C3xC2^3.26D6 - GroupNames

G = C₃xC₂³.₂₆D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₂₆D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃xC₂³.₂₆D₆, C₆².₁₉₄C₂³, (C₆xC₁₂):₁₄C₄, (C₂xC₁₂):₆C₁₂, C₄:Dic₃:₁₇C₆, (C₂xC₁₂):₉Dic₃, C₁₂.₄₄(C₂xC₁₂), (C₄xDic₃):₁₅C₆, (C₂xC₁₂).₄₄₇D₆, C₂³.₃₁(S₃xC₆), C₄.₁₅(C₆xDic₃), (Dic₃xC₁₂):₃₁C₂, (C₂²xC₁₂).₄₂S₃, (C₂²xC₁₂).₂₃C₆, C₆².₁₁₀(C₂xC₄), C₆.₂₄(C₂²xC₁₂), C₁₂.₇₂(C₂xDic₃), C₆.D₄.₅C₆, (C₂²xC₆).₁₂₆D₆, C₆.₁₂₅(C₄oD₁₂), C₂².₆(C₆xDic₃), (C₆xC₁₂).₃₂₆C₂², (C₂xC₆²).₉₇C₂², C₆.₄₄(C₂²xDic₃), C₃²:₁₇(C₄²:C₂), (C₆xDic₃).₁₃₄C₂², (C₂xC₆xC₁₂).₁₄C₂, C₂.₅(Dic₃xC₂xC₆), (C₂xC₄):₄(C₃xDic₃), C₆.₁₅(C₃xC₄oD₄), C₂.₄(C₃xC₄oD₁₂), C₂².₂₂(S₃xC₂xC₆), (C₂xC₄).₁₀₂(S₃xC₆), (C₂xC₆).₄₄(C₂xC₁₂), (C₃xC₄:Dic₃):₃₅C₂, C₃:₄(C₃xC₄²:C₂), (C₂xC₁₂).₁₁₀(C₂xC₆), (C₃xC₁₂).₁₃₇(C₂xC₄), (C₂xC₆).₄₉(C₂²xC₆), (C₂²xC₆).₆₁(C₂xC₆), (C₂²xC₄).₁₁(C₃xS₃), (C₂xC₆).₂₇(C₂xDic₃), (C₃xC₆).₁₀₃(C₄oD₄), (C₃xC₆).₁₁₅(C₂²xC₄), (C₂xC₆).₃₂₇(C₂²xS₃), (C₂xDic₃).₃₄(C₂xC₆), (C₃xC₆.D₄).₁₀C₂, SmallGroup(288,697)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₃xC₂³.₂₆D₆

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₆² — C₆xDic₃ — Dic₃xC₁₂ — C₃xC₂³.₂₆D₆

Lower central

C₃ — C₆ — C₃xC₂³.₂₆D₆

Upper central

C₁ — C₂xC₁₂ — C₂²xC₁₂

Generators and relations for C₃xC₂³.₂₆D₆
G = < a,b,c,d,e,f | a³=b²=c²=d²=1, e⁶=d, f²=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf^-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e⁵ >

Subgroups: 314 in 179 conjugacy classes, 98 normal (30 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², C₂², C₆, C₆, C₆, C₂xC₄, C₂xC₄, C₂xC₄, C₂³, C₃², Dic₃, C₁₂, C₁₂, C₂xC₆, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, C₄:C₄, C₂²xC₄, C₃xC₆, C₃xC₆, C₃xC₆, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₂xC₁₂, C₂²xC₆, C₂²xC₆, C₄²:C₂, C₃xDic₃, C₃xC₁₂, C₆², C₆², C₆², C₄xDic₃, C₄:Dic₃, C₆.D₄, C₄xC₁₂, C₃xC₂²:C₄, C₃xC₄:C₄, C₂²xC₁₂, C₂²xC₁₂, C₆xDic₃, C₆xC₁₂, C₆xC₁₂, C₂xC₆², C₂³.₂₆D₆, C₃xC₄²:C₂, Dic₃xC₁₂, C₃xC₄:Dic₃, C₃xC₆.D₄, C₂xC₆xC₁₂, C₃xC₂³.₂₆D₆
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₂xC₄, C₂³, Dic₃, C₁₂, D₆, C₂xC₆, C₂²xC₄, C₄oD₄, C₃xS₃, C₂xDic₃, C₂xC₁₂, C₂²xS₃, C₂²xC₆, C₄²:C₂, C₃xDic₃, S₃xC₆, C₄oD₁₂, C₂²xDic₃, C₂²xC₁₂, C₃xC₄oD₄, C₆xDic₃, S₃xC₂xC₆, C₂³.₂₆D₆, C₃xC₄²:C₂, C₃xC₄oD₁₂, Dic₃xC₂xC₆, C₃xC₂³.₂₆D₆

Smallest permutation representation of C₃xC₂³.₂₆D₆
►On 48 points

Generators in S₄₈

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

(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)

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

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

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

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

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

108 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	4F	4G	···	4N	6A	···	6F	6G	···	6AE	12A	···	12H	12I	···	12AJ	12AK	···	12AZ
order	1	2	2	2	2	2	3	3	3	3	3	4	4	4	4	4	4	4	···	4	6	···	6	6	···	6	12	···	12	12	···	12	12	···	12
size	1	1	1	1	2	2	1	1	2	2	2	1	1	1	1	2	2	6	···	6	1	···	1	2	···	2	1	···	1	2	···	2	6	···	6

108 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	2	2
type	+	+	+	+	+								+	-	+	+
image	C₁	C₂	C₂	C₂	C₂	C₃	C₄	C₆	C₆	C₆	C₆	C₁₂	S₃	Dic₃	D₆	D₆	C₄oD₄	C₃xS₃	C₃xDic₃	S₃xC₆	S₃xC₆	C₄oD₁₂	C₃xC₄oD₄	C₃xC₄oD₁₂
kernel	C₃xC₂³.₂₆D₆	Dic₃xC₁₂	C₃xC₄:Dic₃	C₃xC₆.D₄	C₂xC₆xC₁₂	C₂³.₂₆D₆	C₆xC₁₂	C₄xDic₃	C₄:Dic₃	C₆.D₄	C₂²xC₁₂	C₂xC₁₂	C₂²xC₁₂	C₂xC₁₂	C₂xC₁₂	C₂²xC₆	C₃xC₆	C₂²xC₄	C₂xC₄	C₂xC₄	C₂³	C₆	C₆	C₂
# reps	1	2	2	2	1	2	8	4	4	4	2	16	1	4	2	1	4	2	8	4	2	8	8	16

Matrix representation of C₃xC₂³.₂₆D₆ ►in GL₄(F₁₃) generated by

3	0	0	0
0	3	0	0
0	0	1	0
0	0	0	1

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

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

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

4	0	0	0
0	10	0	0
0	0	8	0
0	0	0	8

0	10	0	0
9	0	0	0
0	0	12	12
0	0	2	1

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

C₃xC₂³.₂₆D₆ in GAP, Magma, Sage, TeX

C_3\times C_2^3._{26}D_6

% in TeX

G:=Group("C3xC2^3.26D6");

// GroupNames label

G:=SmallGroup(288,697);

// by ID

G=gap.SmallGroup(288,697);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,344,1094,9414]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^6=d,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,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*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 = C3xC23.26D6 order 288 = 25·32

Direct product of C3 and C23.26D6

G = C₃xC₂³.₂₆D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₂₆D₆