S3xC4:1D4 - GroupNames

G = S₃×C₄⋊₁D₄ order 192 = 2⁶·3

Direct product of S₃ and C₄⋊₁D₄

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

Aliases: S₃×C₄⋊₁D₄, C₄²⋊₃₆D₆, C₄⋊₁(S₃×D₄), (C₄×S₃)⋊₇D₄, C₁₂⋊₂(C₂×D₄), (C₂×D₄)⋊₂₅D₆, Dic₃⋊₁(C₂×D₄), D₆.₆₁(C₂×D₄), C₁₂⋊₃D₄⋊₂₅C₂, C₄⋊D₁₂⋊₁₆C₂, (S₃×C₄²)⋊₁₂C₂, (C₄×C₁₂)⋊₂₅C₂², (C₆×D₄)⋊₁₇C₂², C₆.₉₂(C₂²×D₄), (C₂×D₁₂)⋊₃₀C₂², (C₂×C₆).₂₅₈C₂⁴, (C₂×C₁₂).₅₀₇C₂³, (C₄×Dic₃)⋊₆₅C₂², C₂³.₇₄(C₂²×S₃), (C₂²×C₆).₇₂C₂³, (S₃×C₂³).₇₁C₂², C₂².₂₇₉(S₃×C₂³), (C₂²×S₃).₂₅₉C₂³, (C₂×Dic₃).₂₆₈C₂³, (C₂×S₃×D₄)⋊₁₈C₂, C₃⋊₂(C₂×C₄⋊₁D₄), C₂.₆₅(C₂×S₃×D₄), (C₃×C₄⋊₁D₄)⋊₅C₂, (C₂×C₃⋊D₄)⋊₂₅C₂², (S₃×C₂×C₄).₂₅₀C₂², (C₂×C₄).₅₉₆(C₂²×S₃), SmallGroup(192,1273)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — S₃×C₄⋊₁D₄

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×S₃ — S₃×C₂×C₄ — C₂×S₃×D₄ — S₃×C₄⋊₁D₄

Lower central

C₃ — C₂×C₆ — S₃×C₄⋊₁D₄

Upper central

C₁ — C₂² — C₄⋊₁D₄

Generators and relations for S₃×C₄⋊₁D₄
G = < a,b,c,d,e | a³=b²=c⁴=d⁴=e²=1, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c^-1, ede=d^-1 >

Subgroups: 1584 in 498 conjugacy classes, 131 normal (12 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, S₃, C₆, C₆, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₂×C₆, C₄², C₄², C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₄×S₃, D₁₂, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₃×D₄, C₂²×S₃, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₂×C₄², C₄⋊₁D₄, C₄⋊₁D₄, C₂²×D₄, C₄×Dic₃, C₄×C₁₂, S₃×C₂×C₄, C₂×D₁₂, S₃×D₄, C₂×C₃⋊D₄, C₆×D₄, S₃×C₂³, C₂×C₄⋊₁D₄, S₃×C₄², C₄⋊D₁₂, C₁₂⋊₃D₄, C₃×C₄⋊₁D₄, C₂×S₃×D₄, S₃×C₄⋊₁D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₂⁴, C₂²×S₃, C₄⋊₁D₄, C₂²×D₄, S₃×D₄, S₃×C₂³, C₂×C₄⋊₁D₄, C₂×S₃×D₄, S₃×C₄⋊₁D₄

Smallest permutation representation of S₃×C₄⋊₁D₄
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	4
type	+	+	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₆	D₆	S₃×D₄
kernel	S₃×C₄⋊₁D₄	S₃×C₄²	C₄⋊D₁₂	C₁₂⋊₃D₄	C₃×C₄⋊₁D₄	C₂×S₃×D₄	C₄⋊₁D₄	C₄×S₃	C₄²	C₂×D₄	C₄
# reps	1	1	1	6	1	6	1	12	1	6	6

Matrix representation of S₃×C₄⋊₁D₄ ►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	0	12
0	0	0	0	1	12

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

1	11	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	12	0
0	0	0	0	0	12

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

12	2	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	3	12	0	0
0	0	0	0	12	0
0	0	0	0	0	12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,12,10,0,0,0,0,5,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,2,1,0,0,0,0,0,0,1,3,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

S₃×C₄⋊₁D₄ in GAP, Magma, Sage, TeX

S_3\times C_4\rtimes_1D_4

% in TeX

G:=Group("S3xC4:1D4");

// GroupNames label

G:=SmallGroup(192,1273);

// by ID

G=gap.SmallGroup(192,1273);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,570,185,6278]);

// Polycyclic

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

// generators/relations

G = S3×C4⋊1D4 order 192 = 26·3

Direct product of S3 and C4⋊1D4

G = S₃×C₄⋊₁D₄ order 192 = 2⁶·3

Direct product of S₃ and C₄⋊₁D₄