S3xC8.C2^2 - GroupNames

G = S₃×C₈.C₂² order 192 = 2⁶·3

Direct product of S₃ and C₈.C₂²

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

Aliases: S₃×C₈.C₂², Q₁₆⋊₃D₆, SD₁₆⋊₅D₆, C₂₄.₄C₂³, M₄(2)⋊₁₁D₆, C₁₂.₂₃C₂⁴, Dic₁₂⋊₃C₂², D₁₂.₁₆C₂³, Dic₆.₁₆C₂³, (S₃×Q₁₆)⋊₁C₂, (C₂×Q₈)⋊₂₅D₆, (S₃×SD₁₆)⋊₃C₂, C₄○D₄.₄₅D₆, (C₄×S₃).₄₄D₄, D₆.₆₈(C₂×D₄), C₈.D₆⋊₃C₂, D₄.D₆⋊₃C₂, Q₁₆⋊S₃⋊₁C₂, C₄.₁₉₁(S₃×D₄), C₃⋊C₈.₁₁C₂³, C₈.₄(C₂²×S₃), C₂₄⋊C₂⋊₅C₂², C₈⋊S₃⋊₅C₂², C₁₂.₂₄₄(C₂×D₄), (S₃×M₄(2))⋊₃C₂, (S₃×C₈).₁C₂², C₄.₂₃(S₃×C₂³), D₄.S₃⋊₆C₂², (S₃×Q₈)⋊₁₁C₂², (C₆×Q₈)⋊₂₀C₂², (C₃×Q₁₆)⋊₁C₂², (S₃×D₄).₇C₂², C₃⋊Q₁₆⋊₄C₂², C₂².₄₈(S₃×D₄), Q₈.₁₄D₆⋊₁₀C₂, Q₈.₁₁D₆⋊₉C₂, (C₄×S₃).₁₅C₂³, (C₂×Dic₃).₈₂D₄, Dic₃.₆₁(C₂×D₄), Q₈⋊₂S₃⋊₅C₂², (C₃×SD₁₆)⋊₅C₂², D₄.₁₆(C₂²×S₃), (C₃×D₄).₁₆C₂³, C₆.₁₂₄(C₂²×D₄), Q₈.₂₆(C₂²×S₃), (C₃×Q₈).₁₆C₂³, (C₂×C₁₂).₁₁₄C₂³, C₄○D₁₂.₃₀C₂², (C₂×Dic₆)⋊₄₁C₂², (C₂²×S₃).₁₀₂D₄, D₄⋊₂S₃.₆C₂², (C₃×M₄(2))⋊₅C₂², C₄.Dic₃⋊₁₄C₂², Q₈⋊₃S₃.₆C₂², (C₂×S₃×Q₈)⋊₁₇C₂, C₂.₉₇(C₂×S₃×D₄), C₃⋊₄(C₂×C₈.C₂²), (S₃×C₄○D₄).₄C₂, (C₂×C₆).₆₉(C₂×D₄), (C₃×C₈.C₂²)⋊₁C₂, (S₃×C₂×C₄).₁₆₂C₂², (C₂×C₄).₉₈(C₂²×S₃), (C₃×C₄○D₄).₂₅C₂², SmallGroup(192,1335)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — S₃×C₈.C₂²

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₄×S₃ — S₃×C₂×C₄ — C₂×S₃×Q₈ — S₃×C₈.C₂²

Lower central

C₃ — C₆ — C₁₂ — S₃×C₈.C₂²

Upper central

C₁ — C₂ — C₂×C₄ — C₈.C₂²

Generators and relations for S₃×C₈.C₂²
G = < a,b,c,d,e | a³=b²=c⁸=d²=e²=1, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c³, ece=c⁵, ede=c⁴d >

