S3xC4.10D4 - GroupNames

G = S₃×C₄.₁₀D₄ order 192 = 2⁶·3

Direct product of S₃ and C₄.₁₀D₄

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

Aliases: S₃×C₄.₁₀D₄, M₄(2).₂₀D₆, (C₄×S₃).₂D₄, C₁₂.₉₅(C₂×D₄), C₄.₁₅₀(S₃×D₄), (C₂×C₁₂).₇C₂³, (C₂×Q₈).₁₁₈D₆, (C₂×Dic₆).₄C₄, (C₆×Q₈).₅C₂², C₁₂.₁₀D₄⋊₃C₂, (S₃×M₄(2)).₁C₂, C₁₂.₄₇D₄⋊₁₁C₂, D₆.₁₇(C₂²⋊C₄), C₄.Dic₃.₄C₂², Dic₃.₅(C₂²⋊C₄), (C₂×Dic₆).₄₆C₂², (C₃×M₄(2)).₂₀C₂², (S₃×C₂×C₄).₂C₄, (C₂×S₃×Q₈).₁C₂, (C₂×C₄).₆(C₄×S₃), (C₂×C₁₂).₆(C₂×C₄), C₃⋊₁(C₂×C₄.₁₀D₄), (S₃×C₂×C₄).₃C₂², C₂².₁₆(S₃×C₂×C₄), C₆.₁₄(C₂×C₂²⋊C₄), C₂.₁₅(S₃×C₂²⋊C₄), (C₂×C₄).₇(C₂²×S₃), (C₃×C₄.₁₀D₄)⋊₉C₂, (C₂×C₆).₁₀(C₂²×C₄), (C₂×Dic₃).₃(C₂×C₄), (C₂²×S₃).₅₆(C₂×C₄), SmallGroup(192,309)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — S₃×C₄.₁₀D₄

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂×C₁₂ — S₃×C₂×C₄ — C₂×S₃×Q₈ — S₃×C₄.₁₀D₄

Lower central

C₃ — C₆ — C₂×C₆ — S₃×C₄.₁₀D₄

Upper central

C₁ — C₂ — C₂×C₄ — C₄.₁₀D₄

Generators and relations for S₃×C₄.₁₀D₄
G = < a,b,c,d,e | a³=b²=c⁴=1, d⁴=c², e²=dcd^-1=c^-1, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ce=ec, ede^-1=c^-1d³ >

Subgroups: 368 in 146 conjugacy classes, 53 normal (21 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, S₃, C₆, C₆, C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂×C₆, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂×Q₈, C₂×Q₈, C₃⋊C₈, C₂₄, Dic₆, C₄×S₃, C₄×S₃, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₃×Q₈, C₂²×S₃, C₄.₁₀D₄, C₄.₁₀D₄, C₂×M₄(2), C₂²×Q₈, S₃×C₈, C₈⋊S₃, C₄.Dic₃, C₃×M₄(2), C₂×Dic₆, C₂×Dic₆, S₃×C₂×C₄, S₃×C₂×C₄, S₃×Q₈, C₆×Q₈, C₂×C₄.₁₀D₄, C₁₂.₄₇D₄, C₁₂.₁₀D₄, C₃×C₄.₁₀D₄, S₃×M₄(2), C₂×S₃×Q₈, S₃×C₄.₁₀D₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, C₂³, D₆, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₄×S₃, C₂²×S₃, C₄.₁₀D₄, C₂×C₂²⋊C₄, S₃×C₂×C₄, S₃×D₄, C₂×C₄.₁₀D₄, S₃×C₂²⋊C₄, S₃×C₄.₁₀D₄

Smallest permutation representation of S₃×C₄.₁₀D₄
►On 48 points

Generators in S₄₈

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

(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	4	4	8
type	+	+	+	+	+	+			+	+	+	+		-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	S₃	D₄	D₆	D₆	C₄×S₃	C₄.₁₀D₄	S₃×D₄	S₃×C₄.₁₀D₄
kernel	S₃×C₄.₁₀D₄	C₁₂.₄₇D₄	C₁₂.₁₀D₄	C₃×C₄.₁₀D₄	S₃×M₄(2)	C₂×S₃×Q₈	C₂×Dic₆	S₃×C₂×C₄	C₄.₁₀D₄	C₄×S₃	M₄(2)	C₂×Q₈	C₂×C₄	S₃	C₄	C₁
# reps	1	2	1	1	2	1	4	4	1	4	2	1	4	2	2	1

Matrix representation of S₃×C₄.₁₀D₄ ►in GL₆(𝔽₇₃)

72	72	0	0	0	0
1	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
72	72	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	0	1	0	0
0	0	72	0	0	0
0	0	0	0	0	72
0	0	0	0	1	0

72	0	0	0	0	0
0	72	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	72	0	0	0

72	0	0	0	0	0
0	72	0	0	0	0
0	0	0	0	53	16
0	0	0	0	16	20
0	0	57	53	0	0
0	0	53	16	0	0

G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,57,53,0,0,0,0,53,16,0,0,53,16,0,0,0,0,16,20,0,0] >;

S₃×C₄.₁₀D₄ in GAP, Magma, Sage, TeX

S_3\times C_4._{10}D_4

% in TeX

G:=Group("S3xC4.10D4");

// GroupNames label

G:=SmallGroup(192,309);

// by ID

G=gap.SmallGroup(192,309);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,219,58,570,136,438,6278]);

// Polycyclic

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

// generators/relations

G = S3×C4.10D4 order 192 = 26·3

Direct product of S3 and C4.10D4

G = S₃×C₄.₁₀D₄ order 192 = 2⁶·3

Direct product of S₃ and C₄.₁₀D₄