂²⋊₃(S₃×Q₈), D₆.₅₉(C₂×D₄), (C₄×S₃).₄₁D₄, C₄.₁₈₅(S₃×D₄), (C₆×Q₈)⋊₅C₂², (C₂²×S₃)⋊₆Q₈, C₄.D₁₂⋊₂₃C₂, D₆⋊Q₈⋊₁₇C₂, D₆⋊₃Q₈⋊₁₂C₂, C₁₂.₂₃₀(C₂×D₄), C₂²⋊C₄.₅₅D₆, C₆.₇₂(C₂²×D₄), D₆.₃₈(C₄○D₄), C₆.₃₄(C₂²×Q₈), (C₂×C₆).₁₇₀C₂⁴, (C₂×C₁₂).₅₁C₂³, D₆⋊C₄.₂₀C₂², C₄⋊Dic₃⋊₃₄C₂², Dic₃.₄₇(C₂×D₄), (C₂²×C₄).₃₈₇D₆, C₁₂.₄₈D₄⋊₃₅C₂, Dic₃⋊C₄⋊₃₀C₂², (C₂×Dic₆)⋊₂₇C₂², Dic₃.D₄⋊₂₂C₂, (C₂²×C₆).₁₉₈C₂³, C₂².₁₉₁(S₃×C₂³), C₂³.₁₉₇(C₂²×S₃), (C₂×Dic₃).₈₅C₂³, (S₃×C₂³).₁₀₈C₂², (C₂²×S₃).₁₉₂C₂³, (C₂²×C₁₂).₂₅₀C₂², C₆.D₄.₃₂C₂², (C₂²×Dic₃).₂₂₄C₂², (C₂×S₃×Q₈)⋊₅C₂, (S₃×C₄⋊C₄)⋊₂₄C₂, (C₂×C₆)⋊₂(C₂×Q₈), C₂.₄₅(C₂×S₃×D₄), C₃⋊₄(C₂×C₂²⋊Q₈), C₂.₁₇(C₂×S₃×Q₈), C₂.₄₇(S₃×C₄○D₄), (S₃×C₂²×C₄).₇C₂, (C₃×C₂²⋊Q₈)⋊₆C₂, (C₃×C₄⋊C₄)⋊₁₇C₂², C₆.₁₅₉(C₂×C₄○D₄), (S₃×C₂²⋊C₄).₁C₂, (S₃×C₂×C₄).₉₂C₂², (C₂×C₄).₄₅(C₂²×S₃), (C₃×C₂²⋊C₄).₂₅C₂², SmallGroup(192,1185)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — S₃×C₂²⋊Q₈

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×S₃ — S₃×C₂³ — S₃×C₂²×C₄ — S₃×C₂²⋊Q₈

Lower central

C₃ — C₂×C₆ — S₃×C₂²⋊Q₈

Upper central

C₁ — C₂² — C₂²⋊Q₈

Generators and relations for S₃×C₂²⋊Q₈
G = < a,b,c,d,e,f | a³=b²=c²=d²=e⁴=1, f²=e², bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf^-1=cd=dc, ce=ec, de=ed, df=fd, fef^-1=e^-1 >

Subgroups: 800 in 322 conjugacy classes, 121 normal (43 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², S₃, S₃, C₆, C₆, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂×C₆, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂⁴, Dic₆, C₄×S₃, C₄×S₃, C₂×Dic₃, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₂×C₁₂, C₃×Q₈, C₂²×S₃, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂³×C₄, C₂²×Q₈, Dic₃⋊C₄, C₄⋊Dic₃, C₄⋊Dic₃, D₆⋊C₄, C₆.D₄, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₃×C₄⋊C₄, C₂×Dic₆, C₂×Dic₆, S₃×C₂×C₄, S₃×C₂×C₄, S₃×C₂×C₄, S₃×Q₈, C₂²×Dic₃, C₂²×C₁₂, C₆×Q₈, S₃×C₂³, C₂×C₂²⋊Q₈, Dic₃.D₄, S₃×C₂²⋊C₄, S₃×C₄⋊C₄, S₃×C₄⋊C₄, D₆⋊Q₈, C₄.D₁₂, C₁₂.₄₈D₄, D₆⋊₃Q₈, C₃×C₂²⋊Q₈, S₃×C₂²×C₄, C₂×S₃×Q₈, S₃×C₂²⋊Q₈
Quotients: C₁, C₂, C₂², S₃, D₄, Q₈, C₂³, D₆, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×S₃, C₂²⋊Q₈, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, S₃×D₄, S₃×Q₈, S₃×C₂³, C₂×C₂²⋊Q₈, C₂×S₃×D₄, C₂×S₃×Q₈, S₃×C₄○D₄, S₃×C₂²⋊Q₈

Smallest permutation representation of S₃×C₂²⋊Q₈
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

42 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	3	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	4N	4O	4P	6A	6B	6C	6D	6E	12A	12B	12C	12D	12E	12F	12G	12H
order	1	2	2	2	2	2	2	2	2	2	2	2	3	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	6	6	6	6	6	12	12	12	12	12	12	12	12
size	1	1	1	1	2	2	3	3	3	3	6	6	2	2	2	2	2	4	4	4	4	6	6	6	6	12	12	12	12	2	2	2	4	4	4	4	4	4	8	8	8	8

42 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+	-	+	+	+	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	S₃	D₄	Q₈	D₆	D₆	D₆	D₆	C₄○D₄	S₃×D₄	S₃×Q₈	S₃×C₄○D₄
kernel	S₃×C₂²⋊Q₈	Dic₃.D₄	S₃×C₂²⋊C₄	S₃×C₄⋊C₄	D₆⋊Q₈	C₄.D₁₂	C₁₂.₄₈D₄	D₆⋊₃Q₈	C₃×C₂²⋊Q₈	S₃×C₂²×C₄	C₂×S₃×Q₈	C₂²⋊Q₈	C₄×S₃	C₂²×S₃	C₂²⋊C₄	C₄⋊C₄	C₂²×C₄	C₂×Q₈	D₆	C₄	C₂²	C₂
# reps	1	2	2	3	2	1	1	1	1	1	1	1	4	4	2	3	1	1	4	2	2	2

Matrix representation of S₃×C₂²⋊Q₈ ►in GL₆(𝔽₁₃)

1	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	12	12
0	0	0	0	1	0

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	2	12	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	12	0	0	0
0	0	0	12	0	0
0	0	0	0	1	0
0	0	0	0	0	1

7	5	0	0	0	0
3	6	0	0	0	0
0	0	5	0	0	0
0	0	10	8	0	0
0	0	0	0	12	0
0	0	0	0	0	12

1	6	0	0	0	0
4	12	0	0	0	0
0	0	5	8	0	0
0	0	0	8	0	0
0	0	0	0	12	0
0	0	0	0	0	12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,3,0,0,0,0,5,6,0,0,0,0,0,0,5,10,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,4,0,0,0,0,6,12,0,0,0,0,0,0,5,0,0,0,0,0,8,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

S₃×C₂²⋊Q₈ in GAP, Magma, Sage, TeX

S_3\times C_2^2\rtimes Q_8

% in TeX

G:=Group("S3xC2^2:Q8");

// GroupNames label

G:=SmallGroup(192,1185);

// by ID

G=gap.SmallGroup(192,1185);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,100,794,297,136,6278]);

// Polycyclic

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

// generators/relations

G = S3×C22⋊Q8 order 192 = 26·3

Direct product of S3 and C22⋊Q8

G = S₃×C₂²⋊Q₈ order 192 = 2⁶·3

Direct product of S₃ and C₂²⋊Q₈