C3xC2^3.23D6 - GroupNames

G = C₃xC₂³.₂₃D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₂₃D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃xC₂³.₂₃D₆, C₆².₆₅D₄, C₆².₂₀₀C₂³, C₆.₄₇(C₆xD₄), (C₆xD₄).₂₂C₆, (C₆xD₄).₂₇S₃, Dic₃:C₄:₁₄C₆, C₆.D₄:₈C₆, (C₂xC₁₂).₂₄₁D₆, C₂³.₂₈(S₃xC₆), (C₂²xC₆).₃₀D₆, (C₂²xDic₃):₈C₆, (C₆xC₁₂).₂₆₂C₂², (C₂xC₆²).₅₅C₂², C₆.₁₂₃(D₄:₂S₃), (C₆xDic₃).₁₃₇C₂², C₃²:₂₃(C₂².D₄), (Dic₃xC₂xC₆):₉C₂, (D₄xC₃xC₆).₁₆C₂, (C₂xC₆).₈(C₃xD₄), (C₂xC₄).₁₆(S₃xC₆), (C₂xD₄).₅(C₃xS₃), C₆.₂₉(C₃xC₄oD₄), C₂.₁₁(C₆xC₃:D₄), C₂².₅₇(S₃xC₂xC₆), (C₂xC₁₂).₇₁(C₂xC₆), (C₃xC₆).₂₅₇(C₂xD₄), C₆.₁₄₈(C₂xC₃:D₄), C₂².₄(C₃xC₃:D₄), (C₃xDic₃:C₄):₃₆C₂, C₂.₁₅(C₃xD₄:₂S₃), (C₂xC₆).₄₅(C₃:D₄), (C₂²xC₆).₂₉(C₂xC₆), (C₂xC₆).₅₅(C₂²xC₆), C₃:₅(C₃xC₂².D₄), (C₃xC₆).₁₃₇(C₄oD₄), (C₃xC₆.D₄):₂₄C₂, (C₂xC₆).₃₃₃(C₂²xS₃), (C₂xDic₃).₃₆(C₂xC₆), SmallGroup(288,706)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₃xC₂³.₂₃D₆

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₆² — C₆xDic₃ — Dic₃xC₂xC₆ — C₃xC₂³.₂₃D₆

Lower central

C₃ — C₂xC₆ — C₃xC₂³.₂₃D₆

Upper central

C₁ — C₂xC₆ — C₆xD₄

Generators and relations for C₃xC₂³.₂₃D₆
G = < a,b,c,d,e,f | a³=b²=c²=d²=e⁶=1, f²=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe^-1=fbf^-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef^-1=ce^-1 >

Subgroups: 394 in 183 conjugacy classes, 66 normal (34 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₄, C₂², C₂², C₂², C₆, C₆, C₆, C₂xC₄, C₂xC₄, D₄, C₂³, C₃², Dic₃, C₁₂, C₂xC₆, C₂xC₆, C₂xC₆, C₂²:C₄, C₄:C₄, C₂²xC₄, C₂xD₄, C₃xC₆, C₃xC₆, C₃xC₆, C₂xDic₃, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₂²xC₆, C₂²xC₆, C₂².D₄, C₃xDic₃, C₃xC₁₂, C₆², C₆², C₆², Dic₃:C₄, C₆.D₄, C₆.D₄, C₃xC₂²:C₄, C₃xC₄:C₄, C₂²xDic₃, C₂²xC₁₂, C₆xD₄, C₆xD₄, C₆xDic₃, C₆xDic₃, C₆xC₁₂, D₄xC₃², C₂xC₆², C₂³.₂₃D₆, C₃xC₂².D₄, C₃xDic₃:C₄, C₃xC₆.D₄, C₃xC₆.D₄, Dic₃xC₂xC₆, D₄xC₃xC₆, C₃xC₂³.₂₃D₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₂³, D₆, C₂xC₆, C₂xD₄, C₄oD₄, C₃xS₃, C₃:D₄, C₃xD₄, C₂²xS₃, C₂²xC₆, C₂².D₄, S₃xC₆, D₄:₂S₃, C₂xC₃:D₄, C₆xD₄, C₃xC₄oD₄, C₃xC₃:D₄, S₃xC₂xC₆, C₂³.₂₃D₆, C₃xC₂².D₄, C₃xD₄:₂S₃, C₆xC₃:D₄, C₃xC₂³.₂₃D₆

Smallest permutation representation of C₃xC₂³.₂₃D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

72 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	4F	4G	6A	···	6F	6G	···	6S	6T	···	6AG	12A	···	12H	12I	···	12P	12Q	12R	12S	12T
order	1	2	2	2	2	2	2	3	3	3	3	3	4	4	4	4	4	4	4	6	···	6	6	···	6	6	···	6	12	···	12	12	···	12	12	12	12	12
size	1	1	1	1	2	2	4	1	1	2	2	2	4	6	6	6	6	12	12	1	···	1	2	···	2	4	···	4	4	···	4	6	···	6	12	12	12	12

72 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+						+	+	+	+									-
image	C₁	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	S₃	D₄	D₆	D₆	C₄oD₄	C₃xS₃	C₃:D₄	C₃xD₄	S₃xC₆	S₃xC₆	C₃xC₄oD₄	C₃xC₃:D₄	D₄:₂S₃	C₃xD₄:₂S₃
kernel	C₃xC₂³.₂₃D₆	C₃xDic₃:C₄	C₃xC₆.D₄	Dic₃xC₂xC₆	D₄xC₃xC₆	C₂³.₂₃D₆	Dic₃:C₄	C₆.D₄	C₂²xDic₃	C₆xD₄	C₆xD₄	C₆²	C₂xC₁₂	C₂²xC₆	C₃xC₆	C₂xD₄	C₂xC₆	C₂xC₆	C₂xC₄	C₂³	C₆	C₂²	C₆	C₂
# reps	1	2	3	1	1	2	4	6	2	2	1	2	1	2	4	2	4	4	2	4	8	8	2	4

Matrix representation of C₃xC₂³.₂₃D₆ ►in GL₄(F₁₃) generated by

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

1	0	0	0
0	1	0	0
0	0	1	5
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

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

0	3	0	0
4	0	0	0
0	0	5	0
0	0	11	8

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

C₃xC₂³.₂₃D₆ in GAP, Magma, Sage, TeX

C_3\times C_2^3._{23}D_6

% in TeX

G:=Group("C3xC2^3.23D6");

// GroupNames label

G:=SmallGroup(288,706);

// by ID

G=gap.SmallGroup(288,706);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,590,555,9414]);

// Polycyclic

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

// generators/relations

G = C3xC23.23D6 order 288 = 25·32

Direct product of C3 and C23.23D6

G = C₃xC₂³.₂₃D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₂₃D₆