C3xC2^3.8D6 - GroupNames

G = C₃xC₂³.₈D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₈D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃xC₂³.₈D₆, C₆².₁₇₀C₂³, C₄:Dic₃:₃C₆, Dic₃:C₄:₈C₆, C₂³.₈(S₃xC₆), (C₄xDic₃):₁₀C₆, (C₂xC₁₂).₂₆₄D₆, (C₂²xC₆).₂₆D₆, (Dic₃xC₁₂):₂₈C₂, C₆.D₄.₃C₆, C₆.₁₁₆(C₄oD₁₂), (C₆xC₁₂).₁₈₉C₂², (C₂xC₆²).₄₆C₂², C₆.₁₁₁(D₄:₂S₃), C₃²:₁₀(C₄²:₂C₂), (C₆xDic₃).₉₂C₂², C₆.₇(C₃xC₄oD₄), (C₂xC₄).₂₅(S₃xC₆), (C₂xC₁₂).₂(C₂xC₆), C₂.₉(C₃xC₄oD₁₂), C₂².₄₀(S₃xC₂xC₆), (C₃xC₄:Dic₃):₂₇C₂, C₃:₂(C₃xC₄²:₂C₂), C₂.₇(C₃xD₄:₂S₃), (C₃xC₂²:C₄).₃C₆, C₂²:C₄.₂(C₃xS₃), (C₃xDic₃:C₄):₂₇C₂, (C₃xC₂²:C₄).₁₆S₃, (C₂²xC₆).₂₀(C₂xC₆), (C₂xC₆).₂₅(C₂²xC₆), (C₃xC₆).₁₂₇(C₄oD₄), (C₂xC₆).₃₀₃(C₂²xS₃), (C₂xDic₃).₁₉(C₂xC₆), (C₃xC₆.D₄).₈C₂, (C₃²xC₂²:C₄).₃C₂, SmallGroup(288,650)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₃xC₂³.₈D₆

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₆² — C₆xDic₃ — Dic₃xC₁₂ — C₃xC₂³.₈D₆

Lower central

C₃ — C₂xC₆ — C₃xC₂³.₈D₆

Upper central

C₁ — C₂xC₆ — C₃xC₂²:C₄

Generators and relations for C₃xC₂³.₈D₆
G = < a,b,c,d,e,f | a³=b²=c²=d²=1, e⁶=c, f²=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf^-1=bc=cb, ebe^-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e⁵ >

Subgroups: 290 in 137 conjugacy classes, 58 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², C₆, C₆, C₂xC₄, C₂xC₄, C₂³, C₃², Dic₃, C₁₂, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, C₂²:C₄, C₄:C₄, C₃xC₆, C₃xC₆, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₂²xC₆, C₂²xC₆, C₄²:₂C₂, C₃xDic₃, C₃xC₁₂, C₆², C₆², C₄xDic₃, Dic₃:C₄, C₄:Dic₃, C₆.D₄, C₄xC₁₂, C₃xC₂²:C₄, C₃xC₂²:C₄, C₃xC₄:C₄, C₆xDic₃, C₆xC₁₂, C₂xC₆², C₂³.₈D₆, C₃xC₄²:₂C₂, Dic₃xC₁₂, C₃xDic₃:C₄, C₃xC₄:Dic₃, C₃xC₆.D₄, C₃²xC₂²:C₄, C₃xC₂³.₈D₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₂³, D₆, C₂xC₆, C₄oD₄, C₃xS₃, C₂²xS₃, C₂²xC₆, C₄²:₂C₂, S₃xC₆, C₄oD₁₂, D₄:₂S₃, C₃xC₄oD₄, S₃xC₂xC₆, C₂³.₈D₆, C₃xC₄²:₂C₂, C₃xC₄oD₁₂, C₃xD₄:₂S₃, C₃xC₂³.₈D₆

Smallest permutation representation of C₃xC₂³.₈D₆
►On 48 points

Generators in S₄₈

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

(2 36)(4 26)(6 28)(8 30)(10 32)(12 34)(13 39)(14 20)(15 41)(16 22)(17 43)(18 24)(19 45)(21 47)(23 37)(38 44)(40 46)(42 48)

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

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

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

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

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

72 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	4F	4G	4H	4I	6A	···	6F	6G	···	6O	6P	···	6W	12A	12B	12C	12D	12E	···	12R	12S	···	12Z	12AA	12AB	12AC	12AD
order	1	2	2	2	2	3	3	3	3	3	4	4	4	4	4	4	4	4	4	6	···	6	6	···	6	6	···	6	12	12	12	12	12	···	12	12	···	12	12	12	12	12
size	1	1	1	1	4	1	1	2	2	2	2	2	4	6	6	6	6	12	12	1	···	1	2	···	2	4	···	4	2	2	2	2	4	···	4	6	···	6	12	12	12	12

72 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+							+	+	+								-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	S₃	D₆	D₆	C₄oD₄	C₃xS₃	S₃xC₆	S₃xC₆	C₄oD₁₂	C₃xC₄oD₄	C₃xC₄oD₁₂	D₄:₂S₃	C₃xD₄:₂S₃
kernel	C₃xC₂³.₈D₆	Dic₃xC₁₂	C₃xDic₃:C₄	C₃xC₄:Dic₃	C₃xC₆.D₄	C₃²xC₂²:C₄	C₂³.₈D₆	C₄xDic₃	Dic₃:C₄	C₄:Dic₃	C₆.D₄	C₃xC₂²:C₄	C₃xC₂²:C₄	C₂xC₁₂	C₂²xC₆	C₃xC₆	C₂²:C₄	C₂xC₄	C₂³	C₆	C₆	C₂	C₆	C₂
# reps	1	1	2	1	2	1	2	2	4	2	4	2	1	2	1	6	2	4	2	4	12	8	2	4

Matrix representation of C₃xC₂³.₈D₆ ►in GL₄(F₁₃) generated by

1	0	0	0
0	1	0	0
0	0	9	0
0	0	0	9

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

12	0	0	0
0	12	0	0
0	0	12	0
0	0	0	12

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

0	1	0	0
12	0	0	0
0	0	7	0
0	0	7	11

0	8	0	0
5	0	0	0
0	0	11	10
0	0	6	2

G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,12,0,0,0,0,1,3,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,1,0,0,0,0,0,7,7,0,0,0,11],[0,5,0,0,8,0,0,0,0,0,11,6,0,0,10,2] >;

C₃xC₂³.₈D₆ in GAP, Magma, Sage, TeX

C_3\times C_2^3._8D_6

% in TeX

G:=Group("C3xC2^3.8D6");

// GroupNames label

G:=SmallGroup(288,650);

// by ID

G=gap.SmallGroup(288,650);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,176,1598,555,9414]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^6=c,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;

// generators/relations

G = C3xC23.8D6 order 288 = 25·32

Direct product of C3 and C23.8D6

G = C₃xC₂³.₈D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₈D₆