C3xC2^3.14D6 - GroupNames

G = C₃xC₂³.₁₄D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₁₄D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃xC₂³.₁₄D₆, C₆²:₁₁D₄, C₆².₂₀₄C₂³, (C₆xD₄):₉C₆, D₆:C₄:₁₅C₆, (C₆xD₄):₁₂S₃, C₆.₅₁(C₆xD₄), Dic₃:₃(C₃xD₄), C₆.₂₀₀(S₃xD₄), Dic₃:C₄:₁₅C₆, (C₃xDic₃):₁₈D₄, (C₂xC₁₂).₂₄₂D₆, C₂³.₁₄(S₃xC₆), (C₂²xC₆).₃₃D₆, C₆.D₄:₁₂C₆, C₃²:₂₃(C₄:D₄), (C₂²xDic₃):₉C₆, (C₆xC₁₂).₂₆₃C₂², (C₂xC₆²).₅₈C₂², C₆.₁₂₆(D₄:₂S₃), (C₆xDic₃).₁₃₈C₂², (D₄xC₃xC₆):₁₆C₂, (C₂xC₆):₅(C₃xD₄), C₂.₂₇(C₃xS₃xD₄), (C₂xD₄):₅(C₃xS₃), (C₂xC₃:D₄):₆C₆, C₃:₅(C₃xC₄:D₄), (C₃xD₆:C₄):₃₆C₂, (C₆xC₃:D₄):₂₀C₂, (C₂xC₄).₁₇(S₃xC₆), (Dic₃xC₂xC₆):₁₈C₂, C₆.₃₂(C₃xC₄oD₄), C₂.₁₅(C₆xC₃:D₄), C₂²:₃(C₃xC₃:D₄), C₂².₆₁(S₃xC₂xC₆), (C₂xC₆):₁₀(C₃:D₄), (C₂xC₁₂).₇₂(C₂xC₆), (C₃xC₆).₂₂₆(C₂xD₄), C₆.₁₅₂(C₂xC₃:D₄), (S₃xC₂xC₆).₆₁C₂², (C₃xDic₃:C₄):₃₇C₂, C₂.₁₈(C₃xD₄:₂S₃), (C₂xC₆).₅₉(C₂²xC₆), (C₂²xC₆).₃₂(C₂xC₆), (C₃xC₆).₁₄₀(C₄oD₄), (C₃xC₆.D₄):₂₈C₂, (C₂²xS₃).₁₁(C₂xC₆), (C₂xC₆).₃₃₇(C₂²xS₃), (C₂xDic₃).₃₈(C₂xC₆), SmallGroup(288,710)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₃xC₂³.₁₄D₆

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₆² — S₃xC₂xC₆ — C₆xC₃:D₄ — C₃xC₂³.₁₄D₆

Lower central

C₃ — C₂xC₆ — C₃xC₂³.₁₄D₆

Upper central

C₁ — C₂xC₆ — C₆xD₄

Generators and relations for C₃xC₂³.₁₄D₆
G = < a,b,c,d,e,f | a³=b²=c²=d²=e⁶=1, f²=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe^-1=bd=db, fbf^-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=de^-1 >

Subgroups: 522 in 215 conjugacy classes, 70 normal (58 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₂xC₄, C₂xC₄, D₄, C₂³, C₂³, C₃², Dic₃, Dic₃, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₂²:C₄, C₄:C₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₃xS₃, C₃xC₆, C₃xC₆, C₂xDic₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₄:D₄, C₃xDic₃, C₃xDic₃, C₃xC₁₂, S₃xC₆, C₆², C₆², C₆², Dic₃:C₄, D₆:C₄, C₆.D₄, C₃xC₂²:C₄, C₃xC₄:C₄, C₂²xDic₃, C₂xC₃:D₄, C₂²xC₁₂, C₆xD₄, C₆xD₄, C₆xDic₃, C₆xDic₃, C₃xC₃:D₄, C₆xC₁₂, D₄xC₃², S₃xC₂xC₆, C₂xC₆², C₂³.₁₄D₆, C₃xC₄:D₄, C₃xDic₃:C₄, C₃xD₆:C₄, C₃xC₆.D₄, Dic₃xC₂xC₆, C₆xC₃:D₄, D₄xC₃xC₆, C₃xC₂³.₁₄D₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₂³, D₆, C₂xC₆, C₂xD₄, C₄oD₄, C₃xS₃, C₃:D₄, C₃xD₄, C₂²xS₃, C₂²xC₆, C₄:D₄, S₃xC₆, S₃xD₄, D₄:₂S₃, C₂xC₃:D₄, C₆xD₄, C₃xC₄oD₄, C₃xC₃:D₄, S₃xC₂xC₆, C₂³.₁₄D₆, C₃xC₄:D₄, C₃xS₃xD₄, C₃xD₄:₂S₃, C₆xC₃:D₄, C₃xC₂³.₁₄D₆

Smallest permutation representation of C₃xC₂³.₁₄D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

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

72 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	4F	6A	···	6F	6G	···	6S	6T	···	6AG	6AH	6AI	12A	···	12H	12I	···	12P	12Q	12R
order	1	2	2	2	2	2	2	2	3	3	3	3	3	4	4	4	4	4	4	6	···	6	6	···	6	6	···	6	6	6	12	···	12	12	···	12	12	12
size	1	1	1	1	2	2	4	12	1	1	2	2	2	4	6	6	6	6	12	1	···	1	2	···	2	4	···	4	12	12	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	2	2	4	4	4	4
type	+	+	+	+	+	+	+								+	+	+	+	+										+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	C₆	S₃	D₄	D₄	D₆	D₆	C₄oD₄	C₃xS₃	C₃xD₄	C₃:D₄	C₃xD₄	S₃xC₆	S₃xC₆	C₃xC₄oD₄	C₃xC₃:D₄	S₃xD₄	D₄:₂S₃	C₃xS₃xD₄	C₃xD₄:₂S₃
kernel	C₃xC₂³.₁₄D₆	C₃xDic₃:C₄	C₃xD₆:C₄	C₃xC₆.D₄	Dic₃xC₂xC₆	C₆xC₃:D₄	D₄xC₃xC₆	C₂³.₁₄D₆	Dic₃:C₄	D₆:C₄	C₆.D₄	C₂²xDic₃	C₂xC₃:D₄	C₆xD₄	C₆xD₄	C₃xDic₃	C₆²	C₂xC₁₂	C₂²xC₆	C₃xC₆	C₂xD₄	Dic₃	C₂xC₆	C₂xC₆	C₂xC₄	C₂³	C₆	C₂²	C₆	C₆	C₂	C₂
# reps	1	1	1	1	1	2	1	2	2	2	2	2	4	2	1	2	2	1	2	2	2	4	4	4	2	4	4	8	1	1	2	2

Matrix representation of C₃xC₂³.₁₄D₆ ►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
2	12	0	0
0	0	0	1
0	0	1	0

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

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

4	0	0	0
7	10	0	0
0	0	12	0
0	0	0	1

11	2	0	0
5	2	0	0
0	0	0	8
0	0	8	0

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

C₃xC₂³.₁₄D₆ in GAP, Magma, Sage, TeX

C_3\times C_2^3._{14}D_6

% in TeX

G:=Group("C3xC2^3.14D6");

// GroupNames label

G:=SmallGroup(288,710);

// by ID

G=gap.SmallGroup(288,710);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,590,555,9414]);

// Polycyclic

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

// generators/relations

G = C3xC23.14D6 order 288 = 25·32

Direct product of C3 and C23.14D6

G = C₃xC₂³.₁₄D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₁₄D₆