Subgroups: 656 in 258 conjugacy classes, 101 normal (51 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, S₃, C₆, C₆, C₈, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, Q₈, C₂³, Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂×C₆, C₂×C₆, C₂×C₈, M₄(2), M₄(2), SD₁₆, SD₁₆, Q₁₆, Q₁₆, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₃⋊C₈, C₂₄, Dic₆, Dic₆, Dic₆, C₄×S₃, C₄×S₃, D₁₂, D₁₂, C₂×Dic₃, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×D₄, C₃×Q₈, C₃×Q₈, C₃×Q₈, C₂²×S₃, C₂²×S₃, C₂×M₄(2), C₂×SD₁₆, C₂×Q₁₆, C₈.C₂², C₈.C₂², C₂²×Q₈, C₂×C₄○D₄, S₃×C₈, C₈⋊S₃, C₂₄⋊C₂, Dic₁₂, C₄.Dic₃, D₄.S₃, Q₈⋊₂S₃, C₃⋊Q₁₆, C₃×M₄(2), C₃×SD₁₆, C₃×Q₁₆, C₂×Dic₆, C₂×Dic₆, S₃×C₂×C₄, S₃×C₂×C₄, C₄○D₁₂, C₄○D₁₂, S₃×D₄, S₃×D₄, D₄⋊₂S₃, D₄⋊₂S₃, S₃×Q₈, S₃×Q₈, S₃×Q₈, Q₈⋊₃S₃, C₆×Q₈, C₃×C₄○D₄, C₂×C₈.C₂², S₃×M₄(2), C₈.D₆, S₃×SD₁₆, D₄.D₆, S₃×Q₁₆, Q₁₆⋊S₃, Q₈.₁₁D₆, Q₈.₁₄D₆, C₃×C₈.C₂², C₂×S₃×Q₈, S₃×C₄○D₄, S₃×C₈.C₂²
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₂⁴, C₂²×S₃, C₈.C₂², C₂²×D₄, S₃×D₄, S₃×C₂³, C₂×C₈.C₂², C₂×S₃×D₄, S₃×C₈.C₂²

Smallest permutation representation of S₃×C₈.C₂²
►On 48 points

Generators in S₄₈

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

(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 33)(16 34)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)

(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)

(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 19)(18 22)(21 23)(25 27)(26 30)(29 31)(33 35)(34 38)(37 39)(42 44)(43 47)(46 48)

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	8
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	-	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₄	D₄	D₆	D₆	D₆	D₆	D₆	C₈.C₂²	S₃×D₄	S₃×D₄	S₃×C₈.C₂²
kernel	S₃×C₈.C₂²	S₃×M₄(2)	C₈.D₆	S₃×SD₁₆	D₄.D₆	S₃×Q₁₆	Q₁₆⋊S₃	Q₈.₁₁D₆	Q₈.₁₄D₆	C₃×C₈.C₂²	C₂×S₃×Q₈	S₃×C₄○D₄	C₈.C₂²	C₄×S₃	C₂×Dic₃	C₂²×S₃	M₄(2)	SD₁₆	Q₁₆	C₂×Q₈	C₄○D₄	S₃	C₄	C₂²	C₁
# reps	1	1	1	2	2	2	2	1	1	1	1	1	1	2	1	1	1	2	2	1	1	2	1	1	1

Matrix representation of S₃×C₈.C₂² ►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	22	51	66	14
0	0	11	0	0	7
0	0	7	0	0	51
0	0	7	66	11	51

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	66	14	51	22
0	0	66	7	62	22
0	0	22	51	66	14
0	0	11	51	66	7

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,11,7,7,0,0,51,0,0,66,0,0,66,0,0,11,0,0,14,7,51,51],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,66,66,22,11,0,0,14,7,51,51,0,0,51,62,66,66,0,0,22,22,14,7] >;

S₃×C₈.C₂² in GAP, Magma, Sage, TeX

S_3\times C_8.C_2^2

% in TeX

G:=Group("S3xC8.C2^2");

// GroupNames label

G:=SmallGroup(192,1335);

// by ID

G=gap.SmallGroup(192,1335);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,570,185,438,235,102,6278]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^8=d^2=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,d*c*d=c^3,e*c*e=c^5,e*d*e=c^4*d>;

// generators/relations

G = S3×C8.C22 order 192 = 26·3

Direct product of S3 and C8.C22

G = S₃×C₈.C₂² order 192 = 2⁶·3

Direct product of S₃ and C₈.C₂²