C3xC2^3.11D6 - GroupNames

G = C₃xC₂³.₁₁D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₁₁D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃xC₂³.₁₁D₆, C₆².₁₇₆C₂³, D₆:C₄:₁₁C₆, C₆.₂₁(C₆xD₄), (C₂xDic₆):₃C₆, C₆.₁₈₀(S₃xD₄), (C₄xDic₃):₁₂C₆, (C₆xDic₆):₂₇C₂, (C₂xC₁₂).₂₆₇D₆, C₆.D₄:₅C₆, C₂³.₁₁(S₃xC₆), Dic₃.₁(C₃xD₄), (C₂²xC₆).₂₉D₆, (Dic₃xC₁₂):₃₃C₂, (C₃xDic₃).₂₈D₄, C₆.₁₁₉(C₄oD₁₂), (C₆xC₁₂).₂₄₄C₂², C₃²:₁₃(C₄.₄D₄), (C₂xC₆²).₅₂C₂², C₆.₁₁₄(D₄:₂S₃), (C₆xDic₃).₁₂₂C₂², C₂.₁₀(C₃xS₃xD₄), C₆.₉(C₃xC₄oD₄), (C₃xD₆:C₄):₃₀C₂, (C₃xC₂²:C₄):₇C₆, C₂²:C₄:₅(C₃xS₃), (C₂xC₄).₂₈(S₃xC₆), (C₂xC₃:D₄).₄C₆, C₃:₂(C₃xC₄.₄D₄), C₂².₄₄(S₃xC₂xC₆), (C₃xC₂²:C₄):₁₃S₃, (C₂xC₁₂).₅₄(C₂xC₆), C₂.₁₂(C₃xC₄oD₁₂), (C₃xC₆).₂₀₉(C₂xD₄), C₂.₉(C₃xD₄:₂S₃), (C₆xC₃:D₄).₁₁C₂, (S₃xC₂xC₆).₅₅C₂², (C₃xC₆).₉₉(C₄oD₄), (C₂²xS₃).₅(C₂xC₆), (C₂²xC₆).₂₆(C₂xC₆), (C₂xC₆).₃₁(C₂²xC₆), (C₃xC₆.D₄):₂₂C₂, (C₃²xC₂²:C₄):₁₁C₂, (C₂xC₆).₃₀₉(C₂²xS₃), (C₂xDic₃).₂₂(C₂xC₆), SmallGroup(288,656)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₃xC₂³.₁₁D₆

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₆² — S₃xC₂xC₆ — C₃xD₆:C₄ — C₃xC₂³.₁₁D₆

Lower central

C₃ — C₂xC₆ — C₃xC₂³.₁₁D₆

Upper central

C₁ — C₂xC₆ — C₃xC₂²:C₄

Generators and relations for C₃xC₂³.₁₁D₆
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=de⁵ >

Subgroups: 418 in 169 conjugacy classes, 62 normal (58 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, C₃², Dic₃, Dic₃, C₁₂, D₆, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, C₂²:C₄, C₂xD₄, C₂xQ₈, C₃xS₃, C₃xC₆, C₃xC₆, Dic₆, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₄.₄D₄, C₃xDic₃, C₃xDic₃, C₃xC₁₂, S₃xC₆, C₆², C₆², C₄xDic₃, D₆:C₄, C₆.D₄, C₄xC₁₂, C₃xC₂²:C₄, C₃xC₂²:C₄, C₂xDic₆, C₂xC₃:D₄, C₆xD₄, C₆xQ₈, C₃xDic₆, C₆xDic₃, C₃xC₃:D₄, C₆xC₁₂, S₃xC₂xC₆, C₂xC₆², C₂³.₁₁D₆, C₃xC₄.₄D₄, Dic₃xC₁₂, C₃xD₆:C₄, C₃xC₆.D₄, C₃²xC₂²:C₄, C₆xDic₆, C₆xC₃:D₄, C₃xC₂³.₁₁D₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₂³, D₆, C₂xC₆, C₂xD₄, C₄oD₄, C₃xS₃, C₃xD₄, C₂²xS₃, C₂²xC₆, C₄.₄D₄, S₃xC₆, C₄oD₁₂, S₃xD₄, D₄:₂S₃, C₆xD₄, C₃xC₄oD₄, S₃xC₂xC₆, C₂³.₁₁D₆, C₃xC₄.₄D₄, C₃xC₄oD₁₂, C₃xS₃xD₄, C₃xD₄:₂S₃, C₃xC₂³.₁₁D₆

Smallest permutation representation of C₃xC₂³.₁₁D₆
►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 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)

(2 46)(4 48)(6 38)(8 40)(10 42)(12 44)(13 32)(14 20)(15 34)(16 22)(17 36)(18 24)(19 26)(21 28)(23 30)(25 31)(27 33)(29 35)

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

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

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

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

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

72 conjugacy classes

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

72 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+								+	+	+	+									+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	C₆	S₃	D₄	D₆	D₆	C₄oD₄	C₃xS₃	C₃xD₄	S₃xC₆	S₃xC₆	C₄oD₁₂	C₃xC₄oD₄	C₃xC₄oD₁₂	S₃xD₄	D₄:₂S₃	C₃xS₃xD₄	C₃xD₄:₂S₃
kernel	C₃xC₂³.₁₁D₆	Dic₃xC₁₂	C₃xD₆:C₄	C₃xC₆.D₄	C₃²xC₂²:C₄	C₆xDic₆	C₆xC₃:D₄	C₂³.₁₁D₆	C₄xDic₃	D₆:C₄	C₆.D₄	C₃xC₂²:C₄	C₂xDic₆	C₂xC₃:D₄	C₃xC₂²:C₄	C₃xDic₃	C₂xC₁₂	C₂²xC₆	C₃xC₆	C₂²:C₄	Dic₃	C₂xC₄	C₂³	C₆	C₆	C₂	C₆	C₆	C₂	C₂
# reps	1	1	2	1	1	1	1	2	2	4	2	2	2	2	1	2	2	1	4	2	4	4	2	4	8	8	1	1	2	2

Matrix representation of C₃xC₂³.₁₁D₆ ►in GL₆(F₁₃)

9	0	0	0	0	0
0	9	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	1	0	0	0
0	0	5	12	0	0
0	0	0	0	1	0
0	0	0	0	10	12

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

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	0
0	0	0	0	0	12

9	0	0	0	0	0
0	3	0	0	0	0
0	0	5	0	0	0
0	0	0	5	0	0
0	0	0	0	1	5
0	0	0	0	0	12

0	3	0	0	0	0
9	0	0	0	0	0
0	0	8	2	0	0
0	0	0	5	0	0
0	0	0	0	8	0
0	0	0	0	2	5

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

C₃xC₂³.₁₁D₆ in GAP, Magma, Sage, TeX

C_3\times C_2^3._{11}D_6

% in TeX

G:=Group("C3xC2^3.11D6");

// GroupNames label

G:=SmallGroup(288,656);

// by ID

G=gap.SmallGroup(288,656);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,176,1598,555,9414]);

// 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=d*e^5>;

// generators/relations

G = C3xC23.11D6 order 288 = 25·32

Direct product of C3 and C23.11D6

G = C₃xC₂³.₁₁D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₁₁D₆